E. Canada 2002 Problem 5

Determine all functions $f: \mathbb{N} \to \mathbb{N}$ such that $$xf(y) + yf(x) = (x + y)f(x^2 + y^2)$$ for all $x, y \in \mathbb{N}$.

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.Nat.Defs
import Mathlib.Data.Set.Defs

def answer : Set (ℕ → ℕ) := sorry

theorem solution (f : ℕ → ℕ) : (∀ x y, x * f y + y * f x = (x + y) * f (x ^ 2 + y ^ 2)) ↔
    f ∈ answer := sorry

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