import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Geometry.Euclidean.Basic
import Mathlib.Geometry.Euclidean.Circumcenter
import Mathlib.Geometry.Euclidean.Triangle

/-!
# International Mathematical Olympiad 2001, Problem 1

Let ABC be an acute-angled triangle with O as its circumcenter. Let P
on line BC be the foot of the altitude from A. Assume that
∠BCA ≥ ∠ABC + 30°. Prove that ∠CAB + ∠COP < 90°.
-/

namespace Imo2001P1

open scoped EuclideanGeometry

theorem imo2001_p1
    (A B C : EuclideanSpace ℝ (Fin 2))
    (hABC : AffineIndependent ℝ ![A, B, C])
    (hAcuteA : ∠ C A B < Real.pi / 2)
    (hAcuteB : ∠ A B C < Real.pi / 2)
    (hAcuteC : ∠ B C A < Real.pi / 2)
    (hAB : ∠ A B C + Real.pi / 6 ≤ ∠ B C A)
    : let ABC : Affine.Triangle _ _ := ⟨![A, B, C], hABC⟩
      let P := EuclideanGeometry.orthogonalProjection line[ℝ, B, C] A
      ∠ C A B + ∠ C ABC.circumcenter P < Real.pi / 2 := sorry

This problem does not yet have a complete solution.