import Mathlib

set_option maxHeartbeats 0

open BigOperators Real Nat Topology Rat

theorem aime_1984_p15 (x y z w : ℝ)
    (h₀ :
      x ^ 2 / (2 ^ 2 - 1) + y ^ 2 / (2 ^ 2 - 3 ^ 2) + z ^ 2 / (2 ^ 2 - 5 ^ 2) +
          w ^ 2 / (2 ^ 2 - 7 ^ 2) =
        1)
    (h₁ :
      x ^ 2 / (4 ^ 2 - 1) + y ^ 2 / (4 ^ 2 - 3 ^ 2) + z ^ 2 / (4 ^ 2 - 5 ^ 2) +
          w ^ 2 / (4 ^ 2 - 7 ^ 2) =
        1)
    (h₂ :
      x ^ 2 / (6 ^ 2 - 1) + y ^ 2 / (6 ^ 2 - 3 ^ 2) + z ^ 2 / (6 ^ 2 - 5 ^ 2) +
          w ^ 2 / (6 ^ 2 - 7 ^ 2) =
        1)
    (h₃ :
      x ^ 2 / (8 ^ 2 - 1) + y ^ 2 / (8 ^ 2 - 3 ^ 2) + z ^ 2 / (8 ^ 2 - 5 ^ 2) +
          w ^ 2 / (8 ^ 2 - 7 ^ 2) =
        1) :
    x ^ 2 + y ^ 2 + z ^ 2 + w ^ 2 = 36 := by sorry