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Determine all f with f(f(x + y)f(x - y)) = x^2 + \alpha yf(y) for some alpha

Source: Austria National Competition 2017, Final Round, Part 2, Day 1, Problem 1

June 10, 2017
algebrafunctional equationalgebra proposed

Problem Statement

Let α\alpha be a fixed real number. Find all functions f:R→Rf:\mathbb R \to \mathbb R such that f(f(x+y)f(x−y))=x2+αyf(y)f(f(x + y)f(x - y)) = x^2 + \alpha yf(y)for all x,y∈Rx,y \in \mathbb R.
Proposed by Walther Janous
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