Nemotron-IMO-Bench 2026 A47

A polynomial $P=a_0+a_1X+...+a_nX^n$ with complex coefficients and degree $n$ has the property that $|P(z)|\leq 1$ for all $|z|=1$. Prove that $|a_k|\leq 1-|a_n|^2$ for all $0\leq k\leq n-1$.