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f(x^2 y - y) = f(x)^2 f(y) + f(x)^2 - 1

Source: Brazilian Mathematical Olympiad 2024, Level 3, Problem 5

October 12, 2024
functional equationreal numberalgebra

Problem Statement

Let R \mathbb{R} be the set of real numbers. Determine all functions f:R→R f: \mathbb{R} \to \mathbb{R} such that, for any real numbers x x and y y , f(x2y−y)=f(x)2f(y)+f(x)2−1. f(x^2 y - y) = f(x)^2 f(y) + f(x)^2 - 1.
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