Question 115
Let $P(x)$ be a polynomial of degree $n \ge 2$ with nonnegative coefficients. Let $a$, $b$, and $c$ be the side lengths of some triangle. Prove that the numbers $\sqrt[n]{P(a)}$, $\sqrt[n]{P(b)}$, and $\sqrt[n]{P(c)}$ are also the side lengths of some triangle.