D. Latvia 2021 Problem 5

Find all integer pairs $(a, b)$ such that $$(19 a + b) ^ {18} + (a + b) ^ {18} + (a + 19 b) ^ {18}$$ is a perfect square.

Replace sorry in the template below with your solution. See Answer Bank for acceptible answer declarations. Mathlib version used by the checker is v4.29.0.

import Mathlib.Algebra.Group.Int.Defs
import Mathlib.Data.Set.Defs

def answer : Set (ℤ × ℤ) := sorry

theorem solution (a b : ℤ) : (∃ c, c ^ 2 = (19 * a + b) ^ 18 + (a + b) ^ 18 + (a + 19 * b) ^ 18) ↔
    (a, b) ∈ answer := sorry

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