KöMaL 2016 A675

Let $r(x)$ be a polynomial with real coefficients whose degree $n$ is odd. Prove that the number of pairs of polynomials $p(x)$ and $q(x)$ with real coefficients satisfying the equation $\big(p(x)\big)^3 + q(x^2) = r(x)$, is smaller than $2^n$.