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import Mathlib |
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set_option linter.unusedVariables.analyzeTactics true |
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open Real |
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theorem imo_1962_p2_1 |
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(x : ℝ) |
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(h₂ : 1 / 2 < Real.sqrt (x - 3) - Real.sqrt (x + 1)) : |
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-1 ≤ x := by |
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refine neg_le_iff_add_nonneg.mpr ?_ |
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contrapose! h₂ |
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have h₃: x - 3 < 0 := by linarith [h₂] |
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have h₄: Real.sqrt (x + 1) = 0 := by |
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refine Real.sqrt_eq_zero'.mpr ?_ |
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exact le_of_lt h₂ |
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have h₅: Real.sqrt (x -3) = 0 := by |
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refine Real.sqrt_eq_zero'.mpr ?_ |
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exact le_of_lt h₃ |
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rw [h₄, h₅, sub_zero] |
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refine div_nonneg ?_ ?_ |
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all_goals try linarith |
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theorem imo_1962_p2_2 |
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(x : ℝ) |
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(h₀ : 0 ≤ 3 - x) |
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(h₁ : 0 ≤ x + 1) |
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(h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1)) : |
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(2 * √(3 - x) * √(x + 1)) ^ 2 < (4 - 1 / 4) ^ 2 := by |
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refine' pow_lt_pow_left₀ _ _ (by norm_num) |
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. refine lt_tsub_iff_left.mpr ?_ |
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refine lt_tsub_iff_right.mp ?_ |
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suffices g₀: 4 - 2 * sqrt (3 - x) * sqrt (x + 1) = (sqrt (3 - x) - sqrt (x + 1)) ^ 2 |
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. rw [g₀] |
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have g₁: (1:ℝ) / 4 = (1/2)^2 := by norm_num |
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rw [g₁] |
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exact pow_lt_pow_left₀ h₂ (by norm_num) (by norm_num) |
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rw [sub_sq] |
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rw [sq_sqrt h₀, sq_sqrt h₁] |
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ring_nf |
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. refine' mul_nonneg _ _ |
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. refine mul_nonneg (by linarith) ?_ |
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exact sqrt_nonneg (3 - x) |
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. exact sqrt_nonneg (x + 1) |
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theorem imo_1962_p2_3 |
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(x : ℝ) |
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(h₀ : 0 ≤ 3 - x) |
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(h₁ : 0 ≤ x + 1) |
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(h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1)) : |
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2 * √(3 - x) * √(x + 1) < 4 - 1 / 4 := by |
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refine lt_tsub_iff_left.mpr ?refine'_1.a |
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refine lt_tsub_iff_right.mp ?refine'_1.a.a |
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suffices g₀: 4 - 2 * sqrt (3 - x) * sqrt (x + 1) = (sqrt (3 - x) - sqrt (x + 1)) ^ 2 |
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. rw [g₀] |
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have g₁: (1:ℝ) / 4 = (1/2)^2 := by norm_num |
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rw [g₁] |
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exact pow_lt_pow_left₀ h₂ (by norm_num) (by norm_num) |
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rw [sub_sq] |
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rw [sq_sqrt h₀, sq_sqrt h₁] |
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ring_nf |
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theorem imo_1962_p2_4 |
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(x : ℝ) : |
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0 ≤ 2 * √(3 - x) * √(x + 1) := by |
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refine' mul_nonneg ?_ ?_ |
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. refine mul_nonneg (by linarith) ?_ |
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exact sqrt_nonneg (3 - x) |
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. exact sqrt_nonneg (x + 1) |
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theorem imo_1962_p2_5 |
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(x : ℝ) |
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(h₀ : 0 ≤ 3 - x) |
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(h₁ : 0 ≤ x + 1) : |
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-- (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1)) : |
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4 - 2 * √(3 - x) * √(x + 1) = (√(3 - x) - √(x + 1)) ^ 2 := by |
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rw [sub_sq] |
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rw [sq_sqrt h₀, sq_sqrt h₁] |
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ring_nf |
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theorem imo_1962_p2_6 |
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(x : ℝ) |
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-- (h₀ : 0 ≤ 3 - x) |
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-- (h₁ : 0 ≤ x + 1) |
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(h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1)) |
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(h₃: 4 - 2 * √(3 - x) * √(x + 1) = (√(3 - x) - √(x + 1)) ^ 2) : |
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1 / 4 < 4 - 2 * √(3 - x) * √(x + 1) := by |
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rw [h₃] |
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have g₁: (1:ℝ) / 4 = (1/2) ^ 2 := by norm_num |
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rw [g₁] |
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exact pow_lt_pow_left₀ h₂ (by norm_num) (by norm_num) |
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theorem imo_1962_p2_7 |
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(x : ℝ) |
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(h₀ : 0 ≤ 3 - x) |
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(h₁ : 0 ≤ x + 1) |
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-- (h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1)) |
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(h₃: (2 *sqrt (3 - x) * sqrt (x + 1)) ^ 2 < (4 - 1 / 4) ^ 2) : |
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4 * (x + 1) * (3 - x) < 225 / 16 := by |
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norm_num at h₃ |
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suffices g₀: 4 * (x + 1) * (3 - x) = (2 * sqrt (3 - x) * sqrt (x + 1)) ^ 2 |
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. exact Eq.trans_lt g₀ h₃ |
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. rw [mul_pow, mul_pow, sq_sqrt h₀, sq_sqrt h₁] |
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norm_num |
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exact mul_right_comm 4 (x + 1) (3 - x) |
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theorem imo_1962_p2_8 |
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(x : ℝ) |
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(h₀ : 0 ≤ 3 - x) |
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(h₁ : 0 ≤ x + 1) |
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(h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1)) : |
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x < 1 := by |
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suffices g₀: x + 1 < 3 - x |
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. linarith |
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. rw [← sq_sqrt h₀, ← sq_sqrt h₁] |
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refine' pow_lt_pow_left₀ ?_ ?_ (by norm_num) |
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. linarith |
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. exact sqrt_nonneg (x + 1) |
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theorem imo_1962_p2_9 |
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(x : ℝ) |
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(h₄: 4 * (x + 1) * (3 - x) < 225 / 16) : |
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x < 1 - sqrt 31 / 8 ∨ 1 + sqrt 31 / 8 < x := by |
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ring_nf at h₄ |
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have g₀: 0 < x * x + -2 * x + 33 / 64 := by linarith |
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let a:ℝ := sqrt 31 / 8 |
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have g₁: x * x + -2 * x + 33 / 64 = (x - (1 + a)) * (x - (1 - a)) := by |
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simp |
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ring_nf |
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norm_num |
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linarith |
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rw [g₁] at g₀ |
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by_cases g₂: (x - (1 - a)) < 0 |
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. left |
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exact sub_neg.mp g₂ |
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. push_neg at g₂ |
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right |
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have g₃: 0 < (x - (1 + a)) := by exact pos_of_mul_pos_left g₀ g₂ |
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exact sub_pos.mp g₃ |
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theorem imo_1962_p2_10 |
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(x : ℝ) |
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(h₄: x < 1) |
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(h₅: x < 1 - sqrt 31 / 8 ∨ 1 + sqrt 31 / 8 < x) : |
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x < 1 - Real.sqrt 31 / 8 := by |
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cases h₅ with |
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| inl h₅ => exact h₅ |
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| inr h₅ => linarith |
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theorem imo_1962_p2_11 |
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(x a : ℝ) |
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(ha: a = √31 / 8) |
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(h₃: 0 < (x - (1 + a)) * (x - (1 - a))) : |
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x < 1 - √31 / 8 ∨ 1 + √31 / 8 < x := by |
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by_cases g₂: (x - (1 - a)) < 0 |
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. left |
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rw [ha] at g₂ |
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exact sub_neg.mp g₂ |
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. push_neg at g₂ |
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right |
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have g₃: 0 < (x - (1 + a)) := by exact pos_of_mul_pos_left h₃ g₂ |
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rw [ha] at g₃ |
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exact sub_pos.mp g₃ |
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theorem imo_1962_p2_12 |
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(x a : ℝ) |
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(ha: a = 0.5) |
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(h₃: 0 < (x - (1 + a)) * (x - (1 - a))) : |
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x < 1 - 0.5 ∨ 1 + 0.5 < x := by |
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by_cases g₂: (x - (1 - a)) < 0 |
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. left |
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rw [ha] at g₂ |
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exact sub_neg.mp g₂ |
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. push_neg at g₂ |
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right |
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have g₃: 0 < (x - (1 + a)) := by exact pos_of_mul_pos_left h₃ g₂ |
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rw [ha] at g₃ |
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exact sub_pos.mp g₃ |
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theorem imo_1962_p2_13 |
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(x a : ℝ) |
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(ha: a = √31 / 8) : |
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x ^ 2 - 2 * x + 33 / 64 = (x - (1 + a)) * (x - (1 - a)) := by |
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rw [ha] |
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ring_nf |
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norm_num |
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linarith |
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theorem imo_1962_p2_14 |
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(x : ℝ) |
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(h₄ : 12 + (x * 8 - x ^ 2 * 4) < 225 / 16) : |
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0 < x * x + -2 * x + 33 / 64 := by |
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ring_nf at h₄ |
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linarith |
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