import Mathlib set_option linter.unusedVariables.analyzeTactics true theorem imo_2022_p2_simple (g: ℝ → ℝ) (h₀: ∀ x, 0 < x → ∃ y:ℝ , (0 < y ∧ g (x) + g (y) ≤ 2 * x * y ∧ (∀ z:ℝ, (0 < z ∧ z ≠ y) → ¬ g (x) + g (z) ≤ 2 * x * z) )) : (∀ x:ℝ , 0 < x → g x = x^2) := by have h₁: ∀ x y:ℝ , 0 < x ∧ 0 < y → g x + g y ≤ 2 * x * y → x = y := by intros x y hp h₁ by_contra! hc have g₁: 2 * x * x < g x + g x := by let ⟨p,h₁₁⟩ := h₀ x hp.1 cases' h₁₁ with h₁₁ h₁₂ cases' h₁₂ with h₁₂ h₁₃ by_cases hxp: x ≠ p . have h₁₄: ¬ g x + g x ≤ 2 * x * x := by refine h₁₃ x ?_ constructor . exact hp.1 . exact hxp exact not_le.mp h₁₄ . push_neg at hxp exfalso have hpy: y ≠ p := by exact Ne.trans_eq (id (Ne.symm hc)) hxp have hcy: ¬g x + g y ≤ 2 * x * y := by refine h₁₃ y ?_ constructor . exact hp.2 . exact hpy linarith have g₂: 2 * y * y < g y + g y := by let ⟨p,h₁₁⟩ := h₀ y hp.2 cases' h₁₁ with h₁₁ h₁₂ cases' h₁₂ with h₁₂ h₁₃ by_cases hyp: y ≠ p . have h₁₄: ¬ g y + g y ≤ 2 * y * y := by refine h₁₃ y ?_ constructor . exact hp.2 . exact hyp exact not_le.mp h₁₄ . push_neg at hyp exfalso have hpx: x ≠ p := by exact Ne.trans_eq hc hyp have hcy: ¬g x + g y ≤ 2 * x * y := by rw [add_comm, mul_right_comm] refine h₁₃ x ?_ constructor . exact hp.1 . exact hpx linarith ring_nf at g₁ g₂ simp at g₁ g₂ have g₃: x ^ 2 + y ^ 2 < g x + g y := by exact add_lt_add g₁ g₂ have g₄: x ^ 2 + y ^ 2 < 2 * x * y := by exact LT.lt.trans_le g₃ h₁ have g₅: (x - y) ^ 2 < 0 := by rw [sub_sq, sub_add_eq_add_sub] exact sub_neg.mpr g₄ have g₆: (x - y) ≠ 0 := by exact sub_ne_zero.mpr hc have g₇: 0 < (x - y) ^ 2 := by exact (sq_pos_iff).mpr g₆ have g₈: (0:ℝ) ≠ 0 := by refine ne_of_lt ?_ exact lt_trans g₇ g₅ refine false_of_ne g₈ have h₂: ∀ x:ℝ , 0 < x → g x ≤ x ^ 2 := by intros x hxp let ⟨y,h₁₁⟩ := h₀ x hxp cases' h₁₁ with h₁₁ h₁₂ cases' h₁₂ with h₁₂ h₁₃ have hxy: x = y := by apply h₁ x y . exact { left := hxp, right := h₁₁ } . exact h₁₂ rw [← hxy] at h₁₂ linarith have h₃: ∀ x:ℝ , 0 < x → ¬ g x < x ^ 2 := by simp by_contra! hc let ⟨x,hxp⟩ := hc cases' hxp with hxp h₃ let d₁:ℝ := x ^ 2 - g x have hd₁ : g x = x ^ 2 - d₁ := by exact (sub_sub_self (x ^ 2) (g x)).symm let z:ℝ := x + Real.sqrt d₁ have hz: z = x + Real.sqrt d₁ := by exact rfl have hzp: 0 < z := by refine add_pos hxp ?_ refine Real.sqrt_pos_of_pos ?_ exact sub_pos.mpr h₃ have hxz: z ≠ x := by rw [hz] simp push_neg refine Real.sqrt_ne_zero'.mpr ?_ exact sub_pos.mpr h₃ have h₅: g x + g z ≤ 2 * x * z := by rw [hd₁] have h₅₁: - d₁ + Real.sqrt (x ^ 2 - (x ^ 2 - d₁)) ^ 2 ≤ 0 := by simp rw [Real.sq_sqrt] exact sub_nonneg_of_le (h₂ x hxp) have h₅₂: x ^ 2 - d₁ + z ^ 2 ≤ 2 * x * z := by rw [hz, mul_add, add_sq] ring_nf repeat rw [add_assoc] refine add_le_add_left ?_ (x * Real.sqrt (x ^ 2 - g x) * 2) rw [hd₁] linarith exact add_le_of_add_le_left h₅₂ (h₂ z hzp) let ⟨y,hyp⟩ := h₀ x hxp cases' hyp with hyp h₆ cases' h₆ with h₆ h₇ have hxy: x = y := by apply h₁ . exact { left := hxp, right := hyp } . exact h₆ have h₈: ¬g x + g z ≤ 2 * x * z := by refine h₇ z ?_ constructor . exact hzp . exact Ne.trans_eq hxz hxy linarith[h₅,h₈] intros x hxp have g₂: g x ≤ x ^ 2 := by exact h₂ x hxp have g₃: ¬ g x < x ^ 2 := by exact h₃ x hxp linarith theorem imo_2022_p2 (f: ℝ → ℝ) (hfp: ∀ x:ℝ, 0 < x → 0 < f x) (h₀: ∀ x:ℝ , 0 < x → ∃! y:ℝ , 0 < y ∧ (x * f (y) + y * f (x) ≤ 2) ): ∀ x:ℝ , 0 < x → f (x) = 1 / x := by have h₁: ∀ x y:ℝ , (0 < x ∧ 0 < y) → (x * f (y) + y * f (x) ≤ 2) → x = y := by intros x y hp h₁ by_contra! hc have h₁₀: x * f x + x * f x > 2 := by let ⟨z,h₁₁⟩ := h₀ x hp.1 cases' h₁₁ with h₁₁ h₁₂ have h₁₄: y = z := by apply h₁₂ y constructor . exact hp.2 . exact h₁ have hxz: ¬ x = z := by exact Ne.trans_eq hc h₁₄ have h₁₆: ¬ (fun y => 0 < y ∧ x * f y + y * f x ≤ 2) x := by exact mt (h₁₂ x) hxz have h₁₇: ¬ (0 < x ∧ x * f x + x * f x ≤ 2) := by exact h₁₆ push_neg at h₁₇ exact h₁₇ hp.1 have h₁₁: y * f y + y * f y > 2 := by let ⟨z,h₁₁⟩ := h₀ y hp.2 cases' h₁₁ with h₁₁ h₁₂ have h₁₄: x = z := by apply h₁₂ x constructor . exact hp.1 . rw [add_comm] exact h₁ have hxz: ¬ y = z := by exact Ne.trans_eq (id (Ne.symm hc)) h₁₄ have h₁₆: ¬ (fun y_2 => 0 < y_2 ∧ y * f y_2 + y_2 * f y ≤ 2) y := by exact mt (h₁₂ y) hxz have h₁₇: ¬ (0 < y ∧ y * f y + y * f y ≤ 2) := by exact h₁₆ push_neg at h₁₇ exact h₁₇ hp.2 ring_nf at h₁₀ h₁₁ simp at h₁₀ h₁₁ have h₁₅: 1 / x < f x := by exact (div_lt_iff₀' hp.1).mpr (h₁₀) have h₁₆: 1 / y < f y := by exact (div_lt_iff₀' hp.2).mpr (h₁₁) have h₁₂: x / y + y / x < 2 := by refine lt_of_le_of_lt' h₁ ?_ refine add_lt_add ?_ ?_ . rw [← mul_one_div] exact (mul_lt_mul_left hp.1).mpr h₁₆ . rw [← mul_one_div] exact (mul_lt_mul_left hp.2).mpr h₁₅ have h₁₃: 2 < x / y + y / x := by refine lt_of_mul_lt_mul_right ?_ (le_of_lt hp.1) refine lt_of_mul_lt_mul_right ?_ (le_of_lt hp.2) repeat rw [add_mul, mul_assoc] rw [mul_comm x y, ← mul_assoc (x/y)] rw [div_mul_comm x y y, div_mul_comm y x x, div_self, div_self] . ring_nf refine lt_of_sub_pos ?_ rw [mul_comm _ 2, ← mul_assoc] rw [← sub_sq'] refine sq_pos_of_ne_zero ?_ exact sub_ne_zero.mpr hc.symm . exact ne_of_gt hp.1 . exact ne_of_gt hp.2 linarith have h₂: ∀ x:ℝ , 0 < x → x * f x ≤ 1 := by intros x hxp let ⟨y,h₂₁⟩ := h₀ x hxp cases' h₂₁ with h₂₁ h₂₂ have hxy: x = y := by apply h₁ x y . constructor . exact hxp . exact h₂₁.1 . exact h₂₁.2 rw [← hxy] at h₂₁ linarith have h₃: ∀ x:ℝ , 0 < x → ¬ x * f x < 1 := by by_contra! hc let ⟨x,hxp⟩ := hc cases' hxp with hxp h₃ let d₁:ℝ := 1 - x * f x have hd₁ : x * f x = 1 - d₁ := by exact (sub_sub_self 1 (x * f x)).symm let z:ℝ := x + d₁ / f x have hz: z = x + d₁ / f x := by exact rfl have hzp: 0 < z := by refine add_pos hxp ?_ refine div_pos ?_ ?_ . exact sub_pos.mpr h₃ . exact hfp x hxp have hxz: ¬ x = z := by by_contra! hcz₀ rw [← hcz₀] at hz have hcz₁: 0 < d₁ / f x := by refine div_pos ?_ (hfp x hxp) exact sub_pos.mpr h₃ linarith have h₄: ¬ (x * f z + z * f x ≤ 2) := by have h₄₁: x * f z + z * f x ≤ 2 → x = z := by exact h₁ x z { left := hxp, right := hzp } exact mt h₄₁ hxz have h₅: x * f z < 1 := by suffices h₅₁: z * f z ≤ 1 by refine lt_of_lt_of_le ?_ h₅₁ refine (mul_lt_mul_right (hfp z hzp)).mpr ?_ rw [hz] refine lt_add_of_pos_right x ?_ refine div_pos ?_ (hfp x hxp) exact sub_pos.mpr h₃ exact h₂ z hzp have h₆: x * f z + z * f x < 2 := by suffices h₇: z * f x ≤ 1 by linarith rw [hz, add_mul, hd₁] rw [div_mul_comm d₁ (f x) (f x)] rw [div_self] . rw [one_mul, sub_add_cancel] . exact Ne.symm (ne_of_lt (hfp x hxp)) linarith intros x hxp have h₄: x * f x ≤ 1 := by exact h₂ x hxp have h₅: ¬ x * f x < 1 := by exact h₃ x hxp refine eq_div_of_mul_eq ?_ ?_ . exact ne_of_gt hxp . push_neg at h₅ linarith