RO-TST 1987 Problem 4

Let $P(X) = a_{n}X^{n} + a_{n - 1}X^{n - 1} + \ldots + a_{1}X + a_{0}$ be a real polynomial of degree $n$ . Suppose $n$ is an even number and:

a) $a_{0} > 0$ , $a_{n} > 0$ ;

b) $a_{1}^{2} + a_{2}^{2} + \ldots + a_{n - 1}^{2}\leq\frac {4\min(a_{0}^{2} , a_{n}^{2})}{n - 1}$ .

Prove that $P(x)\geq 0$ for all real values $x$ .

Laurentiu Panaitopol