#olympiad

Find all polynomials $P$ with real coefficients such that $P(x^2)=P(x{)}^2$ for every real $x$.

Let $a$ and $b$ be positive integers such that $ab+1$ divides $a^2+b^2$. Prove that $\frac{a^2+b^2}{ab+1}$ is the square of an integer.