Switzerland 2003

Given that $x, y \in \mathbb{R}$ satisfy $x + y = x^3 + y^3 = x^5 + y^5$, what are the possible values of $x + y$?

Replace sorry in the template below with your solution. See Answer Bank for acceptible answer declarations. Mathlib version used by the checker is v4.31.0.

import Mathlib.Data.Real.Basic

def answer : Set ℝ := sorry

theorem solution (s : ℝ) :
    (∃ x y, x + y = s ∧ x ^ 3 + y ^ 3 = s ∧ x ^ 5 + y ^ 5 = s) ↔ s ∈ answer := sorry

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