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import Mathlib
open Real
set_option linter.unusedVariables.analyzeTactics true
theorem imo_1962_p2
(x : ℝ)
(h₀ : 0 ≤ 3 - x)
(h₁ : 0 ≤ x + 1)
(h₂ : 1 / 2 < Real.sqrt (3 - x) - Real.sqrt (x + 1)) :
-1 ≤ x ∧ x < 1 - Real.sqrt 31 / 8 := by
constructor
. exact neg_le_iff_add_nonneg.mpr h₁
have h₃: (2 *sqrt (3 - x) * sqrt (x + 1)) ^ 2 < (4 - 1 / 4) ^ 2 := by
refine' pow_lt_pow_left₀ _ _ (by norm_num)
. refine lt_tsub_iff_left.mpr ?_
refine lt_tsub_iff_right.mp ?_
suffices g₀: 4 - 2 * sqrt (3 - x) * sqrt (x + 1) = (sqrt (3 - x) - sqrt (x + 1)) ^ 2
. rw [g₀]
have g₁: (1:ℝ) / 4 = (1/2)^2 := by norm_num
rw [g₁]
exact pow_lt_pow_left₀ h₂ (by norm_num) (by norm_num)
rw [sub_sq]
rw [sq_sqrt h₀, sq_sqrt h₁]
ring_nf
. refine' mul_nonneg _ _
. refine mul_nonneg (by norm_num) ?_
exact sqrt_nonneg (3 - x)
. exact sqrt_nonneg (x + 1)
have h₄: 4 * (x + 1) * (3 - x) < 225 / 16 := by
norm_num at h₃
suffices g₀: 4 * (x + 1) * (3 - x) = (2 * sqrt (3 - x) * sqrt (x + 1)) ^ 2
. exact Eq.trans_lt g₀ h₃
. rw [mul_pow, mul_pow, sq_sqrt h₀, sq_sqrt h₁]
norm_num
exact mul_right_comm 4 (x + 1) (3 - x)
have hx1: x < 1 := by
suffices g₀: x + 1 < 3 - x
. linarith
. rw [← sq_sqrt h₀, ← sq_sqrt h₁]
refine' pow_lt_pow_left₀ _ _ (by norm_num)
. linarith
exact sqrt_nonneg (x + 1)
have h₅: x < 1 - sqrt 31 / 8 ∨ 1 + sqrt 31 / 8 < x := by
ring_nf at h₄
have g₀: 0 < x * x + -2 * x + 33 / 64 := by linarith
let a:ℝ := sqrt 31 / 8
have g₁: x * x + -2 * x + 33 / 64 = (x - (1 + a)) * (x - (1 - a)) := by
simp
ring_nf
norm_num
linarith
rw [g₁] at g₀
by_cases g₂: (x - (1 - a)) < 0
. left
exact sub_neg.mp g₂
push_neg at g₂
right
have g₃: 0 < (x - (1 + a)) := by exact pos_of_mul_pos_left g₀ g₂
exact sub_pos.mp g₃
cases h₅ with
| inl h₅ => exact h₅
| inr h₅ => linarith
|