import Mathlib import Mathlib.Analysis.SpecialFunctions.Pow.Real set_option linter.unusedVariables.analyzeTactics true open Real Set lemma mylemma_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 mylemma_amgm (b1 b2 b3 b4 :ℝ) (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 mylemma_2 (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 mylemma_amgm 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 -- rw [g₂, ← sqrt_eq_rpow (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => x x_1)] -- rw [g₃, ← sqrt_eq_rpow (Finset.sum (Finset.Ico 1 (1 + n)) fun x_1 => (x x_1)⁻¹)] 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) exact Eq.trans_le h₃₃ h₃₂ lemma mylemma_3 (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 mylemma_2 (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) theorem imo_2023_p4 (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 = Real.sqrt ( (Finset.sum (Finset.Ico 1 (n + 1)) fun (k : ℕ) => (x k)) * (Finset.sum (Finset.Ico 1 (n + 1)) fun (k : ℕ) => 1 / (x k)) ) ) (h₁: ∀ (n:ℕ), (1 ≤ n ∧ n ≤ 2023) → ∃ (kz:ℤ), (a n = ↑kz )) : (3034 ≤ a 2023) := by have ha1: 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) 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 => mylemma_3 (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)