Balti Tee 1998 · Ülesanne 6

Solution: The polynomial $Q(x) = P(x) - P(-x)$, of degree at most $5$, has roots at $-b$, $-a$, $0$, $a$ and $b$; these are five distinct numbers. Moreover, $Q'(0) = 0$, showing that $Q$ has a multiple root at $0$. Thus $Q$ must be the constant $0$, i.e. $P(x) = P(-x)$ for all $x$.