Putnam 1947 B4

Given $P(z)= z^2 +az +b,$ where $a,b \in \mathbb{C}.$ Suppose that $|P(z)|=1$ for every complex number $z$ with $|z|=1.$ Prove that $a=b=0.$

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.