ILL 1972 Problem 25

We consider $n$ real variables $x_i(1 \le i \le n)$ , where $n$ is an integer and $n \ge 2$ . The product of these variables will be denoted by $p$ , their sum by $s$ , and the sum of their squares by $S$ . Furthermore, let $\alpha$ be a positive constant. We now study the inequality $ps \le S\alpha$ . Prove that it holds for every $n$ -tuple $(x_i)$ if and only if $\alpha=\frac{n+1}{2}$