Let $a_1,\ldots,a_n$ be points on the unit circle. Let $P(z)=(z-a_1)\cdots(z-a_n)$. Prove that there exists a point $b$ on the unit circle such that $|P(b)|=1$.
My solution: $|P(0)|=1$, and $P$ is holomorphic in $|z|\leq 1$. By the maximum principle, there exists $z$ on the unit circle such that $|P(z)|>1$. But $P(a_1)=0$. By intermediate value theorem, there exists $b$ on the unit circle such that $|P(b)|=1$.
Question: Is there a way to avoid the maximum principle?