Kazakhstan Regional 2026 Grade 11 Problem 1

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for any reals $x, y$, $$f(f(y) + x - y) + f(x - y) = f(xf(y) - y).$$

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

import Mathlib.Data.Real.Basic

def P (f : ℝ → ℝ) := ∀ x y, f (f y + x - y) + f (x - y) = f (x * f y - y)

def answer : Set (ℝ → ℝ) := sorry

theorem solution (f : ℝ → ℝ) : P f ↔ f ∈ answer := sorry

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