diff --git "a/Lemmas/imo_2023_p4_lemmas.lean" "b/Lemmas/imo_2023_p4_lemmas.lean" new file mode 100644--- /dev/null +++ "b/Lemmas/imo_2023_p4_lemmas.lean" @@ -0,0 +1,5554 @@ +import Mathlib +import Mathlib.Analysis.SpecialFunctions.Pow.Real + +set_option linter.unusedVariables.analyzeTactics true + +open Real + +lemma imo_2023_p4_1 + (x a: ℕ → ℝ) + (hxp: ∀ (i : ℕ), 0 < x i) + (h₀: ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : + ∀ (n : ℕ), (1 ≤ n ∧ n ≤ 2022) → a (n) < a (n + 1) := by + intros n hn + have h₂: a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + refine h₀ n ?_ + constructor + . exact hn.1 + . linarith + have h₃: a (n + 1) = sqrt ((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k) := by + refine h₀ (n + 1) ?_ + constructor + . linarith + . linarith + rw [h₂,h₃] + refine sqrt_lt_sqrt ?_ ?_ + . refine le_of_lt ?_ + refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + . have g₀: 1 ≤ n + 1 := by linarith + rw [Finset.sum_Ico_succ_top g₀ _, Finset.sum_Ico_succ_top g₀ _] + repeat rw [add_mul, mul_add] + have h₄: 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by + refine add_pos ?_ ?_ + . refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + . exact one_div_pos.mpr (hxp (n + 1)) + . refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . refine add_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + . exact one_div_pos.mpr (hxp (n + 1)) + linarith + + +lemma imo_2023_p4_1_1 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2022) : + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + refine h₀ n ?_ + constructor + . exact hn.1 + . linarith + + +lemma imo_2023_p4_1_2 + -- (x a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * + -- Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2022) : + 1 ≤ n ∧ n ≤ 2023 := by + constructor + . exact hn.1 + . linarith + + +lemma imo_2023_p4_1_3 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2022) + (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : + a n < a (n + 1) := by + have h₃: a (n + 1) = sqrt ((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k) := by + refine h₀ (n + 1) ?_ + constructor + . linarith + . linarith + rw [h₂,h₃] + refine sqrt_lt_sqrt ?_ ?_ + . refine le_of_lt ?_ + refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + . have g₀: 1 ≤ n + 1 := by linarith + rw [Finset.sum_Ico_succ_top g₀ _, Finset.sum_Ico_succ_top g₀ _] + repeat rw [add_mul, mul_add] + have h₄: 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by + refine add_pos ?_ ?_ + . refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + . exact one_div_pos.mpr (hxp (n + 1)) + . refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . refine add_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + . exact one_div_pos.mpr (hxp (n + 1)) + linarith + + +lemma imo_2023_p4_1_4 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2022) : + -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : + a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k) := by + refine h₀ (n + 1) ?_ + constructor + . linarith + . linarith + + +lemma imo_2023_p4_1_5 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2022) + (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) : + a n < a (n + 1) := by + rw [h₂,h₃] + refine sqrt_lt_sqrt ?_ ?_ + . refine le_of_lt ?_ + refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + . have g₀: 1 ≤ n + 1 := by linarith + rw [Finset.sum_Ico_succ_top g₀ _, Finset.sum_Ico_succ_top g₀ _] + repeat rw [add_mul, mul_add] + have h₄: 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by + refine add_pos ?_ ?_ + . refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + . exact one_div_pos.mpr (hxp (n + 1)) + . refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . refine add_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + . exact one_div_pos.mpr (hxp (n + 1)) + linarith + + +lemma imo_2023_p4_1_6 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2022) : + -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) : + √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * + Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) < + √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * + Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k) := by + refine sqrt_lt_sqrt ?_ ?_ + . refine le_of_lt ?_ + refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + . have g₀: 1 ≤ n + 1 := by linarith + rw [Finset.sum_Ico_succ_top g₀ _, Finset.sum_Ico_succ_top g₀ _] + repeat rw [add_mul, mul_add] + have h₄: 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by + refine add_pos ?_ ?_ + . refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + . exact one_div_pos.mpr (hxp (n + 1)) + . refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . refine add_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + . exact one_div_pos.mpr (hxp (n + 1)) + linarith + + +lemma imo_2023_p4_1_7 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2022) : + -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) : + 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k := by + refine le_of_lt ?_ + refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + + +lemma imo_2023_p4_1_8 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2022) : + -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) : + 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k := by + refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + + +lemma imo_2023_p4_1_9 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2022) : + -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) : + 0 < Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k := by + refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + + +lemma imo_2023_p4_1_10 + -- (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2022) : + -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) : + (Finset.Ico 1 (n + 1)).Nonempty := by + simp + linarith + + +lemma imo_2023_p4_1_11 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2022) : + -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) : + 0 < Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k := by + refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + + +lemma imo_2023_p4_1_12 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2022) + -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) : + ∀ i ∈ Finset.Ico 1 (n + 1), 0 < 1 / x i := by + intros i _ + exact one_div_pos.mpr (hxp i) + + +lemma imo_2023_p4_1_13 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2022) : + -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) : + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) < + (Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k := by + have g₀: 1 ≤ n + 1 := by linarith + rw [Finset.sum_Ico_succ_top g₀ _, Finset.sum_Ico_succ_top g₀ _] + repeat rw [add_mul, mul_add] + have h₄: 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by + refine add_pos ?_ ?_ + . refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + . exact one_div_pos.mpr (hxp (n + 1)) + . refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . refine add_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + . exact one_div_pos.mpr (hxp (n + 1)) + linarith + + +lemma imo_2023_p4_1_14 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2022) + -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) + (g₀ : 1 ≤ n + 1) : + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) < + (Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k := by + rw [Finset.sum_Ico_succ_top g₀ _, Finset.sum_Ico_succ_top g₀ _] + repeat rw [add_mul, mul_add] + have h₄: 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by + refine add_pos ?_ ?_ + . refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + . exact one_div_pos.mpr (hxp (n + 1)) + . refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . refine add_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + . exact one_div_pos.mpr (hxp (n + 1)) + linarith + + +lemma imo_2023_p4_1_15 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2022) : + -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) + -- (g₀ : 1 ≤ n + 1) : + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) < + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + x (n + 1)) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by + repeat rw [add_mul, mul_add] + have h₄: 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by + refine add_pos ?_ ?_ + . refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + . exact one_div_pos.mpr (hxp (n + 1)) + . refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . refine add_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + . exact one_div_pos.mpr (hxp (n + 1)) + linarith + + +lemma imo_2023_p4_1_16 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2022) : + -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) + -- (g₀ : 1 ≤ n + 1) : + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) < + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by + have h₄: 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by + refine add_pos ?_ ?_ + . refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + . exact one_div_pos.mpr (hxp (n + 1)) + . refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . refine add_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + . exact one_div_pos.mpr (hxp (n + 1)) + linarith + + +lemma imo_2023_p4_1_17 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2022) : + -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) + -- (g₀ : 1 ≤ n + 1) : + 0 < + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + x (n + 1) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 1 / x (n + 1)) := by + refine add_pos ?_ ?_ + . refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + . exact one_div_pos.mpr (hxp (n + 1)) + . refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . refine add_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + . exact one_div_pos.mpr (hxp (n + 1)) + + +lemma imo_2023_p4_1_18 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2022) : + -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) + -- (g₀ : 1 ≤ n + 1) : + 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) := by + refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + . exact one_div_pos.mpr (hxp (n + 1)) + + +lemma imo_2023_p4_1_19 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2022) : + -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) + -- (g₀ : 1 ≤ n + 1) : + 0 < Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k := by + refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + + +lemma imo_2023_p4_1_20 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2022) + -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) + -- (g₀ : 1 ≤ n + 1) : + ∀ i ∈ Finset.Ico 1 (n + 1), 0 < 1 / x i := by + intros i _ + exact one_div_pos.mpr (hxp i) + + +lemma imo_2023_p4_1_21 + -- (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2022) : + -- (h₂ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₃ : a (n + 1) = √((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k)) + -- (g₀ : 1 ≤ n + 1) : + (Finset.Ico 1 (n + 1)).Nonempty := by + simp + linarith + + + + + + +lemma imo_2023_p4_2 +-- my_amgm + (b1 b2 b3 b4 :���) + (hb1: 0 ≤ b1) + (hb2: 0 ≤ b2) + (hb3: 0 ≤ b3) + (hb4: 0 ≤ b4) : + (4 * (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ b1 + b2 + b3 + b4) := by + let w1 : ℝ := (4:ℝ)⁻¹ + let w2 : ℝ := w1 + let w3 : ℝ := w2 + let w4 : ℝ := w3 + rw [mul_comm] + refine mul_le_of_le_div₀ ?_ (by norm_num) ?_ + . refine add_nonneg ?_ hb4 + refine add_nonneg ?_ hb3 + exact add_nonneg hb1 hb2 + . have h₀: (b1^w1 * b2^w2 * b3^w3 * b4^w4) ≤ w1 * b1 + w2 * b2 + w3 * b3 + w4 * b4 := by + refine geom_mean_le_arith_mean4_weighted (by norm_num) ?_ ?_ ?_ hb1 hb2 hb3 hb4 ?_ + . norm_num + . norm_num + . norm_num + . norm_num + repeat rw [mul_rpow _] + . ring_nf at * + linarith + repeat { assumption } + . exact mul_nonneg hb1 hb2 + . exact hb4 + . refine mul_nonneg ?_ hb3 + exact mul_nonneg hb1 hb2 + + +lemma imo_2023_p4_2_1 + (b1 b2 b3 b4 : ℝ) + (w1 w2 w3 w4 : ℝ) + (hb1 : 0 ≤ b1) + (hb2 : 0 ≤ b2) + (hb3 : 0 ≤ b3) + (hb4 : 0 ≤ b4) + (hw1 : w1 = (4:ℝ)⁻¹) + (hw2 : w2 = w1) + (hw3 : w3 = w1) + (hw4 : w4 = w1) : + (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ * (4:ℝ) ≤ b1 + b2 + b3 + b4 := by + refine mul_le_of_le_div₀ ?_ (by norm_num) ?_ + . refine add_nonneg ?_ hb4 + refine add_nonneg ?_ hb3 + exact add_nonneg hb1 hb2 + . have h₀: (b1^w1 * b2^w2 * b3^w3 * b4^w4) ≤ w1 * b1 + w2 * b2 + w3 * b3 + w4 * b4 := by + have g₀ : 0 < w1 := by + rw [hw1] + norm_num + refine geom_mean_le_arith_mean4_weighted ?_ (by linarith) (by linarith) ?_ hb1 hb2 hb3 hb4 ?_ + . exact le_of_lt g₀ + . linarith + . rw [hw4, hw3, hw2, hw1] + norm_num + repeat rw [mul_rpow _] + . rw [hw4, hw3, hw2, hw1] at * + refine le_trans h₀ ?_ + ring_nf at * + linarith + repeat { assumption } + . exact mul_nonneg hb1 hb2 + . exact hb4 + . refine mul_nonneg ?_ hb3 + exact mul_nonneg hb1 hb2 + + +lemma imo_2023_p4_2_2 + (b1 b2 b3 b4 : ℝ) + (hb1 : 0 ≤ b1) + (hb2 : 0 ≤ b2) + (hb3 : 0 ≤ b3) + (hb4 : 0 ≤ b4) : + -- (hw1 : w1 = (4:ℝ)⁻¹) + -- (hw2 : w2 = w1) + -- (hw3 : w3 = w2) + -- (hw4 : w4 = w3) + 0 ≤ b1 + b2 + b3 + b4 := by + refine add_nonneg ?_ hb4 + refine add_nonneg ?_ hb3 + exact add_nonneg hb1 hb2 + + +lemma imo_2023_p4_2_3 + (b1 b2 b3 b4 : ℝ) + (w1 w2 w3 w4 : ℝ) + (hb1 : 0 ≤ b1) + (hb2 : 0 ≤ b2) + (hb3 : 0 ≤ b3) + (hb4 : 0 ≤ b4) + (hw1 : w1 = (4:ℝ)⁻¹) + (hw2 : w2 = w1) + (hw3 : w3 = w2) + (hw4 : w4 = w3) : + (b1 * b2 * b3 * b4) ^ ((4:ℝ)⁻¹) ≤ (b1 + b2 + b3 + b4) / 4 := by + have h₀: (b1^w1 * b2^w2 * b3^w3 * b4^w4) ≤ w1 * b1 + w2 * b2 + w3 * b3 + w4 * b4 := by + have g₀ : 0 < w1 := by + rw [hw1] + norm_num + refine geom_mean_le_arith_mean4_weighted ?_ (by linarith) (by linarith) ?_ hb1 hb2 hb3 hb4 ?_ + . exact le_of_lt g₀ + . linarith + . rw [hw4, hw3, hw2, hw1] + norm_num + repeat rw [mul_rpow _] + . rw [hw4, hw3, hw2, hw1] at * + refine le_trans h₀ ?_ + ring_nf at * + linarith + repeat { assumption } + . exact mul_nonneg hb1 hb2 + . exact hb4 + . refine mul_nonneg ?_ hb3 + exact mul_nonneg hb1 hb2 + + +lemma imo_2023_p4_2_4 + (b1 b2 b3 b4 : ℝ) + (w1 w2 w3 w4 : ℝ) + (hb1 : 0 ≤ b1) + (hb2 : 0 ≤ b2) + (hb3 : 0 ≤ b3) + (hb4 : 0 ≤ b4) + (hw1 : w1 = (4:ℝ)⁻¹) + (hw2 : w2 = w1) + (hw3 : w3 = w2) + (hw4 : w4 = w3) : + b1 ^ w1 * b2 ^ w2 * b3 ^ w3 * b4 ^ w4 ≤ w1 * b1 + w2 * b2 + w3 * b3 + w4 * b4 := by + have g₀ : 0 < w1 := by + rw [hw1] + norm_num + refine geom_mean_le_arith_mean4_weighted ?_ (by linarith) (by linarith) ?_ hb1 hb2 hb3 hb4 ?_ + . exact le_of_lt g₀ + . linarith + . rw [hw4, hw3, hw2, hw1] + norm_num + + +lemma imo_2023_p4_2_5 + (b1 b2 b3 b4 : ℝ) + (w1 w2 w3 w4 : ℝ) + (hb1 : 0 ≤ b1) + (hb2 : 0 ≤ b2) + (hb3 : 0 ≤ b3) + (hb4 : 0 ≤ b4) + (hw1 : w1 = ((4:ℝ)⁻¹)) + (hw2 : w2 = w1) + (hw3 : w3 = w2) + (hw4 : w4 = w3) + (h₀ : b1 ^ w1 * b2 ^ w2 * b3 ^ w3 * b4 ^ w4 ≤ w1 * b1 + w2 * b2 + w3 * b3 + w4 * b4) : + (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ (b1 + b2 + b3 + b4) / 4 := by + repeat rw [mul_rpow _] + . rw [hw4, hw3, hw2, hw1] at * + refine le_trans h₀ ?_ + ring_nf at * + linarith + repeat { assumption } + . exact mul_nonneg hb1 hb2 + . exact hb4 + . refine mul_nonneg ?_ hb3 + exact mul_nonneg hb1 hb2 + + +lemma imo_2023_p4_2_6 + (b1 b2 b3 b4 : ℝ) + (w1 w2 w3 w4 : ℝ) + -- (hb1 : 0 ≤ b1) + -- (hb2 : 0 ≤ b2) + -- (hb3 : 0 ≤ b3) + -- (hb4 : 0 ≤ b4) + (hw1 : w1 = ((4:ℝ)⁻¹)) + (hw2 : w2 = w1) + (hw3 : w3 = w2) + (hw4 : w4 = w3) + (h₀ : b1 ^ w1 * b2 ^ w2 * b3 ^ w3 * b4 ^ w4 ≤ w1 * b1 + w2 * b2 + w3 * b3 + w4 * b4) : + b1 ^ (4:ℝ)⁻¹ * b2 ^ (4:ℝ)⁻¹ * b3 ^ (4:ℝ)⁻¹ * b4 ^ (4:ℝ)⁻¹ ≤ (b1 + b2 + b3 + b4) / 4 := by + rw [hw4, hw3, hw2, hw1] at * + refine le_trans h₀ ?_ + ring_nf at * + linarith + + +lemma imo_2023_p4_3 + (x a: ℕ → ℝ) + (hxp: ∀ (i : ℕ), 0 < x i) + (h₀: ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k))) + (n: ℕ) + (hn: 1 ≤ n ∧ n ≤ 2021) : + (4 * a n ≤ + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + + (x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + repeat rw [mul_add, add_mul] + have g₁₁: 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1 := by + refine le_of_lt ?_ + refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + have g₁₂: 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹ := by + refine le_of_lt ?_ + refine Finset.sum_pos ?_ ?_ + . intros i _ + exact inv_pos.mpr (hxp i) + . simp + linarith + have h₃₂: 4 * ( ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹ + ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + let b1:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + let b2:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + let b3:ℝ := (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + let b4:ℝ := x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + have hb1: b1 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) := by + exact rfl + have hb2: b2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) := by + exact rfl + have hb3: b3 = (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact rfl + have hb4: b4 = x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact rfl + rw [← hb1, ← hb2, ← hb3, ← hb4] + have g₀: 4 * (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ b1 + b2 + b3 + b4 := by + have b1p: 0 ≤ b1 := by + rw [hb1] + refine mul_nonneg ?_ ?_ + . ring_nf + exact g₁₁ + . refine le_of_lt ?_ + exact one_div_pos.mpr (hxp (n + 1)) + have b2p: 0 ≤ b2 := by + rw [hb2] + refine mul_nonneg ?_ ?_ + . ring_nf + exact g₁₁ + . refine le_of_lt ?_ + exact one_div_pos.mpr (hxp (n + 2)) + have b3p: 0 ≤ b3 := by + rw [hb3] + refine mul_nonneg ?_ ?_ + . exact LT.lt.le (hxp (n + 1)) + . ring_nf + exact g₁₂ + have b4p: 0 ≤ b4 := by + rw [hb4] + refine mul_nonneg ?_ ?_ + . exact LT.lt.le (hxp (n + 2)) + . ring_nf + exact g₁₂ + exact imo_2023_p4_2 b1 b2 b3 b4 b1p b2p b3p b4p + linarith + have h₃₃: 4 * a (n) = 4 * (((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹ := by + simp + ring_nf + have g₀: (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 + * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 + = x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 := by + linarith + have g₁: x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1 := by + rw [mul_assoc] + have gg₁: x (1 + n) * (x (1 + n))⁻¹ = 1 := by + refine div_self ?_ + exact ne_of_gt (hxp (1 + n)) + have gg₂: x (2 + n) * (x (2 + n))⁻¹ = 1 := by + refine div_self ?_ + exact ne_of_gt (hxp (2 + n)) + rw [gg₁, gg₂] + norm_num + rw [g₁] at g₀ + rw [g₀] + simp + repeat rw [mul_rpow] + . have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by + rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹] + norm_num + have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by + rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹] + norm_num + have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + refine h₀ n ?_ + constructor + . exact hn.1 + . linarith + norm_cast at * + rw [g₂, g₃, ← mul_rpow g₁₁ g₁₂] + rw [← sqrt_eq_rpow] + ring_nf at g₄ + exact g₄ + . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + exact Eq.trans_le h₃₃ h₃₂ + + +lemma imo_2023_p4_3_1 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) : + 4 * a n ≤ + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + have g₁₁: 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1 := by + refine le_of_lt ?_ + refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + have g₁₂: 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹ := by + refine le_of_lt ?_ + refine Finset.sum_pos ?_ ?_ + . intros i _ + exact inv_pos.mpr (hxp i) + . simp + linarith + have h₃₂: 4 * ( ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹ + ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + let b1:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + let b2:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + let b3:ℝ := (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + let b4:ℝ := x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + have hb1: b1 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) := by + exact rfl + have hb2: b2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) := by + exact rfl + have hb3: b3 = (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact rfl + have hb4: b4 = x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact rfl + rw [← hb1, ← hb2, ← hb3, ← hb4] + have g₀: 4 * (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ b1 + b2 + b3 + b4 := by + have b1p: 0 ≤ b1 := by + rw [hb1] + refine mul_nonneg ?_ ?_ + . ring_nf + exact g₁₁ + . refine le_of_lt ?_ + exact one_div_pos.mpr (hxp (n + 1)) + have b2p: 0 ≤ b2 := by + rw [hb2] + refine mul_nonneg ?_ ?_ + . ring_nf + exact g₁₁ + . refine le_of_lt ?_ + exact one_div_pos.mpr (hxp (n + 2)) + have b3p: 0 ≤ b3 := by + rw [hb3] + refine mul_nonneg ?_ ?_ + . exact LT.lt.le (hxp (n + 1)) + . ring_nf + exact g₁₂ + have b4p: 0 ≤ b4 := by + rw [hb4] + refine mul_nonneg ?_ ?_ + . exact LT.lt.le (hxp (n + 2)) + . ring_nf + exact g₁₂ + exact imo_2023_p4_2 b1 b2 b3 b4 b1p b2p b3p b4p + linarith + have h₃₃: 4 * a (n) = 4 * (((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹ := by + simp + ring_nf + have g₀: (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 + * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 + = x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 := by + linarith + have g₁: x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1 := by + rw [mul_assoc] + have gg₁: x (1 + n) * (x (1 + n))⁻¹ = 1 := by + refine div_self ?_ + exact ne_of_gt (hxp (1 + n)) + have gg₂: x (2 + n) * (x (2 + n))⁻¹ = 1 := by + refine div_self ?_ + exact ne_of_gt (hxp (2 + n)) + rw [gg₁, gg₂] + norm_num + rw [g₁] at g₀ + rw [g₀] + simp + repeat rw [mul_rpow] + . have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by + rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹] + norm_num + have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by + rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹] + norm_num + have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + refine h₀ n ?_ + constructor + . exact hn.1 + . linarith + norm_cast at * + rw [g₂, g₃, ← mul_rpow g₁₁ g₁₂] + rw [← sqrt_eq_rpow] + ring_nf at g₄ + exact g₄ + . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + exact Eq.trans_le h₃₃ h₃₂ + + +lemma imo_2023_p4_3_2 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) : + 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1 := by + refine le_of_lt ?_ + refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + + +lemma imo_2023_p4_3_3 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) : + 4 * a n ≤ + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + have g₁₂: 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹ := by + refine le_of_lt ?_ + refine Finset.sum_pos ?_ ?_ + . intros i _ + exact inv_pos.mpr (hxp i) + . simp + linarith + have h₃₂: 4 * ( ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹ + ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + let b1:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + let b2:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + let b3:ℝ := (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + let b4:ℝ := x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + have hb1: b1 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) := by + exact rfl + have hb2: b2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) := by + exact rfl + have hb3: b3 = (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact rfl + have hb4: b4 = x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact rfl + rw [← hb1, ← hb2, ← hb3, ← hb4] + have g₀: 4 * (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ b1 + b2 + b3 + b4 := by + have b1p: 0 ≤ b1 := by + rw [hb1] + refine mul_nonneg ?_ ?_ + . ring_nf + exact g₁₁ + . refine le_of_lt ?_ + exact one_div_pos.mpr (hxp (n + 1)) + have b2p: 0 ≤ b2 := by + rw [hb2] + refine mul_nonneg ?_ ?_ + . ring_nf + exact g₁₁ + . refine le_of_lt ?_ + exact one_div_pos.mpr (hxp (n + 2)) + have b3p: 0 ≤ b3 := by + rw [hb3] + refine mul_nonneg ?_ ?_ + . exact LT.lt.le (hxp (n + 1)) + . ring_nf + exact g₁₂ + have b4p: 0 ≤ b4 := by + rw [hb4] + refine mul_nonneg ?_ ?_ + . exact LT.lt.le (hxp (n + 2)) + . ring_nf + exact g₁₂ + exact imo_2023_p4_2 b1 b2 b3 b4 b1p b2p b3p b4p + linarith + have h₃₃: 4 * a (n) = 4 * (((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹ := by + simp + ring_nf + have g₀: (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 + * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 + = x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 := by + linarith + have g₁: x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1 := by + rw [mul_assoc] + have gg₁: x (1 + n) * (x (1 + n))⁻¹ = 1 := by + refine div_self ?_ + exact ne_of_gt (hxp (1 + n)) + have gg₂: x (2 + n) * (x (2 + n))⁻¹ = 1 := by + refine div_self ?_ + exact ne_of_gt (hxp (2 + n)) + rw [gg₁, gg₂] + norm_num + rw [g₁] at g₀ + rw [g₀] + simp + repeat rw [mul_rpow] + . have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by + rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹] + norm_num + have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by + rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹] + norm_num + have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + refine h₀ n ?_ + constructor + . exact hn.1 + . linarith + norm_cast at * + rw [g₂, g₃, ← mul_rpow g₁₁ g₁₂] + rw [← sqrt_eq_rpow] + ring_nf at g₄ + exact g₄ + . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + exact Eq.trans_le h₃₃ h₃₂ + + +lemma imo_2023_p4_3_4 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) : + -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) : + 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹ := by + refine le_of_lt ?_ + refine Finset.sum_pos ?_ ?_ + . intros i _ + exact inv_pos.mpr (hxp i) + . simp + linarith + + +lemma imo_2023_p4_3_5 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) : + ∀ i ∈ Finset.Ico 1 (1 + n), 0 < (x i)⁻¹ := by + intros i _ + exact inv_pos.mpr (hxp i) + + +lemma imo_2023_p4_3_6 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) : + 4 * a n ≤ + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + have h₃₂: 4 * ( ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹ + ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + let b1:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + let b2:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + let b3:ℝ := (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + let b4:ℝ := x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + have hb1: b1 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) := by + exact rfl + have hb2: b2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) := by + exact rfl + have hb3: b3 = (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact rfl + have hb4: b4 = x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact rfl + rw [← hb1, ← hb2, ← hb3, ← hb4] + have g₀: 4 * (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ b1 + b2 + b3 + b4 := by + have b1p: 0 ≤ b1 := by + rw [hb1] + refine mul_nonneg ?_ ?_ + . ring_nf + exact g₁₁ + . refine le_of_lt ?_ + exact one_div_pos.mpr (hxp (n + 1)) + have b2p: 0 ≤ b2 := by + rw [hb2] + refine mul_nonneg ?_ ?_ + . ring_nf + exact g₁₁ + . refine le_of_lt ?_ + exact one_div_pos.mpr (hxp (n + 2)) + have b3p: 0 ≤ b3 := by + rw [hb3] + refine mul_nonneg ?_ ?_ + . exact LT.lt.le (hxp (n + 1)) + . ring_nf + exact g₁₂ + have b4p: 0 ≤ b4 := by + rw [hb4] + refine mul_nonneg ?_ ?_ + . exact LT.lt.le (hxp (n + 2)) + . ring_nf + exact g₁₂ + exact imo_2023_p4_2 b1 b2 b3 b4 b1p b2p b3p b4p + linarith + have h₃₃: 4 * a (n) = 4 * (((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹ := by + simp + ring_nf + have g₀: (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 + * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 + = x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 := by + linarith + have g₁: x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1 := by + rw [mul_assoc] + have gg₁: x (1 + n) * (x (1 + n))⁻¹ = 1 := by + refine div_self ?_ + exact ne_of_gt (hxp (1 + n)) + have gg₂: x (2 + n) * (x (2 + n))⁻¹ = 1 := by + refine div_self ?_ + exact ne_of_gt (hxp (2 + n)) + rw [gg₁, gg₂] + norm_num + rw [g₁] at g₀ + rw [g₀] + simp + repeat rw [mul_rpow] + . have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by + rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹] + norm_num + have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by + rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹] + norm_num + have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + refine h₀ n ?_ + constructor + . exact hn.1 + . linarith + norm_cast at * + rw [g₂, g₃, ← mul_rpow g₁₁ g₁₂] + rw [← sqrt_eq_rpow] + ring_nf at g₄ + exact g₄ + . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + exact Eq.trans_le h₃₃ h₃₂ + + +lemma imo_2023_p4_3_7 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2021) + (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) : + 4 * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ + (4:ℝ)⁻¹ ≤ + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + let b1:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + let b2:ℝ := (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + let b3:ℝ := (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + let b4:ℝ := x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + have hb1: b1 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) := by + exact rfl + have hb2: b2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) := by + exact rfl + have hb3: b3 = (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact rfl + have hb4: b4 = x (n + 2) * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact rfl + rw [← hb1, ← hb2, ← hb3, ← hb4] + have g₀: 4 * (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ b1 + b2 + b3 + b4 := by + have b1p: 0 ≤ b1 := by + rw [hb1] + refine mul_nonneg ?_ ?_ + . ring_nf + exact g₁₁ + . refine le_of_lt ?_ + exact one_div_pos.mpr (hxp (n + 1)) + have b2p: 0 ≤ b2 := by + rw [hb2] + refine mul_nonneg ?_ ?_ + . ring_nf + exact g₁₁ + . refine le_of_lt ?_ + exact one_div_pos.mpr (hxp (n + 2)) + have b3p: 0 ≤ b3 := by + rw [hb3] + refine mul_nonneg ?_ ?_ + . exact LT.lt.le (hxp (n + 1)) + . ring_nf + exact g₁₂ + have b4p: 0 ≤ b4 := by + rw [hb4] + refine mul_nonneg ?_ ?_ + . exact LT.lt.le (hxp (n + 2)) + . ring_nf + exact g₁₂ + exact imo_2023_p4_2 b1 b2 b3 b4 b1p b2p b3p b4p + linarith + + +lemma imo_2023_p4_3_8 + (x : ℕ → ℝ) + (b1 b2 b3 b4 : ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2021) + (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + (hb1 : b1 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) + (hb2 : b2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) + (hb3 : b3 = x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + (hb4 : b4 = x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) : + 4 * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ + (4:ℝ)⁻¹ ≤ + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + rw [← hb1, ← hb2, ← hb3, ← hb4] + have g₀: 4 * (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ b1 + b2 + b3 + b4 := by + have b1p: 0 ≤ b1 := by + rw [hb1] + refine mul_nonneg ?_ ?_ + . ring_nf + exact g₁₁ + . refine le_of_lt ?_ + exact one_div_pos.mpr (hxp (n + 1)) + have b2p: 0 ≤ b2 := by + rw [hb2] + refine mul_nonneg ?_ ?_ + . ring_nf + exact g₁₁ + . refine le_of_lt ?_ + exact one_div_pos.mpr (hxp (n + 2)) + have b3p: 0 ≤ b3 := by + rw [hb3] + refine mul_nonneg ?_ ?_ + . exact LT.lt.le (hxp (n + 1)) + . ring_nf + exact g₁₂ + have b4p: 0 ≤ b4 := by + rw [hb4] + refine mul_nonneg ?_ ?_ + . exact LT.lt.le (hxp (n + 2)) + . ring_nf + exact g₁₂ + exact imo_2023_p4_2 b1 b2 b3 b4 b1p b2p b3p b4p + linarith + + +lemma imo_2023_p4_3_9 + (x : ℕ → ℝ) + (b1 b2 b3 b4 : ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2021) + (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + (hb1 : b1 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) + (hb2 : b2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) + (hb3 : b3 = x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + (hb4 : b4 = x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) : + 4 * (b1 * b2 * b3 * b4) ^ (4:ℝ)⁻¹ ≤ b1 + b2 + (b3 + b4) := by + have b1p: 0 ≤ b1 := by + rw [hb1] + refine mul_nonneg ?_ ?_ + . ring_nf + exact g₁₁ + . refine le_of_lt ?_ + exact one_div_pos.mpr (hxp (n + 1)) + have b2p: 0 ≤ b2 := by + rw [hb2] + refine mul_nonneg ?_ ?_ + . ring_nf + exact g₁₁ + . refine le_of_lt ?_ + exact one_div_pos.mpr (hxp (n + 2)) + have b3p: 0 ≤ b3 := by + rw [hb3] + refine mul_nonneg ?_ ?_ + . exact LT.lt.le (hxp (n + 1)) + . ring_nf + exact g₁₂ + have b4p: 0 ≤ b4 := by + rw [hb4] + refine mul_nonneg ?_ ?_ + . exact LT.lt.le (hxp (n + 2)) + . ring_nf + exact g₁₂ + rw [← add_assoc] + exact imo_2023_p4_2 b1 b2 b3 b4 b1p b2p b3p b4p + + +lemma imo_2023_p4_3_10 + (x : ℕ → ℝ) + (b1 : ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2021) + (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + (hb1 : b1 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) : + -- (hb2 : b2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) + -- (hb3 : b3 = x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (hb4 : b4 = x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) : + 0 ≤ b1 := by + rw [hb1] + refine mul_nonneg ?_ ?_ + . ring_nf + exact g₁₁ + . refine le_of_lt ?_ + exact one_div_pos.mpr (hxp (n + 1)) + + +lemma imo_2023_p4_3_11 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + -- (hb1 : b1 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) + -- (hb2 : b2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) + -- (hb3 : b3 = x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (hb4 : b4 = x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + 0 ≤ 1 / x (n + 1) := by + refine le_of_lt ?_ + exact one_div_pos.mpr (hxp (n + 1)) + + +lemma imo_2023_p4_3_12 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + (h₃₂ : 4 * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ + (4:ℝ)⁻¹ ≤ + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : + 4 * a n ≤ + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + have h₃₃: 4 * a (n) = 4 * (((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1))) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ) ^ (4:ℝ)⁻¹ := by + simp + ring_nf + have g₀: (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 + * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 + = x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 := by + linarith + have g₁: x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1 := by + rw [mul_assoc] + have gg₁: x (1 + n) * (x (1 + n))⁻¹ = 1 := by + refine div_self ?_ + exact ne_of_gt (hxp (1 + n)) + have gg₂: x (2 + n) * (x (2 + n))⁻¹ = 1 := by + refine div_self ?_ + exact ne_of_gt (hxp (2 + n)) + rw [gg₁, gg₂] + norm_num + rw [g₁] at g₀ + rw [g₀] + simp + repeat rw [mul_rpow] + . have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by + rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹] + norm_num + have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by + rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹] + norm_num + have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + refine h₀ n ?_ + constructor + . exact hn.1 + . linarith + norm_cast at * + rw [g₂, g₃, ← mul_rpow g₁₁ g₁₂] + rw [← sqrt_eq_rpow] + ring_nf at g₄ + exact g₄ + . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + exact Eq.trans_le h₃₃ h₃₂ + + +lemma imo_2023_p4_3_13 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + (h₃₂ : 4 * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ + (4:ℝ)⁻¹ ≤ + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : + 4 * a n = + 4 * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ (4:ℝ)⁻¹ := by + simp + ring_nf + have g₀: (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 + * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 + = x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 := by + linarith + have g₁: x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1 := by + rw [mul_assoc] + have gg₁: x (1 + n) * (x (1 + n))⁻¹ = 1 := by + refine div_self ?_ + exact ne_of_gt (hxp (1 + n)) + have gg₂: x (2 + n) * (x (2 + n))⁻¹ = 1 := by + refine div_self ?_ + exact ne_of_gt (hxp (2 + n)) + rw [gg₁, gg₂] + norm_num + rw [g₁] at g₀ + rw [g₀] + simp + repeat rw [mul_rpow] + . have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by + rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹] + norm_num + have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by + rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹] + norm_num + have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + refine h₀ n ?_ + constructor + . exact hn.1 + . linarith + norm_cast at * + rw [g₂, g₃, ← mul_rpow g₁₁ g₁₂] + rw [← sqrt_eq_rpow] + ring_nf at g₄ + exact g₄ + . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + + +lemma imo_2023_p4_3_14 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + (h₃₂ : 4 * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ + (4:ℝ)⁻¹ ≤ + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : + a n = + ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (1 / (4:ℝ)) := by + have g₀: (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 + * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 + = x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 := by + linarith + have g₁: x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1 := by + rw [mul_assoc] + have gg₁: x (1 + n) * (x (1 + n))⁻¹ = 1 := by + refine div_self ?_ + exact ne_of_gt (hxp (1 + n)) + have gg₂: x (2 + n) * (x (2 + n))⁻¹ = 1 := by + refine div_self ?_ + exact ne_of_gt (hxp (2 + n)) + rw [gg₁, gg₂] + norm_num + rw [g₁] at g₀ + rw [g₀] + simp + repeat rw [mul_rpow] + have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by + rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹] + norm_num + have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by + rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹] + norm_num + have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + refine h₀ n ?_ + constructor + . exact hn.1 + . linarith + norm_cast at * + rw [g₂, g₃] + rw [← mul_rpow] + rw [← sqrt_eq_rpow] + ring_nf at g₄ + exact g₄ + . exact g₁₁ + . exact g₁₂ + . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + + +lemma imo_2023_p4_3_15 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + (h₃₂ : 4 * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ + (4:ℝ)⁻¹ ≤ + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = + x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) : + a n = + ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (1 / (4:ℝ)) := by + have g₁: x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1 := by + rw [mul_assoc] + have gg₁: x (1 + n) * (x (1 + n))⁻¹ = 1 := by + refine div_self ?_ + exact ne_of_gt (hxp (1 + n)) + have gg₂: x (2 + n) * (x (2 + n))⁻¹ = 1 := by + refine div_self ?_ + exact ne_of_gt (hxp (2 + n)) + rw [gg₁, gg₂] + norm_num + rw [g₁] at g₀ + rw [g₀] + simp + repeat rw [mul_rpow] + have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by + rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹] + norm_num + have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by + rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹] + norm_num + have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + refine h₀ n ?_ + constructor + . exact hn.1 + . linarith + norm_cast at * + rw [g₂, g₃] + rw [← mul_rpow] + rw [← sqrt_eq_rpow] + ring_nf at g₄ + exact g₄ + . exact g₁₁ + . exact g₁₂ + . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + + +lemma imo_2023_p4_3_16 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + -- (h₃₂ : 4 * + -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + -- (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * + -- (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ + -- (4:ℝ)⁻¹ ≤ + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + -- ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + -- x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = + -- x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) : + x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1 := by + rw [mul_assoc] + have gg₁: x (1 + n) * (x (1 + n))⁻¹ = 1 := by + refine div_self ?_ + exact ne_of_gt (hxp (1 + n)) + have gg₂: x (2 + n) * (x (2 + n))⁻¹ = 1 := by + refine div_self ?_ + exact ne_of_gt (hxp (2 + n)) + rw [gg₁, gg₂] + norm_num + + +lemma imo_2023_p4_3_17 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + -- (h₃₂ : 4 * + -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + -- (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * + -- (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ + -- (4:ℝ)⁻¹ ≤ + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + -- ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + -- x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = + -- x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) : + x (1 + n) * (x (1 + n))⁻¹ = 1 := by + refine div_self ?_ + exact ne_of_gt (hxp (1 + n)) + + +lemma imo_2023_p4_3_18 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + -- (h₃₂ : 4 * + -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + -- (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * + -- (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ + -- (4:ℝ)⁻¹ ≤ + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + -- ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + -- x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = + -- x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) + (gg₁ : x (1 + n) * (x (1 + n))⁻¹ = 1) + (gg₂ : x (2 + n) * (x (2 + n))⁻¹ = 1) : + x (1 + n) * (x (1 + n))⁻¹ * (x (2 + n) * (x (2 + n))⁻¹) = 1 := by + rw [gg₁, gg₂] + norm_num + + +lemma imo_2023_p4_3_19 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + (h₃₂ : 4 * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ + (4:ℝ)⁻¹ ≤ + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 + * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 + = x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) + (g₁ : x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1) : + a n = + ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (1 / (4:ℝ)) := by + rw [g₁] at g₀ + rw [g₀] + simp + repeat rw [mul_rpow] + have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by + rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹] + norm_num + have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by + rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹] + norm_num + have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + refine h₀ n ?_ + constructor + . exact hn.1 + . linarith + norm_cast at * + rw [g₂, g₃, ← mul_rpow] + rw [← sqrt_eq_rpow] + ring_nf at g₄ + exact g₄ + . exact g₁₁ + . exact g₁₂ + . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + + +lemma imo_2023_p4_3_20 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + (h₃₂ : 4 * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ + (4:ℝ)⁻¹ ≤ + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * (x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹) * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = + 1 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) + (g₁ : x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1) : + a n = + ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (4:ℝ)⁻¹ := by + repeat rw [mul_rpow] + have g₂: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ)) := by + rw [← rpow_mul g₁₁ (2:ℝ) (4:ℝ)⁻¹] + norm_num + have g₃: ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ)) := by + rw [← rpow_mul g₁₂ (2:ℝ) (4:ℝ)⁻¹] + norm_num + have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + refine h₀ n ?_ + constructor + . exact hn.1 + . linarith + norm_cast at * + rw [g₂, g₃, ← mul_rpow] + rw [← sqrt_eq_rpow] + ring_nf at g₄ + exact g₄ + . exact g₁₁ + . exact g₁₂ + . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + . exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + + +lemma imo_2023_p4_3_21 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + (h₃₂ : 4 * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ + (4:ℝ)⁻¹ ≤ + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = + 1 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) + (g₁ : x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1) + (g₂ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1/(2:ℝ))) + (g₃ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (2:ℝ)) ^ (4:ℝ)⁻¹ + = (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1/(2:ℝ))) : + a n = + ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2) ^ (4:ℝ)⁻¹ * + ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (4:ℝ)⁻¹ := by + have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + refine h₀ n ?_ + constructor + . exact hn.1 + . linarith + norm_cast at * + rw [g₂, g₃] + rw [← mul_rpow] + rw [← sqrt_eq_rpow] + ring_nf at g₄ + exact g₄ + . exact g₁₁ + . exact g₁₂ + + +lemma imo_2023_p4_3_22 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) : + -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + -- (h₃₂ : 4 * + -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + -- (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * + -- (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ + -- (4:ℝ)⁻¹ ≤ + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + -- ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + -- x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = + -- 1 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) + -- (g₁ : x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1) + -- (g₂ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2) ^ (4:ℝ)⁻¹ = + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1 / 2)) + -- (g₃ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (4:ℝ)⁻¹ = + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1 / 2)) : + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * + Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + have g₄: a (n) = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + refine h₀ n ?_ + constructor + . exact hn.1 + . linarith + norm_cast + + +lemma imo_2023_p4_3_23 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2021) + (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + (h₃₂ : 4 * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ + (4:ℝ)⁻¹ ≤ + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = + 1 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) + (g₁ : x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1) + (g₂ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2) ^ (4:ℝ)⁻¹ = + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1 / (2:ℝ))) + (g₃ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (4:ℝ)⁻¹ = + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1 / (2:ℝ))) + (g₄ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * + Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : + a n = + ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2) ^ (4:ℝ)⁻¹ * + ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (4:ℝ)⁻¹ := by + norm_cast at * + rw [g₂, g₃, ← mul_rpow] + . rw [← sqrt_eq_rpow] + ring_nf at g₄ + exact g₄ + . exact g₁₁ + . exact g₁₂ + + +lemma imo_2023_p4_3_24 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) + (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + -- (h₃₂ : 4 * + -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + -- (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1) * + -- (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) ^ + -- (4:ℝ)⁻¹ ≤ + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + -- ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1) + + -- x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) + -- (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = + -- 1 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) + -- (g₁ : x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1) + (g₂ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2) ^ (4:ℝ)⁻¹ = + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1 / (2:ℝ))) + (g₃ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (4:ℝ)⁻¹ = + (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1 / (2:ℝ))) + (g₄ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) : + a n = + ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2) ^ (4:ℝ)⁻¹ * + ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (4:ℝ)⁻¹ := by + rw [g₂, g₃, ← mul_rpow] + . rw [← sqrt_eq_rpow] + ring_nf at g₄ + exact g₄ + . exact g₁₁ + . exact g₁₂ + + +lemma imo_2023_p4_3_25 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1) ) + -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + -- (h₃₂ : 4 * + -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + -- (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1) * + -- (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) ^ + -- (4:ℝ)⁻¹ ≤ + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + -- ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1) + + -- x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) + -- (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = + -- 1 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) + -- (g₁ : x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1) + -- (g₂ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2) ^ (4:ℝ)⁻¹ = + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1 / 2)) + -- (g₃ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (4:ℝ)⁻¹ = + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1 / 2)) + (g₄ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) : + a n = + ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) * Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ + (1 / (2:ℝ)) := by + rw [← sqrt_eq_rpow] + ring_nf at g₄ + exact g₄ + + +lemma imo_2023_p4_3_26 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) + -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + -- (h₃₂ : 4 * + -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + -- (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1) * + -- (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) ^ + -- (4:ℝ)⁻¹ ≤ + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + -- ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1) + + -- x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) + -- (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = + -- 1 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) + -- (g₁ : x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1) + -- (g₂ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2) ^ (4:ℝ)⁻¹ = + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ (1 / 2)) + -- (g₃ : ((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) ^ (4:ℝ)⁻¹ = + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ (1 / 2)) + (g₄ : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * + Finset.sum (Finset.Ico 1 (n + 1)) fun x_1 => 1 / x x_1)) : + a n = √((Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) * + Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) := by + ring_nf at g₄ + exact g₄ + + +lemma imo_2023_p4_3_27 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + -- (h₃₂ : 4 * + -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + -- (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * + -- (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ + -- (4:ℝ)⁻¹ ≤ + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + -- ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + -- x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = + -- 1 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) + -- (g₁ : x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1) : + 0 ≤ (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 := by + exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + + +lemma imo_2023_p4_3_28 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + -- (h₃₂ : 4 * + -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + -- ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + -- (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * + -- (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ + -- (4:ℝ)⁻¹ ≤ + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + -- (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + -- ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + -- x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (g₀ : (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ * + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 = + -- 1 * (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) ^ 2 * + -- (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2) + -- (g₁ : x (1 + n) * (x (1 + n))⁻¹ * x (2 + n) * (x (2 + n))⁻¹ = 1) : + 0 ≤ (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) ^ 2 := by + exact sq_nonneg (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + + +lemma imo_2023_p4_3_29 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₁₁ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1) + -- (g₁₂ : 0 ≤ Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹) + (h₃₂ : 4 * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ + (4:ℝ)⁻¹ ≤ + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (h₃₃ : 4 * a n = + 4 * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2))) * + (x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) * + (x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) ^ + (4:ℝ)⁻¹) : + 4 * a n ≤ + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1)) + + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 2)) + + ((x (n + 1) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + x (n + 2) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact Eq.trans_le h₃₃ h₃₂ + + + +lemma imo_2023_p4_4 + (x a: ℕ → ℝ) + (hxp: ∀ (i : ℕ), 0 < x i) + (hx: ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + (h₀: ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k))) + (h₀₁: ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) : + (∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) := by + intros n hn + have g₀: 0 ≤ a n + 2 := by + refine le_of_lt ?_ + refine add_pos ?_ (by norm_num) + refine h₀₁ n ?_ + constructor + . exact hn.1 + . linarith + have g₁: 0 ≤ a (n + 2) := by + refine le_of_lt ?_ + refine h₀₁ (n + 2) ?_ + constructor + . linarith + . linarith + rw [← sqrt_sq g₀, ← sqrt_sq g₁] + have g₂: 0 ≤ (a n + 2) ^ 2 := by exact sq_nonneg (a n + 2) + simp + refine Real.sqrt_lt_sqrt g₂ ?_ + have g₃: 0 ≤ ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by + refine le_of_lt ?_ + refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + have gn₀: a (n) ^ 2 = ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by + rw [← sq_sqrt g₃] + have g₄: 0 ≤ a n := by + refine le_of_lt ?_ + refine h₀₁ n ?_ + constructor + . exact hn.1 + . linarith + refine (sq_eq_sq₀ g₄ ?_).mpr ?_ + . exact + sqrt_nonneg + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * + Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + refine h₀ (n) ?_ + constructor + . exact hn.1 + . linarith + have gn₁: a (n + 2) = sqrt ((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k) := by + refine h₀ (n + 2) ?_ + constructor + . linarith + . linarith + rw [add_sq, gn₁, sq_sqrt] + . have ga₀: 1 ≤ n + 2 := by linarith + rw [Finset.sum_Ico_succ_top ga₀ _, Finset.sum_Ico_succ_top ga₀ _] + have ga₁: 1 ≤ n + 1 := by linarith + rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _] + rw [add_assoc, add_assoc, add_assoc] + rw [add_mul, mul_add] + rw [← gn₀] + repeat rw [add_assoc] + refine add_lt_add_left ?_ (a (n) ^ 2) + rw [mul_add (x (n + 1) + x (n + 2))] + have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by + repeat rw [add_mul, mul_add, mul_add] + repeat rw [mul_div_left_comm _ 1 _, one_mul] + repeat rw [div_self ?_] + . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) + = x (n + 1) * x (n + 1) := by + rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 2)) + have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) + = x (n + 2) * x (n + 2) := by + rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 1)) + have h₂₀: 0 < x (n + 1) * x (n + 2) := by + refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . exact hxp (n + 2) + have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by + refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) + rw [mul_add, hc₂, hc₃, ← sq, ← sq] + refine lt_of_sub_pos ?_ + have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 + = (x (n + 1) - x (n + 2)) ^ 2 := by + rw [sub_sq] + linarith + rw [gh₂₁] + refine (sq_pos_iff).mpr ?_ + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + linarith + . exact ne_of_gt (hxp (n + 2)) + . exact ne_of_gt (hxp (n + 1)) + clear gn₀ gn₁ g₀ g₁ g₂ g₃ ga₀ ga₁ + have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * (1 / x (n + 1) + 1 / x (n + 2)) + + ((x (n + 1) + x (n + 2)) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact imo_2023_p4_3 (fun k => x k) a hxp h₀ n hn + linarith + . refine mul_nonneg ?_ ?_ + . refine Finset.sum_nonneg ?_ + intros i _ + exact LT.lt.le (hxp i) + . refine Finset.sum_nonneg ?_ + intros i _ + simp + exact LT.lt.le (hxp i) + + +lemma imo_2023_p4_4_1 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) : + 0 ≤ a n + 2 := by + refine le_of_lt ?_ + refine add_pos ?_ (by norm_num) + refine h₀₁ n ?_ + constructor + . exact hn.1 + . linarith + + +lemma imo_2023_p4_4_2 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) : + 0 < a n := by + refine h₀₁ n ?_ + constructor + . exact hn.1 + . linarith + + +lemma imo_2023_p4_4_3 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) : + -- (g₀ : 0 ≤ a n + 2) : + 0 ≤ a (n + 2) := by + refine le_of_lt ?_ + refine h₀₁ (n + 2) ?_ + constructor + . linarith + . linarith + + +lemma imo_2023_p4_4_4 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + (g₀ : 0 ≤ a n + 2) + (g₁ : 0 ≤ a (n + 2)) : + a n + 2 < a (n + 2) := by + rw [← sqrt_sq g₀, ← sqrt_sq g₁] + have g₂: 0 ≤ (a n + 2) ^ 2 := by exact sq_nonneg (a n + 2) + simp + refine Real.sqrt_lt_sqrt g₂ ?_ + have g₃: 0 ≤ ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by + refine le_of_lt ?_ + refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + have gn₀: a (n) ^ 2 = ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by + rw [← sq_sqrt g₃] + have g₄: 0 ≤ a n := by + refine le_of_lt ?_ + refine h₀₁ n ?_ + constructor + . exact hn.1 + . linarith + refine (sq_eq_sq₀ g₄ ?_).mpr ?_ + . exact + sqrt_nonneg + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * + Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + refine h₀ (n) ?_ + constructor + . exact hn.1 + . linarith + have gn₁: a (n + 2) = sqrt ((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k) := by + refine h₀ (n + 2) ?_ + constructor + . linarith + . linarith + rw [add_sq, gn₁, sq_sqrt] + . have ga₀: 1 ≤ n + 2 := by linarith + rw [Finset.sum_Ico_succ_top ga₀ _, Finset.sum_Ico_succ_top ga₀ _] + have ga₁: 1 ≤ n + 1 := by linarith + rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _] + rw [add_assoc, add_assoc, add_assoc] + rw [add_mul, mul_add] + rw [← gn₀] + repeat rw [add_assoc] + refine add_lt_add_left ?_ (a (n) ^ 2) + rw [mul_add (x (n + 1) + x (n + 2))] + have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by + repeat rw [add_mul, mul_add, mul_add] + repeat rw [mul_div_left_comm _ 1 _, one_mul] + repeat rw [div_self ?_] + . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) + = x (n + 1) * x (n + 1) := by + rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 2)) + have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) + = x (n + 2) * x (n + 2) := by + rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 1)) + have h₂₀: 0 < x (n + 1) * x (n + 2) := by + refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . exact hxp (n + 2) + have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by + refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) + rw [mul_add, hc₂, hc₃, ← sq, ← sq] + refine lt_of_sub_pos ?_ + have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 + = (x (n + 1) - x (n + 2)) ^ 2 := by + rw [sub_sq] + linarith + rw [gh₂₁] + refine (sq_pos_iff).mpr ?_ + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + linarith + . exact ne_of_gt (hxp (n + 2)) + . exact ne_of_gt (hxp (n + 1)) + clear gn₀ gn₁ g₀ g₁ g₂ g₃ ga₀ ga₁ + have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * (1 / x (n + 1) + 1 / x (n + 2)) + + ((x (n + 1) + x (n + 2)) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact imo_2023_p4_3 (fun k => x k) a hxp h₀ n hn + linarith + . refine mul_nonneg ?_ ?_ + . refine Finset.sum_nonneg ?_ + intros i _ + exact LT.lt.le (hxp i) + . refine Finset.sum_nonneg ?_ + intros i _ + simp + exact LT.lt.le (hxp i) + + +lemma imo_2023_p4_4_5 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + (g₀ : 0 ≤ a n + 2) + (g₁ : 0 ≤ a (n + 2)) : + √((a n + 2) ^ 2) < √(a (n + 2) ^ 2) := by + have g₂: 0 ≤ (a n + 2) ^ 2 := by exact sq_nonneg (a n + 2) + simp + refine Real.sqrt_lt_sqrt g₂ ?_ + have g₃: 0 ≤ ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by + refine le_of_lt ?_ + refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + have gn₀: a (n) ^ 2 = ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by + rw [← sq_sqrt g₃] + have g₄: 0 ≤ a n := by + refine le_of_lt ?_ + refine h₀₁ n ?_ + constructor + . exact hn.1 + . linarith + refine (sq_eq_sq₀ g₄ ?_).mpr ?_ + . exact + sqrt_nonneg + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * + Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + refine h₀ (n) ?_ + constructor + . exact hn.1 + . linarith + have gn₁: a (n + 2) = sqrt ((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k) := by + refine h₀ (n + 2) ?_ + constructor + . linarith + . linarith + rw [add_sq, gn₁, sq_sqrt] + . have ga₀: 1 ≤ n + 2 := by linarith + rw [Finset.sum_Ico_succ_top ga₀ _, Finset.sum_Ico_succ_top ga₀ _] + have ga₁: 1 ≤ n + 1 := by linarith + rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _] + rw [add_assoc, add_assoc, add_assoc] + rw [add_mul, mul_add] + rw [← gn₀] + repeat rw [add_assoc] + refine add_lt_add_left ?_ (a (n) ^ 2) + rw [mul_add (x (n + 1) + x (n + 2))] + have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by + repeat rw [add_mul, mul_add, mul_add] + repeat rw [mul_div_left_comm _ 1 _, one_mul] + repeat rw [div_self ?_] + . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) + = x (n + 1) * x (n + 1) := by + rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 2)) + have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) + = x (n + 2) * x (n + 2) := by + rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 1)) + have h₂₀: 0 < x (n + 1) * x (n + 2) := by + refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . exact hxp (n + 2) + have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by + refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) + rw [mul_add, hc₂, hc₃, ← sq, ← sq] + refine lt_of_sub_pos ?_ + have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 + = (x (n + 1) - x (n + 2)) ^ 2 := by + rw [sub_sq] + linarith + rw [gh₂₁] + refine (sq_pos_iff).mpr ?_ + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + linarith + . exact ne_of_gt (hxp (n + 2)) + . exact ne_of_gt (hxp (n + 1)) + clear gn₀ gn₁ g₀ g₁ g₂ g₃ ga₀ ga₁ + have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * (1 / x (n + 1) + 1 / x (n + 2)) + + ((x (n + 1) + x (n + 2)) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact imo_2023_p4_3 (fun k => x k) a hxp h₀ n hn + linarith + . refine mul_nonneg ?_ ?_ + . refine Finset.sum_nonneg ?_ + intros i _ + exact LT.lt.le (hxp i) + . refine Finset.sum_nonneg ?_ + intros i _ + simp + exact LT.lt.le (hxp i) + + +lemma imo_2023_p4_4_6 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + (g₀ : 0 ≤ a n + 2) + (g₁ : 0 ≤ a (n + 2)) + (g₂ : 0 ≤ (a n + 2) ^ 2) : + (a n + 2) ^ 2 < a (n + 2) ^ 2 := by + have g₃: 0 ≤ ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by + refine le_of_lt ?_ + refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + have gn₀: a (n) ^ 2 = ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by + rw [← sq_sqrt g₃] + have g₄: 0 ≤ a n := by + refine le_of_lt ?_ + refine h₀₁ n ?_ + constructor + . exact hn.1 + . linarith + refine (sq_eq_sq₀ g₄ ?_).mpr ?_ + . exact + sqrt_nonneg + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * + Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + . refine h₀ (n) ?_ + constructor + . exact hn.1 + . linarith + have gn₁: a (n + 2) = sqrt ((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k) := by + refine h₀ (n + 2) ?_ + constructor + . linarith + . linarith + rw [add_sq, gn₁, sq_sqrt] + . have ga₀: 1 ≤ n + 2 := by linarith + rw [Finset.sum_Ico_succ_top ga₀ _, Finset.sum_Ico_succ_top ga₀ _] + have ga₁: 1 ≤ n + 1 := by linarith + rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _] + rw [add_assoc, add_assoc, add_assoc] + rw [add_mul, mul_add] + rw [← gn₀] + repeat rw [add_assoc] + refine add_lt_add_left ?_ (a (n) ^ 2) + rw [mul_add (x (n + 1) + x (n + 2))] + have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by + repeat rw [add_mul, mul_add, mul_add] + repeat rw [mul_div_left_comm _ 1 _, one_mul] + repeat rw [div_self ?_] + . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) + = x (n + 1) * x (n + 1) := by + rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 2)) + have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) + = x (n + 2) * x (n + 2) := by + rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 1)) + have h₂₀: 0 < x (n + 1) * x (n + 2) := by + refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . exact hxp (n + 2) + have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by + refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) + rw [mul_add, hc₂, hc₃, ← sq, ← sq] + refine lt_of_sub_pos ?_ + have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 + = (x (n + 1) - x (n + 2)) ^ 2 := by + rw [sub_sq] + linarith + rw [gh₂₁] + refine (sq_pos_iff).mpr ?_ + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + linarith + . exact ne_of_gt (hxp (n + 2)) + . exact ne_of_gt (hxp (n + 1)) + clear gn₀ gn₁ g₀ g₁ g₂ g₃ ga₀ ga₁ + have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * (1 / x (n + 1) + 1 / x (n + 2)) + + ((x (n + 1) + x (n + 2)) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact imo_2023_p4_3 (fun k => x k) a hxp h₀ n hn + linarith + . refine mul_nonneg ?_ ?_ + . refine Finset.sum_nonneg ?_ + intros i _ + exact LT.lt.le (hxp i) + . refine Finset.sum_nonneg ?_ + intros i _ + simp + exact LT.lt.le (hxp i) + + +lemma imo_2023_p4_4_7 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) : + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) : + 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k := by + refine le_of_lt ?_ + refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . exact fun i _ => hxp i + . simp + linarith + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + + +lemma imo_2023_p4_4_8 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) : + a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k := by + rw [← sq_sqrt g₃] + have g₄: 0 ≤ a n := by + refine le_of_lt ?_ + refine h₀₁ n ?_ + constructor + . exact hn.1 + . linarith + refine (sq_eq_sq₀ g₄ ?_).mpr ?_ + . exact + sqrt_nonneg + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * + Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + refine h₀ (n) ?_ + constructor + . exact hn.1 + . linarith + + +lemma imo_2023_p4_4_9 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + (g₀ : 0 ≤ a n + 2) + (g₁ : 0 ≤ a (n + 2)) + (g₂ : 0 ≤ (a n + 2) ^ 2) + (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) : + (a n + 2) ^ 2 < a (n + 2) ^ 2 := by + have gn₀: a (n) ^ 2 = ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * (Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) := by + rw [← sq_sqrt g₃] + have g₄: 0 ≤ a n := by + refine le_of_lt ?_ + refine h₀₁ n ?_ + constructor + . exact hn.1 + . linarith + refine (sq_eq_sq₀ g₄ ?_).mpr ?_ + . exact + sqrt_nonneg + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * + Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + . refine h₀ (n) ?_ + constructor + . exact hn.1 + . linarith + have gn₁: a (n + 2) = sqrt ((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k) := by + refine h₀ (n + 2) ?_ + constructor + . linarith + . linarith + rw [add_sq, gn₁, sq_sqrt] + . have ga₀: 1 ≤ n + 2 := by linarith + rw [Finset.sum_Ico_succ_top ga₀ _, Finset.sum_Ico_succ_top ga₀ _] + have ga₁: 1 ≤ n + 1 := by linarith + rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _] + rw [add_assoc, add_assoc, add_assoc] + rw [add_mul, mul_add] + rw [← gn₀] + repeat rw [add_assoc] + refine add_lt_add_left ?_ (a (n) ^ 2) + rw [mul_add (x (n + 1) + x (n + 2))] + have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by + repeat rw [add_mul, mul_add, mul_add] + repeat rw [mul_div_left_comm _ 1 _, one_mul] + repeat rw [div_self ?_] + . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) + = x (n + 1) * x (n + 1) := by + rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 2)) + have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) + = x (n + 2) * x (n + 2) := by + rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 1)) + have h₂₀: 0 < x (n + 1) * x (n + 2) := by + refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . exact hxp (n + 2) + have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by + refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) + rw [mul_add, hc₂, hc₃, ← sq, ← sq] + refine lt_of_sub_pos ?_ + have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 + = (x (n + 1) - x (n + 2)) ^ 2 := by + rw [sub_sq] + linarith + rw [gh₂₁] + refine (sq_pos_iff).mpr ?_ + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + linarith + . exact ne_of_gt (hxp (n + 2)) + . exact ne_of_gt (hxp (n + 1)) + clear gn₀ gn₁ g₀ g₁ g₂ g₃ ga₀ ga₁ + have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * (1 / x (n + 1) + 1 / x (n + 2)) + + ((x (n + 1) + x (n + 2)) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact imo_2023_p4_3 (fun k => x k) a hxp h₀ n hn + linarith + . refine mul_nonneg ?_ ?_ + . refine Finset.sum_nonneg ?_ + intros i _ + exact LT.lt.le (hxp i) + . refine Finset.sum_nonneg ?_ + intros i _ + simp + exact LT.lt.le (hxp i) + + + +lemma imo_2023_p4_4_10 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + (g₄ : 0 ≤ a n) : + a n ^ 2 = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) ^ 2 := by + refine (sq_eq_sq₀ g₄ ?_).mpr ?_ + . exact + sqrt_nonneg + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * + Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + . refine h₀ (n) ?_ + constructor + . exact hn.1 + . linarith + + +lemma imo_2023_p4_4_11 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (g₄ : 0 ≤ a n) : + 0 ≤ √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * + Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact sqrt_nonneg + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * + Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + +lemma imo_2023_p4_4_12 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + (g₀ : 0 ≤ a n + 2) + (g₁ : 0 ≤ a (n + 2)) + (g₂ : 0 ≤ (a n + 2) ^ 2) + (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + (gn₁ : a (n + 2) = + √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) : + (a n + 2) ^ 2 < a (n + 2) ^ 2 := by + rw [add_sq, gn₁, sq_sqrt] + . have ga₀: 1 ≤ n + 2 := by linarith + rw [Finset.sum_Ico_succ_top ga₀ _, Finset.sum_Ico_succ_top ga₀ _] + have ga₁: 1 ≤ n + 1 := by linarith + rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _] + rw [add_assoc, add_assoc, add_assoc] + rw [add_mul, mul_add] + rw [← gn₀] + repeat rw [add_assoc] + refine add_lt_add_left ?_ (a (n) ^ 2) + rw [mul_add (x (n + 1) + x (n + 2))] + have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by + repeat rw [add_mul, mul_add, mul_add] + repeat rw [mul_div_left_comm _ 1 _, one_mul] + repeat rw [div_self ?_] + . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) + = x (n + 1) * x (n + 1) := by + rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 2)) + have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) + = x (n + 2) * x (n + 2) := by + rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 1)) + have h₂₀: 0 < x (n + 1) * x (n + 2) := by + refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . exact hxp (n + 2) + have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by + refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) + rw [mul_add, hc₂, hc₃, ← sq, ← sq] + refine lt_of_sub_pos ?_ + have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 + = (x (n + 1) - x (n + 2)) ^ 2 := by + rw [sub_sq] + linarith + rw [gh₂₁] + refine (sq_pos_iff).mpr ?_ + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + linarith + . exact ne_of_gt (hxp (n + 2)) + . exact ne_of_gt (hxp (n + 1)) + clear gn₀ gn₁ g₀ g₁ g₂ g₃ ga₀ ga₁ + have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * (1 / x (n + 1) + 1 / x (n + 2)) + + ((x (n + 1) + x (n + 2)) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact imo_2023_p4_3 (fun k => x k) a hxp h₀ n hn + linarith + . refine mul_nonneg ?_ ?_ + . refine Finset.sum_nonneg ?_ + intros i _ + exact LT.lt.le (hxp i) + . refine Finset.sum_nonneg ?_ + intros i _ + simp + exact LT.lt.le (hxp i) + + + +lemma imo_2023_p4_4_13 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + (g₀ : 0 ≤ a n + 2) + (g₁ : 0 ≤ a (n + 2)) + (g₂ : 0 ≤ (a n + 2) ^ 2) + (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + (gn₁ : a (n + 2) = + √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) : + a n ^ 2 + 2 * a n * 2 + 2 ^ 2 < + (Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k := by + have ga₀: 1 ≤ n + 2 := by linarith + rw [Finset.sum_Ico_succ_top ga₀ _, Finset.sum_Ico_succ_top ga₀ _] + have ga₁: 1 ≤ n + 1 := by linarith + rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _] + rw [add_assoc, add_assoc, add_assoc] + rw [add_mul, mul_add] + rw [← gn₀] + repeat rw [add_assoc] + refine add_lt_add_left ?_ (a (n) ^ 2) + rw [mul_add (x (n + 1) + x (n + 2))] + have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by + repeat rw [add_mul, mul_add, mul_add] + repeat rw [mul_div_left_comm _ 1 _, one_mul] + repeat rw [div_self ?_] + . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) + = x (n + 1) * x (n + 1) := by + rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 2)) + have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) + = x (n + 2) * x (n + 2) := by + rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 1)) + have h₂₀: 0 < x (n + 1) * x (n + 2) := by + refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . exact hxp (n + 2) + have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by + refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) + rw [mul_add, hc₂, hc₃, ← sq, ← sq] + refine lt_of_sub_pos ?_ + have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 + = (x (n + 1) - x (n + 2)) ^ 2 := by + rw [sub_sq] + linarith + rw [gh₂₁] + refine (sq_pos_iff).mpr ?_ + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + linarith + . exact ne_of_gt (hxp (n + 2)) + . exact ne_of_gt (hxp (n + 1)) + clear gn₀ gn₁ g₀ g₁ g₂ g₃ ga₀ ga₁ + have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * (1 / x (n + 1) + 1 / x (n + 2)) + + ((x (n + 1) + x (n + 2)) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact imo_2023_p4_3 (fun k => x k) a hxp h₀ n hn + linarith + + +lemma imo_2023_p4_4_14 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + (g₀ : 0 ≤ a n + 2) + (g₁ : 0 ≤ a (n + 2)) + (g₂ : 0 ≤ (a n + 2) ^ 2) + (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + (gn₁ : a (n + 2) = + √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) + (ga₀ : 1 ≤ n + 2) : + a n ^ 2 + 2 * a n * 2 + 2 ^ 2 < + ((Finset.sum (Finset.Ico 1 (n + 2)) fun k => x k) + x (n + 2)) * + ((Finset.sum (Finset.Ico 1 (n + 2)) fun k => 1 / x k) + 1 / x (n + 2)) := by + have ga₁: 1 ≤ n + 1 := by linarith + rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _] + rw [add_assoc, add_assoc, add_assoc] + rw [add_mul, mul_add] + rw [← gn₀] + repeat rw [add_assoc] + refine add_lt_add_left ?_ (a (n) ^ 2) + rw [mul_add (x (n + 1) + x (n + 2))] + have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by + repeat rw [add_mul, mul_add, mul_add] + repeat rw [mul_div_left_comm _ 1 _, one_mul] + repeat rw [div_self ?_] + . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) + = x (n + 1) * x (n + 1) := by + rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 2)) + have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) + = x (n + 2) * x (n + 2) := by + rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 1)) + have h₂₀: 0 < x (n + 1) * x (n + 2) := by + refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . exact hxp (n + 2) + have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by + refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) + rw [mul_add, hc₂, hc₃, ← sq, ← sq] + refine lt_of_sub_pos ?_ + have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 + = (x (n + 1) - x (n + 2)) ^ 2 := by + rw [sub_sq] + linarith + rw [gh₂₁] + refine (sq_pos_iff).mpr ?_ + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + linarith + . exact ne_of_gt (hxp (n + 2)) + . exact ne_of_gt (hxp (n + 1)) + clear gn₀ gn₁ g₀ g₁ g₂ g₃ ga₀ ga₁ + have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * (1 / x (n + 1) + 1 / x (n + 2)) + + ((x (n + 1) + x (n + 2)) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact imo_2023_p4_3 (fun k => x k) a hxp h₀ n hn + linarith + + +lemma imo_2023_p4_4_15 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₁ : a (n + 2) = + -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) + (ga₀ : 1 ≤ n + 2) + (ga₁ : 1 ≤ n + 1) + (ga₂ : a n ^ 2 + (2 * a n * 2 + 2 ^ 2) < + a n ^ 2 + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + + (x (n + 1) + x (n + 2)) * ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + (1 / x (n + 1) + 1 / x (n + 2)))) : + a n ^ 2 + 2 * a n * 2 + 2 ^ 2 < + (Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k := by + rw [Finset.sum_Ico_succ_top ga₀ _, Finset.sum_Ico_succ_top ga₀ _] + rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _] + rw [add_assoc, add_assoc, add_assoc] + rw [add_mul, mul_add] + rw [← gn₀] + repeat rw [add_assoc] + refine add_lt_add_left ?_ (a (n) ^ 2) + linarith + + +lemma imo_2023_p4_4_16 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + (g₀ : 0 ≤ a n + 2) + (g₁ : 0 ≤ a (n + 2)) + (g₂ : 0 ≤ (a n + 2) ^ 2) + (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + (gn₁ : a (n + 2) = + √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) + (ga₀ : 1 ≤ n + 2) + (ga₁ : 1 ≤ n + 1) : + a n ^ 2 + (2 * a n * 2 + 2 ^ 2) < + a n ^ 2 + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + + (x (n + 1) + x (n + 2)) * + ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + (1 / x (n + 1) + 1 / x (n + 2))) := by + repeat rw [add_assoc] + refine add_lt_add_left ?_ (a (n) ^ 2) + rw [mul_add (x (n + 1) + x (n + 2))] + have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by + repeat rw [add_mul, mul_add, mul_add] + repeat rw [mul_div_left_comm _ 1 _, one_mul] + repeat rw [div_self ?_] + . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) + = x (n + 1) * x (n + 1) := by + rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 2)) + have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) + = x (n + 2) * x (n + 2) := by + rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 1)) + have h₂₀: 0 < x (n + 1) * x (n + 2) := by + refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . exact hxp (n + 2) + have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by + refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) + rw [mul_add, hc₂, hc₃, ← sq, ← sq] + refine lt_of_sub_pos ?_ + have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 + = (x (n + 1) - x (n + 2)) ^ 2 := by + rw [sub_sq] + linarith + rw [gh₂₁] + refine (sq_pos_iff).mpr ?_ + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + linarith + . exact ne_of_gt (hxp (n + 2)) + . exact ne_of_gt (hxp (n + 1)) + clear gn₀ gn₁ g₀ g₁ g₂ g₃ ga₀ ga₁ + have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * (1 / x (n + 1) + 1 / x (n + 2)) + + ((x (n + 1) + x (n + 2)) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact imo_2023_p4_3 (fun k => x k) a hxp h₀ n hn + linarith + + +lemma imo_2023_p4_4_17 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₁ : a (n + 2) = + -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) + -- (ga₀ : 1 ≤ n + 2) + -- (ga₁ : 1 ≤ n + 1) + (ga₂ : 2 * a n * 2 + 2 ^ 2 < + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + + (((x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)))) : + a n ^ 2 + 2 * a n * 2 + 2 ^ 2 < + (Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * + Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k := by + have ga₀: 1 ≤ n + 2 := by linarith + rw [Finset.sum_Ico_succ_top ga₀ _, Finset.sum_Ico_succ_top ga₀ _] + have ga₁: 1 ≤ n + 1 := by linarith + rw [Finset.sum_Ico_succ_top ga₁ _, Finset.sum_Ico_succ_top ga₁ _] + rw [add_assoc, add_assoc, add_assoc] + rw [add_mul, mul_add] + rw [← gn₀] + repeat rw [add_assoc] + refine add_lt_add_left ?_ (a (n) ^ 2) + linarith + + +lemma imo_2023_p4_4_18 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₁ : a (n + 2) = + -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) + -- (ga₀ : 1 ≤ n + 2) + -- (ga₁ : 1 ≤ n + 1) : + 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by + repeat rw [add_mul, mul_add, mul_add] + repeat rw [mul_div_left_comm _ 1 _, one_mul] + repeat rw [div_self ?_] + . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) + = x (n + 1) * x (n + 1) := by + rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 2)) + have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) + = x (n + 2) * x (n + 2) := by + rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 1)) + have h₂₀: 0 < x (n + 1) * x (n + 2) := by + refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . exact hxp (n + 2) + have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by + refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) + rw [mul_add, hc₂, hc₃, ← sq, ← sq] + refine lt_of_sub_pos ?_ + have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 + = (x (n + 1) - x (n + 2)) ^ 2 := by + rw [sub_sq] + linarith + rw [gh₂₁] + refine (sq_pos_iff).mpr ?_ + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + linarith + . exact ne_of_gt (hxp (n + 2)) + . exact ne_of_gt (hxp (n + 1)) + + +lemma imo_2023_p4_4_19 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₁ : a (n + 2) = + -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) + -- (ga₀ : 1 ≤ n + 2) + -- (ga₁ : 1 ≤ n + 1) + -- (ga₂: 4 < x (n + 1) / x (n + 1) + x (n + 1) / x (n + 2) + (x (n + 2) / x (n + 1) + x (n + 2) / x (n + 2))) : + 4 < x (n + 1) / x (n + 1) + x (n + 1) / x (n + 2) + (x (n + 2) / x (n + 1) + x (n + 2) / x (n + 2)) := by + repeat rw [div_self ?_] + . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) + = x (n + 1) * x (n + 1) := by + rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 2)) + have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) + = x (n + 2) * x (n + 2) := by + rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 1)) + have h₂₀: 0 < x (n + 1) * x (n + 2) := by + refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . exact hxp (n + 2) + have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by + refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) + rw [mul_add, hc₂, hc₃, ← sq, ← sq] + refine lt_of_sub_pos ?_ + have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 + = (x (n + 1) - x (n + 2)) ^ 2 := by + rw [sub_sq] + linarith + rw [gh₂₁] + refine (sq_pos_iff).mpr ?_ + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + linarith + . exact ne_of_gt (hxp (n + 2)) + . exact ne_of_gt (hxp (n + 1)) + + +lemma imo_2023_p4_4_20 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₁ : a (n + 2) = + -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) + -- (ga₀ : 1 ≤ n + 2) + -- (ga₁ : 1 ≤ n + 1) : + 4 < x (n + 1) / x (n + 1) + x (n + 1) / x (n + 2) + + (x (n + 2) / x (n + 1) + x (n + 2) / x (n + 2)) := by + -- repeat rw [mul_div_left_comm _ 1 _, one_mul] + repeat rw [div_self ?_] + . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) + = x (n + 1) * x (n + 1) := by + rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 2)) + have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) + = x (n + 2) * x (n + 2) := by + rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 1)) + have h₂₀: 0 < x (n + 1) * x (n + 2) := by + refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . exact hxp (n + 2) + have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by + refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) + rw [mul_add, hc₂, hc₃, ← sq, ← sq] + refine lt_of_sub_pos ?_ + have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 + = (x (n + 1) - x (n + 2)) ^ 2 := by + rw [sub_sq] + linarith + rw [gh₂₁] + refine (sq_pos_iff).mpr ?_ + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + linarith + . exact ne_of_gt (hxp (n + 2)) + . exact ne_of_gt (hxp (n + 1)) + + +lemma imo_2023_p4_4_21 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₁ : a (n + 2) = + -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) + -- (ga₀ : 1 ≤ n + 2) + -- (ga₁ : 1 ≤ n + 1) : + x (n + 2) ≠ 0 := by + exact ne_of_gt (hxp (n + 2)) + + +lemma imo_2023_p4_4_22 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₁ : a (n + 2) = + -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) + -- (ga₀ : 1 ≤ n + 2) + -- (ga₁ : 1 ≤ n + 1) : + 4 < 1 + x (n + 1) / x (n + 2) + (x (n + 2) / x (n + 1) + 1) := by + have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) + = x (n + 1) * x (n + 1) := by + rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 2)) + have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) + = x (n + 2) * x (n + 2) := by + rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 1)) + have h₂₀: 0 < x (n + 1) * x (n + 2) := by + refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . exact hxp (n + 2) + have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by + refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) + rw [mul_add, hc₂, hc₃, ← sq, ← sq] + refine lt_of_sub_pos ?_ + have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 + = (x (n + 1) - x (n + 2)) ^ 2 := by + rw [sub_sq] + linarith + rw [gh₂₁] + refine (sq_pos_iff).mpr ?_ + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + linarith + + +lemma imo_2023_p4_4_23 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₁ : a (n + 2) = + -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) + -- (ga₀ : 1 ≤ n + 2) + -- (ga₁ : 1 ≤ n + 1) : + x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1) := by + rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 2)) + + +lemma imo_2023_p4_4_24 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₁ : a (n + 2) = + -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) + -- (ga₀ : 1 ≤ n + 2) + -- (ga₁ : 1 ≤ n + 1) + -- (hc₂ : x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1)) : + x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2) := by + have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) + = x (n + 2) * x (n + 2) := by + rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 1)) + linarith + + +lemma imo_2023_p4_4_25 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₁ : a (n + 2) = + -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) + -- (ga₀ : 1 ≤ n + 2) + -- (ga₁ : 1 ≤ n + 1) + (hc₂ : x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1)) + (hc₃ : x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2)) + (h₂₀ : 0 < x (n + 1) * x (n + 2)) : + 4 < 1 + x (n + 1) / x (n + 2) + (x (n + 2) / x (n + 1) + 1) := by + have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by + refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) + rw [mul_add, hc₂, hc₃, ← sq, ← sq] + refine lt_of_sub_pos ?_ + have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 + = (x (n + 1) - x (n + 2)) ^ 2 := by + rw [sub_sq] + linarith + rw [gh₂₁] + refine (sq_pos_iff).mpr ?_ + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + linarith + + +lemma imo_2023_p4_4_26 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₁ : a (n + 2) = + -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) + -- (ga₀ : 1 ≤ n + 2) + -- (ga₁ : 1 ≤ n + 1) + (hc₂ : x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1)) + (hc₃ : x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2)) + (h₂₀ : 0 < x (n + 1) * x (n + 2)) : + 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by + refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) + rw [mul_add, hc₂, hc₃, ← sq, ← sq] + refine lt_of_sub_pos ?_ + have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 + = (x (n + 1) - x (n + 2)) ^ 2 := by + rw [sub_sq] + linarith + rw [gh₂₁] + refine (sq_pos_iff).mpr ?_ + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + + +lemma imo_2023_p4_4_27 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₁ : a (n + 2) = + -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) + -- (ga₀ : 1 ≤ n + 2) + -- (ga₁ : 1 ≤ n + 1) + (hc₂ : x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1)) + (hc₃ : x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2)) : + -- (h₂₀ : 0 < x (n + 1) * x (n + 2)) : + x (n + 1) * x (n + 2) * 2 < x (n + 1) * x (n + 2) * + (x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1)) := by + rw [mul_add, hc₂, hc₃, ← sq, ← sq] + refine lt_of_sub_pos ?_ + have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 + = (x (n + 1) - x (n + 2)) ^ 2 := by + rw [sub_sq] + linarith + rw [gh₂₁] + refine (sq_pos_iff).mpr ?_ + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + + +lemma imo_2023_p4_4_28 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₁ : a (n + 2) = + -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) + -- (ga₀ : 1 ≤ n + 2) + -- (ga₁ : 1 ≤ n + 1) + (hc₂ : x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1)) + (hc₃ : x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2)) + (h₂₀ : 0 < x (n + 1) * x (n + 2)) : + 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by + refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) + rw [mul_add, hc₂, hc₃, ← sq, ← sq] + refine lt_of_sub_pos ?_ + have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 + = (x (n + 1) - x (n + 2)) ^ 2 := by + rw [sub_sq] + linarith + rw [gh₂₁] + refine (sq_pos_iff).mpr ?_ + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + + +lemma imo_2023_p4_4_29 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₁ : a (n + 2) = + -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) + -- (ga₀ : 1 ≤ n + 2) + -- (ga₁ : 1 ≤ n + 1) + -- (hc₂ : x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1)) + -- (hc₃ : x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2)) + -- (h₂₀ : 0 < x (n + 1) * x (n + 2)) : + x (n + 1) * x (n + 2) * 2 < x (n + 1) ^ 2 + x (n + 2) ^ 2 := by + refine lt_of_sub_pos ?_ + have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 + = (x (n + 1) - x (n + 2)) ^ 2 := by + rw [sub_sq] + linarith + rw [gh₂₁] + refine (sq_pos_iff).mpr ?_ + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + + +lemma imo_2023_p4_4_30 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₁ : a (n + 2) = + -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) + -- (ga₀ : 1 ≤ n + 2) + -- (ga₁ : 1 ≤ n + 1) + -- (hc₂ : x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1)) + -- (hc₃ : x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2)) + -- (h₂₀ : 0 < x (n + 1) * x (n + 2)) : + 0 < x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 := by + have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 + = (x (n + 1) - x (n + 2)) ^ 2 := by + rw [sub_sq] + linarith + rw [gh₂₁] + refine (sq_pos_iff).mpr ?_ + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + + +lemma imo_2023_p4_4_31 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₁ : a (n + 2) = + -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) + -- (ga₀ : 1 ≤ n + 2) + -- (ga₁ : 1 ≤ n + 1) + -- (hc₂ : x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1)) + -- (hc₃ : x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2)) + -- (h₂₀ : 0 < x (n + 1) * x (n + 2)) : + x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2 := by + rw [sub_sq] + linarith + + +lemma imo_2023_p4_4_32 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₁ : a (n + 2) = + -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) + -- (ga₀ : 1 ≤ n + 2) + -- (ga₁ : 1 ≤ n + 1) + -- (hc₂ : x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1)) + -- (hc₃ : x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2)) + -- (h₂₀ : 0 < x (n + 1) * x (n + 2)) + (gh₂₁ : x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2) : + 0 < x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 := by + rw [gh₂₁] + refine (sq_pos_iff).mpr ?_ + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + + +lemma imo_2023_p4_4_33 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₁ : a (n + 2) = + -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) + -- (ga₀ : 1 ≤ n + 2) + -- (ga₁ : 1 ≤ n + 1) + -- (hc₂ : x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) = x (n + 1) * x (n + 1)) + -- (hc₃ : x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) = x (n + 2) * x (n + 2)) + -- (h₂₀ : 0 < x (n + 1) * x (n + 2)) + -- (gh₂₁ : x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 = (x (n + 1) - x (n + 2)) ^ 2) : + x (n + 1) - x (n + 2) ≠ 0 := by + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + + +lemma imo_2023_p4_4_34 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + (h₂ : 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2))) : + 2 * a n * 2 + 2 ^ 2 < + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + + (((x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2))) := by + have h₃: 4 * a (n) ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * (1 / x (n + 1) + 1 / x (n + 2)) + + ((x (n + 1) + x (n + 2)) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact imo_2023_p4_3 (fun k => x k) a hxp h₀ n hn + linarith + + +lemma imo_2023_p4_4_35 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (h₂ : 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2))) + (h₃ : 4 * a n ≤ + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + + (x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) : + 2 * a n * 2 + 2 ^ 2 < + (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * (1 / x (n + 1) + 1 / x (n + 2)) + + (((x (n + 1) + x (n + 2)) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + + (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2))) := by + have h₂: 4 < (x (n + 1) + x (n + 2)) * (1 / x (n + 1) + 1 / x (n + 2)) := by + repeat rw [add_mul, mul_add, mul_add] + repeat rw [mul_div_left_comm _ 1 _, one_mul] + repeat rw [div_self ?_] + . have hc₂: x (n + 1) * x (n + 2) * (x (n + 1) / x (n + 2)) + = x (n + 1) * x (n + 1) := by + rw [mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 2)) + have hc₃: x (n + 1) * x (n + 2) * (x (n + 2) / x (n + 1)) + = x (n + 2) * x (n + 2) := by + rw [mul_comm (x (n + 1)) (x (n + 2)), mul_assoc, ← mul_div_assoc, mul_div_right_comm, div_self ?_] + simp + exact ne_of_gt (hxp (n + 1)) + have h₂₀: 0 < x (n + 1) * x (n + 2) := by + refine mul_pos ?_ ?_ + . exact hxp (n + 1) + . exact hxp (n + 2) + have h₂₁: 2 < x (n + 1) / x (n + 2) + x (n + 2) / x (n + 1) := by + refine lt_of_mul_lt_mul_left ?_ (le_of_lt h₂₀) + rw [mul_add, hc₂, hc₃, ← sq, ← sq] + refine lt_of_sub_pos ?_ + have gh₂₁: x (n + 1) ^ 2 + x (n + 2) ^ 2 - x (n + 1) * x (n + 2) * 2 + = (x (n + 1) - x (n + 2)) ^ 2 := by + rw [sub_sq] + linarith + rw [gh₂₁] + refine (sq_pos_iff).mpr ?_ + refine sub_ne_zero.mpr ?_ + exact hx (n+1) (n+2) (by linarith) + linarith + . exact ne_of_gt (hxp (n + 2)) + . exact ne_of_gt (hxp (n + 1)) + linarith + + +lemma imo_2023_p4_4_36 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₁ : a (n + 2) = + -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) : + 0 ≤ (Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * + Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k := by + refine mul_nonneg ?_ ?_ + . refine Finset.sum_nonneg ?_ + intros i _ + exact LT.lt.le (hxp i) + . refine Finset.sum_nonneg ?_ + intros i _ + simp + exact LT.lt.le (hxp i) + + +lemma imo_2023_p4_4_37 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₁ : a (n + 2) = + -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) : + 0 ≤ Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k := by + refine Finset.sum_nonneg ?_ + intros i _ + exact LT.lt.le (hxp i) + + +lemma imo_2023_p4_4_38 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) : + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : 0 ≤ a n + 2) + -- (g₁ : 0 ≤ a (n + 2)) + -- (g₂ : 0 ≤ (a n + 2) ^ 2) + -- (g₃ : 0 ≤ (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₀ : a n ^ 2 = (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) + -- (gn₁ : a (n + 2) = + -- √((Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k)) : + 0 ≤ Finset.sum (Finset.Ico 1 (n + 2 + 1)) fun k => 1 / x k := by + refine Finset.sum_nonneg ?_ + intros i _ + simp + exact LT.lt.le (hxp i) + + +lemma imo_2023_p4_5 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (h₁ : ∀ (n : ℕ), (1 ≤ n ∧ n ≤ 2023) → ∃ (kz:ℤ), (a n = ↑kz )) + (ha1 : a 1 = 1) : + 3034 ≤ a 2023 := by + have h₀₁: ∀ (n : ℕ), (1 ≤ n ∧ n ≤ 2023) → 0 < a n := by + intros n hn + have ha: a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact h₀ (n) (hn) + rw [ha] + refine Real.sqrt_pos.mpr ?_ + refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact hxp i + . simp + linarith + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + have h₁₁: ∀ (n : ℕ), (1 ≤ n ∧ n ≤ 2023) → ∃ (kn:ℕ), a n = ↑kn := by + intros n hn + have g₁₁: 0 < a n := by + exact h₀₁ n hn + let ⟨p, gp⟩ := h₁ n hn + let q:ℕ := Int.toNat p + have g₁₂: ↑q = p := by + refine Int.toNat_of_nonneg ?_ + rw [gp] at g₁₁ + norm_cast at g₁₁ + exact Int.le_of_lt g₁₁ + use q + rw [gp] + norm_cast + exact id g₁₂.symm + have h₂₁: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2021) → a n + 2 < a (n+2) := by + exact fun n a_1 => imo_2023_p4_4 (fun i => x i) a hxp hx h₀ h₀₁ n a_1 + have h₂: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2021) → a n + 3 ≤ a (n+2) := by + intros n hn + have g₀: a n + 2 < a (n + 2) := by exact h₂₁ n hn + have g₀₁: ∃ (p:ℕ), a n = ↑p := by + apply h₁₁ + constructor + . linarith + . linarith + have g₀₂: ∃ (q:ℕ), a (n + 2) = ↑q := by + apply h₁₁ + constructor + . linarith + . linarith + let ⟨p, _⟩ := g₀₁ + let ⟨q, _⟩ := g₀₂ + have g₁: p + 2 < q := by + suffices g₁₁: ↑p + (2:ℝ) < ↑q + . norm_cast at g₁₁ + . linarith + have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast + linarith + have h₃: ∀ (n:ℕ), (0 ≤ n ∧ n ≤ 1010) → a 1 + 3 * (↑(n) + 1) ≤ a (3 + 2 * n) := by + intros n hn + induction' n with d hd + · simp + exact h₂ (1) (by norm_num) + · rw [mul_add] + simp + have g₀: a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by + refine h₂ (3 + 2 * d) ?_ + constructor + . linarith + . linarith + have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by + refine add_le_add_right ?_ (3) + apply hd + constructor + . linarith + . linarith + refine le_trans (by linarith[g₁]) g₀ + rw [ha1] at h₃ + have h₄: (3034:ℝ) = 1 + 3 * (↑1010 + 1) := by norm_num + rw [h₄] + exact h₃ (1010) (by norm_num) + + + +lemma imo_2023_p4_5_1 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ kz, a n = kz) + -- (ha1 : a 1 = 1) : + ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n := by + intros n hn + have ha: a n = sqrt ((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k) := by + exact h₀ (n) (hn) + rw [ha] + refine Real.sqrt_pos.mpr ?_ + refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact hxp i + . simp + linarith + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + + +lemma imo_2023_p4_5_2 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ kz, a n = kz) + -- (ha1 : a 1 = 1) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2023) + (ha : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : + 0 < a n := by + rw [ha] + refine Real.sqrt_pos.mpr ?_ + refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact hxp i + . simp + linarith + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + + +lemma imo_2023_p4_5_3 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ kz, a n = kz) + -- (ha1 : a 1 = 1) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2023) : + -- (ha : a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : + 0 < (Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * + Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k := by + refine mul_pos ?_ ?_ + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact hxp i + . simp + linarith + . refine Finset.sum_pos ?_ ?_ + . intros i _ + exact one_div_pos.mpr (hxp i) + . simp + linarith + + +lemma imo_2023_p4_5_4 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = kz) + (ha1 : a 1 = 1) + (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) : + 3034 ≤ a 2023 := by + have h₁₁: ∀ (n : ℕ), (1 ≤ n ∧ n ≤ 2023) → ∃ (kn:ℕ), a n = ↑kn := by + intros n hn + have g₁₁: 0 < a n := by + exact h₀₁ n hn + let ⟨p, gp⟩ := h₁ n hn + let q:ℕ := Int.toNat p + have g₁₂: ↑q = p := by + refine Int.toNat_of_nonneg ?_ + rw [gp] at g₁₁ + norm_cast at g₁₁ + exact Int.le_of_lt g₁₁ + use q + rw [gp] + norm_cast + exact id g₁₂.symm + have h₂₁: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2021) → a n + 2 < a (n+2) := by + exact fun n a_1 => imo_2023_p4_4 (fun i => x i) a hxp hx h₀ h₀₁ n a_1 + have h₂: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2021) → a n + 3 ≤ a (n+2) := by + intros n hn + have g₀: a n + 2 < a (n + 2) := by exact h₂₁ n hn + have g₀₁: ∃ (p:ℕ), a n = ↑p := by + apply h₁₁ + constructor + . linarith + . linarith + have g₀₂: ∃ (q:ℕ), a (n + 2) = ↑q := by + apply h₁₁ + constructor + . linarith + . linarith + let ⟨p, _⟩ := g₀₁ + let ⟨q, _⟩ := g₀₂ + have g₁: p + 2 < q := by + suffices g₁₁: ↑p + (2:ℝ) < ↑q + . norm_cast at g₁₁ + . linarith + have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast + linarith + have h₃: ∀ (n:ℕ), (0 ≤ n ∧ n ≤ 1010) → a 1 + 3 * (↑(n) + 1) ≤ a (3 + 2 * n) := by + intros n hn + induction' n with d hd + · simp + exact h₂ (1) (by norm_num) + · rw [mul_add] + simp + have g₀: a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by + refine h₂ (3 + 2 * d) ?_ + constructor + . linarith + . linarith + have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by + refine add_le_add_right ?_ (3) + apply hd + constructor + . linarith + . linarith + refine le_trans (by linarith[g₁]) g₀ + rw [ha1] at h₃ + have h₄: (3034:ℝ) = 1 + 3 * (↑1010 + 1) := by norm_num + rw [h₄] + exact h₃ (1010) (by norm_num) + + +lemma imo_2023_p4_5_5 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = kz) + -- (ha1 : a 1 = 1) + (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) : + ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn := by + intros n hn + have g₁₁: 0 < a n := by + exact h₀₁ n hn + let ⟨p, gp⟩ := h₁ n hn + let q:ℕ := Int.toNat p + have g₁₂: ↑q = p := by + refine Int.toNat_of_nonneg ?_ + rw [gp] at g₁₁ + norm_cast at g₁₁ + exact Int.le_of_lt g₁₁ + use q + rw [gp] + norm_cast + exact id g₁₂.symm + + +lemma imo_2023_p4_5_6 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2023) + (g₁₁ : 0 < a n) : + ∃ (kn:ℕ), a n = ↑kn := by + let ⟨p, gp⟩ := h₁ n hn + let q:ℕ := Int.toNat p + have g₁₂: ↑q = p := by + refine Int.toNat_of_nonneg ?_ + rw [gp] at g₁₁ + norm_cast at g₁₁ + exact Int.le_of_lt g₁₁ + use q + rw [gp] + norm_cast + exact id g₁₂.symm + + +lemma imo_2023_p4_5_7 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n q : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2023) + (g₁₁ : 0 < a n) + (p : ℤ) + (gp : a n = ↑p) + (hq : q = Int.toNat p) : + ∃ kn:ℕ, a n = ↑kn := by + have g₁₂: (↑q:ℤ) = p := by + rw [hq] + refine Int.toNat_of_nonneg ?_ + rw [gp] at g₁₁ + norm_cast at g₁₁ + exact Int.le_of_lt g₁₁ + use q + rw [gp] + exact congrArg Int.cast (id g₁₂.symm) + + +lemma imo_2023_p4_5_8 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n q : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2023) + (g₁₁ : 0 < a n) + (p : ℤ) + (gp : a n = ↑p) + (hq : q = Int.toNat p) : + ↑q = p := by + rw [hq] + refine Int.toNat_of_nonneg ?_ + rw [gp] at g₁₁ + norm_cast at g₁₁ + exact Int.le_of_lt g₁₁ + + +lemma imo_2023_p4_5_9 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n q : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2023) + (g₁₁ : 0 < a n) + (p : ℤ) + (gp : a n = ↑p) + (hq : q = Int.toNat p) : + 0 ≤ p := by + rw [gp] at g₁₁ + norm_cast at g₁₁ + exact Int.le_of_lt g₁₁ + + +lemma imo_2023_p4_5_10 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n q : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2023) + -- (g₁₁ : 0 < a n) + (p : ℤ) + (gp : a n = ↑p) + -- (hq : q = Int.toNat p) + (g₁₂ : ↑q = p) : + ∃ (kn:ℕ), a n = ↑kn := by + use q + rw [gp] + norm_cast + exact id g₁₂.symm + + +lemma imo_2023_p4_5_11 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2023) + -- (g₁₁ : 0 < a n) + (p : ℤ) + (gp : a n = ↑p) + (q : ℕ := Int.toNat p) + (g₁₂ : ↑q = p) : + a n = ↑q := by + rw [gp] + norm_cast + exact id g₁₂.symm + + +lemma imo_2023_p4_5_12 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + (ha1 : a 1 = 1) + (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) : + 3034 ≤ a 2023 := by + have h₂₁: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2021) → a n + 2 < a (n+2) := by + exact fun n a_1 => imo_2023_p4_4 (fun i => x i) a hxp hx h₀ h₀₁ n a_1 + have h₂: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2021) → a n + 3 ≤ a (n+2) := by + intros n hn + have g₀: a n + 2 < a (n + 2) := by exact h₂₁ n hn + have g₀₁: ∃ (p:ℕ), a n = ↑p := by + apply h₁₁ + constructor + . linarith + . linarith + have g₀₂: ∃ (q:ℕ), a (n + 2) = ↑q := by + apply h₁₁ + constructor + . linarith + . linarith + let ⟨p, _⟩ := g₀₁ + let ⟨q, _⟩ := g₀₂ + have g₁: p + 2 < q := by + suffices g₁₁: ↑p + (2:ℝ) < ↑q + . norm_cast at g₁₁ + . linarith + have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast + linarith + have h₃: ∀ (n:ℕ), (0 ≤ n ∧ n ≤ 1010) → a 1 + 3 * (↑(n) + 1) ≤ a (3 + 2 * n) := by + intros n hn + induction' n with d hd + · simp + exact h₂ (1) (by norm_num) + · rw [mul_add] + simp + have g₀: a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by + refine h₂ (3 + 2 * d) ?_ + constructor + . linarith + . linarith + have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by + refine add_le_add_right ?_ (3) + apply hd + constructor + . linarith + . linarith + refine le_trans (by linarith[g₁]) g₀ + rw [ha1] at h₃ + have h₄: (3034:ℝ) = 1 + 3 * (↑1010 + 1) := by norm_num + rw [h₄] + exact h₃ (1010) (by norm_num) + + + +lemma imo_2023_p4_5_13 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) : + 3034 ≤ a 2023 := by + have h₂: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2021) → a n + 3 ≤ a (n+2) := by + intros n hn + have g₀: a n + 2 < a (n + 2) := by exact h₂₁ n hn + have g₀₁: ∃ (p:ℕ), a n = ↑p := by + apply h₁₁ + constructor + . linarith + . linarith + have g₀₂: ∃ (q:ℕ), a (n + 2) = ↑q := by + apply h₁₁ + constructor + . linarith + . linarith + let ⟨p, _⟩ := g₀₁ + let ⟨q, _⟩ := g₀₂ + have g₁: p + 2 < q := by + suffices g₁₁: ↑p + (2:ℝ) < ↑q + . norm_cast at g₁₁ + . linarith + have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast + linarith + have h₃: ∀ (n:ℕ), (0 ≤ n ∧ n ≤ 1010) → a 1 + 3 * (↑(n) + 1) ≤ a (3 + 2 * n) := by + intros n hn + induction' n with d hd + · simp + exact h₂ (1) (by norm_num) + · rw [mul_add] + simp + have g₀: a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by + refine h₂ (3 + 2 * d) ?_ + constructor + . linarith + . linarith + have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by + refine add_le_add_right ?_ (3) + apply hd + constructor + . linarith + . linarith + refine le_trans (by linarith[g₁]) g₀ + rw [ha1] at h₃ + have h₄: (3034:ℝ) = 1 + 3 * (↑1010 + 1) := by norm_num + rw [h₄] + exact h₃ (1010) (by norm_num) + + +lemma imo_2023_p4_5_14 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) : + ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2) := by + intros n hn + have g₀: a n + 2 < a (n + 2) := by exact h₂₁ n hn + have g₀₁: ∃ (p:ℕ), a n = ↑p := by + apply h₁₁ + constructor + . linarith + . linarith + have g₀₂: ∃ (q:ℕ), a (n + 2) = ↑q := by + apply h₁₁ + constructor + . linarith + . linarith + let ⟨p, _⟩ := g₀₁ + let ⟨q, _⟩ := g₀₂ + have g₁: p + 2 < q := by + suffices g₁₁: ↑p + (2:ℝ) < ↑q + . norm_cast at g₁₁ + . linarith + have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast + linarith + + +lemma imo_2023_p4_5_15 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + (g₀ : a n + 2 < a (n + 2)) : + a n + 3 ≤ a (n + 2) := by + have g₀₁: ∃ (p:ℕ), a n = ↑p := by + apply h₁₁ + constructor + . linarith + . linarith + have g₀₂: ∃ (q:ℕ), a (n + 2) = ↑q := by + apply h₁₁ + constructor + . linarith + . linarith + let ⟨p, _⟩ := g₀₁ + let ⟨q, _⟩ := g₀₂ + have g₁: p + 2 < q := by + suffices g₁₁: ↑p + (2:ℝ) < ↑q + . norm_cast at g₁₁ + . linarith + have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast + linarith + + +lemma imo_2023_p4_5_16 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) : + -- (g₀ : a n + 2 < a (n + 2)) : + ∃ (p:ℕ), a n = ↑p := by + apply h₁₁ + constructor + . linarith + . linarith + + +lemma imo_2023_p4_5_17 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) + (n : ℕ) + (hn : 1 ≤ n ∧ n ≤ 2021) + (g₀ : a n + 2 < a (n + 2)) + (g₀₁ : ∃ (p:ℕ), a n = ↑p) : + a n + 3 ≤ a (n + 2) := by + have g₀₂: ∃ (q:ℕ), a (n + 2) = ↑q := by + apply h₁₁ + constructor + . linarith + . linarith + let ⟨p, _⟩ := g₀₁ + let ⟨q, _⟩ := g₀₂ + have g₁: p + 2 < q := by + suffices g₁₁: ↑p + (2:ℝ) < ↑q + . norm_cast at g₁₁ + . linarith + have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast + linarith + + +lemma imo_2023_p4_5_18 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2021) + (g₀ : a n + 2 < a (n + 2)) + (g₀₁ : ∃ (p:ℕ), a n = ↑p) + (g₀₂ : ∃ (q:ℕ), a (n + 2) = ↑q) : + a n + 3 ≤ a (n + 2) := by + let ⟨p, _⟩ := g₀₁ + let ⟨q, _⟩ := g₀₂ + have g₁: p + 2 < q := by + suffices g₁₁: ↑p + (2:ℝ) < ↑q + . norm_cast at g₁₁ + . linarith + have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast + linarith + + +lemma imo_2023_p4_5_19 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2021) + (g₀ : a n + 2 < a (n + 2)) + -- (g₀₁ : ∃ p, a n = ↑p) + -- (g₀₂ : ∃ q, a (n + 2) = ↑q) + (p : ℕ) + (h₈ : a n = ↑p) + (q : ℕ) + (h₉ : a (n + 2) = ↑q) : + a n + 3 ≤ a (n + 2) := by + have g₁: p + 2 < q := by + suffices g₁₁: ↑p + (2:ℝ) < ↑q + . norm_cast at g₁₁ + . linarith + have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast + linarith + + +lemma imo_2023_p4_5_20 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2021) + (g₀ : a n + 2 < a (n + 2)) + -- (g₀₁ : ∃ p, a n = ↑p) + -- (g₀₂ : ∃ q, a (n + 2) = ↑q) + (p : ℕ) + (h₈ : a n = ↑p) + (q : ℕ) + (h₉ : a (n + 2) = ↑q) : + p + 2 < q := by + suffices g₁₁: ↑p + (2:ℝ) < ↑q + . norm_cast at g₁₁ + . linarith + + +lemma imo_2023_p4_5_21 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) + (n : ℕ) + -- (hn : 1 ≤ n ∧ n ≤ 2021) + -- (g₀ : a n + 2 < a (n + 2)) + -- (g₀₁ : ∃ p, a n = ↑p) + -- (g₀₂ : ∃ q, a (n + 2) = ↑q) + (p : ℕ) + (h₈ : a n = ↑p) + (q : ℕ) + (h₉ : a (n + 2) = ↑q) + (g₁ : p + 2 < q) : + a n + 3 ≤ a (n + 2) := by + have g₂: ↑p + (3:ℝ) ≤ ↑q := by norm_cast + linarith + + +lemma imo_2023_p4_5_22 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) + (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) : + 3034 ≤ a 2023 := by + have h₃: ∀ (n:ℕ), (0 ≤ n ∧ n ≤ 1010) → a 1 + 3 * (↑(n) + 1) ≤ a (3 + 2 * n) := by + intros n hn + induction' n with d hd + · simp + exact h₂ (1) (by norm_num) + · rw [mul_add] + simp + have g₀: a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by + refine h₂ (3 + 2 * d) ?_ + constructor + . linarith + . linarith + have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by + refine add_le_add_right ?_ (3) + apply hd + constructor + . linarith + . linarith + refine le_trans (by linarith[g₁]) g₀ + rw [ha1] at h₃ + have h₄: (3034:ℝ) = 1 + 3 * (↑1010 + 1) := by norm_num + rw [h₄] + exact h₃ (1010) (by norm_num) + + +lemma imo_2023_p4_5_23 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) + (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) : + ∀ (n : ℕ), 0 ≤ n ∧ n ≤ 1010 → a 1 + 3 * (↑n + 1) ≤ a (3 + 2 * n) := by + intros n hn + induction' n with d hd + · simp + exact h₂ (1) (by norm_num) + · rw [mul_add] + simp + have g₀: a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by + refine h₂ (3 + 2 * d) ?_ + constructor + . linarith + . linarith + have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by + refine add_le_add_right ?_ (3) + apply hd + constructor + . linarith + . linarith + refine le_trans (by linarith[g₁]) g₀ + + +lemma imo_2023_p4_5_24 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) + (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) + (n : ℕ) + (hn : 0 ≤ n ∧ n ≤ 1010) : + a 1 + 3 * (↑n + 1) ≤ a (3 + 2 * n) := by + induction' n with d hd + · simp + exact h₂ (1) (by norm_num) + · rw [mul_add] + simp + have g₀: a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by + refine h₂ (3 + 2 * d) ?_ + constructor + . linarith + . linarith + have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by + refine add_le_add_right ?_ (3) + apply hd + constructor + . linarith + . linarith + refine le_trans (by linarith[g₁]) g₀ + + +lemma imo_2023_p4_5_25 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) + (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) : + -- (hn : 0 ≤ Nat.zero ∧ Nat.zero ≤ 1010) : + a 1 + 3 * (↑Nat.zero + 1) ≤ a (3 + 2 * Nat.zero) := by + simp + exact h₂ (1) (by norm_num) + + +lemma imo_2023_p4_5_26 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) + (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) + (d : ℕ) + (hd : 0 ≤ d ∧ d ≤ 1010 → a 1 + 3 * (↑d + 1) ≤ a (3 + 2 * d)) + (hn : 0 ≤ Nat.succ d ∧ Nat.succ d ≤ 1010) : + a 1 + 3 * (↑(Nat.succ d) + 1) ≤ a (3 + 2 * Nat.succ d) := by + rw [mul_add, Nat.succ_eq_add_one] + simp + have g₀: a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by + refine h₂ (3 + 2 * d) ?_ + constructor + . linarith + . linarith + have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by + refine add_le_add_right ?_ (3) + apply hd + constructor + . linarith + . linarith + refine le_trans (by linarith[g₁]) g₀ + + +lemma imo_2023_p4_5_27 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) + (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) + (d : ℕ) + (hd : 0 ≤ d ∧ d ≤ 1010 → a 1 + 3 * (↑d + 1) ≤ a (3 + 2 * d)) + (hn : 0 ≤ Nat.succ d ∧ Nat.succ d ≤ 1010) : + a 1 + (3 * (↑d + 1) + 3) ≤ a (3 + 2 * (d + 1)) := by + have g₀: a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by + refine h₂ (3 + 2 * d) ?_ + constructor + . linarith + . linarith + have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by + refine add_le_add_right ?_ (3) + apply hd + constructor + . linarith + . linarith + refine le_trans (by linarith[g₁]) g₀ + + +lemma imo_2023_p4_5_28 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) + (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) + (d : ℕ) + -- (hd : 0 ≤ d ∧ d ≤ 1010 → a 1 + 3 * (↑d + 1) ≤ a (3 + 2 * d)) + (hn : 0 ≤ Nat.succ d ∧ Nat.succ d ≤ 1010) : + a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1)) := by + refine h₂ (3 + 2 * d) ?_ + constructor + . linarith + . linarith + + +lemma imo_2023_p4_5_29 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) + -- (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) + (d : ℕ) + (hd : 0 ≤ d ∧ d ≤ 1010 → a 1 + 3 * (↑d + 1) ≤ a (3 + 2 * d)) + (hn : 0 ≤ Nat.succ d ∧ Nat.succ d ≤ 1010) + (g₀ : a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1))) : + a 1 + (3 * (↑d + 1) + 3) ≤ a (3 + 2 * (d + 1)) := by + have g₁: a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by + refine add_le_add_right ?_ (3) + apply hd + constructor + . linarith + . linarith + refine le_trans (by linarith[g₁]) g₀ + + +lemma imo_2023_p4_5_30 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) + -- (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) + (d : ℕ) + (hd : 0 ≤ d ∧ d ≤ 1010 → a 1 + 3 * (↑d + 1) ≤ a (3 + 2 * d)) + (hn : 0 ≤ Nat.succ d ∧ Nat.succ d ≤ 1010) : + -- (g₀ : a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1))) : + a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3 := by + refine add_le_add_right ?_ (3) + apply hd + constructor + . linarith + . linarith + + +lemma imo_2023_p4_5_31 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) + -- (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) + (d : ℕ) + (hd : 0 ≤ d ∧ d ≤ 1010 → a 1 + 3 * (↑d + 1) ≤ a (3 + 2 * d)) + (hn : 0 ≤ Nat.succ d ∧ Nat.succ d ≤ 1010) : + -- (g₀ : a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1))) : + a 1 + 3 * (↑d + 1) ≤ a (3 + 2 * d) := by + apply hd + constructor + . linarith + . linarith + + +lemma imo_2023_p4_5_32 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) + -- (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) + (d : ℕ) + -- (hd : 0 ≤ d ∧ d ≤ 1010 → a 1 + 3 * (↑d + 1) ≤ a (3 + 2 * d)) + -- (hn : 0 ≤ Nat.succ d ∧ Nat.succ d ≤ 1010) + (g₀ : a (3 + 2 * d) + 3 ≤ a (3 + 2 * (d + 1))) + (g₁ : a 1 + 3 * (↑d + 1) + 3 ≤ a (3 + 2 * d) + 3) : + a 1 + (3 * (↑d + 1) + 3) ≤ a (3 + 2 * (d + 1)) := by + exact le_trans (by linarith[g₁]) g₀ + + +lemma imo_2023_p4_5_33 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) + -- (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) + (h₃ : ∀ (n : ℕ), 0 ≤ n ∧ n ≤ 1010 → a 1 + 3 * (↑n + 1) ≤ a (3 + 2 * n)) : + 3034 ≤ a 2023 := by + rw [ha1] at h₃ + have h₄: (3034:ℝ) = 1 + 3 * (↑1010 + 1) := by norm_num + rw [h₄] + exact h₃ (1010) (by norm_num) + + +lemma imo_2023_p4_5_34 + -- (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + -- (ha1 : a 1 = 1) + -- (h₀₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → 0 < a n) + -- (h₁₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kn:ℕ), a n = ↑kn) + -- (h₂₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 2 < a (n + 2)) + -- (h₂ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2021 → a n + 3 ≤ a (n + 2)) + (h₃ : ∀ (n : ℕ), 0 ≤ n ∧ n ≤ 1010 → 1 + 3 * (↑n + 1) ≤ a (3 + 2 * n)) + (h₄ : (3034:ℝ) = 1 + 3 * (↑1010 + 1)) : + 3034 ≤ a 2023 := by + rw [h₄] + exact h₃ (1010) (by norm_num) + + + +lemma imo_2023_p4_6 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * + Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) : + a 1 = 1 := by + have g₀: sqrt ((Finset.sum (Finset.Ico 1 (1 + 1)) fun k => x k) + * Finset.sum (Finset.Ico 1 (1 + 1)) fun k => 1 / x k) = 1 := by + norm_num + refine div_self ?_ + exact ne_of_gt (hxp 1) + rw [← g₀] + exact h₀ (1) (by norm_num) + + +lemma imo_2023_p4_6_1 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) : + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * + -- Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) : + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) : + √((Finset.sum (Finset.Ico 1 (1 + 1)) fun k => x k) * + Finset.sum (Finset.Ico 1 (1 + 1)) fun k => 1 / x k) = 1 := by + norm_num + refine div_self ?_ + exact ne_of_gt (hxp 1) + + +lemma imo_2023_p4_6_2 + (x : ℕ → ℝ) + -- (a : ℕ → ℝ) + (hxp : ∀ (i : ℕ), 0 < x i) : + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + -- (h₀ : ∀ (n : ℕ), + -- 1 ≤ n ∧ n ≤ 2023 → + -- a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) : + x 1 * (x 1)⁻¹ = 1 := by + refine div_self ?_ + exact ne_of_gt (hxp 1) + + +lemma imo_2023_p4_6_3 + (x : ℕ → ℝ) + (a : ℕ → ℝ) + -- (hxp : ∀ (i : ℕ), 0 < x i) + -- (hx : ∀ (i j : ℕ), i ≠ j → x i ≠ x j) + (h₀ : ∀ (n : ℕ), + 1 ≤ n ∧ n ≤ 2023 → + a n = √((Finset.sum (Finset.Ico 1 (n + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (n + 1)) fun k => 1 / x k)) + -- (h₁ : ∀ (n : ℕ), 1 ≤ n ∧ n ≤ 2023 → ∃ (kz:ℤ), a n = ↑kz) + (g₀ : √((Finset.sum (Finset.Ico 1 (1 + 1)) fun k => x k) * Finset.sum (Finset.Ico 1 (1 + 1)) fun k => 1 / x k) = 1) : + a 1 = 1 := by + rw [← g₀] + exact h₀ (1) (by norm_num)