Nemotron-IMO-Bench 2026 A41

A function $f: \mathbf{R}\to \mathbf{R}$ is called additive if $f(x+y)=f(x)+f(y)$ for all $x,y\in \mathbf{R}$. Let $a$ be a real number, $n$ a positive integer and let $f_1,\ldots, f_n$ be additive functions such that $f_1(x)f_2(x)\ldots f_n(x)=ax^n$ for all $x\in \mathbf{R}$. Prove that at least one of the functions $f_1,\ldots, f_n$ is linear (i.e. of the form $x\mapsto cx$ for some constant $c$).