Friday, November 18th

Problem
Let $p(x)$ be a polynomial with real coefficients such that $p(x)=x$ doesn't have real roots. Prove that $p(p(x))=x$ doesn't have real roots either.

Solution
Without loss of generality suppose that $p(x)>x$ for all $x$. Then $p(p(x))>p(x)>x$ and thus $p(p(x))=x$ does not have any real roots.