BALTIC-WAY 2015 Problem 3

Let $n>1$ be an integer. Find all non-constant real polynomials $P(x)$ satisfying, for any real $x$, the identity

$$P(x) P\left(x^{2}\right) P\left(x^{3}\right) \cdots P\left(x^{n}\right)=P\left(x^{\frac{n(n+1)}{2}}\right) .$$