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Large 3 var real functional equation

Source: 2022 IMOC A5

September 5, 2022
algebrafunctional equation

Problem Statement

Find all functions f:R→Rf:\mathbb R\to \mathbb R such that \begin{align*} \left (x \left (f(x)-\dfrac{f(y)+f(z)}{2} \right) +y \left (f(y)-\dfrac{f(z)+f(x)}{2} \right ) +z\left (f(z)- \dfrac{f(x)+f(y)}{2} \right) \right )f(x+y+z)= \\ f(x^3)+f(y^3)+f(z^3)-3f(xyz) \end{align*} for all x,y,z∈R.x,y,z\in \mathbb R.
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