Question 7
Let $a_{1}, a_{2}, \ldots, a_{n}$ be non-negative real numbers. Define $M$ to be the sum of all products of pairs $a_{i} a_{j}$ ($i\lt j$), i.e.,
$$
M = a_{1}(a_{2} + a_{3} + \cdots + a_{n}) + a_{2}(a_{3} + a_{4} + \cdots + a_{n}) + \cdots + a_{n-1} a_{n}
$$
Prove that the square of at least one of the numbers $a_{1}, a_{2}, \ldots, a_{n}$ does not exceed $2M / n(n-1)$.
$$
M = a_{1}(a_{2} + a_{3} + \cdots + a_{n}) + a_{2}(a_{3} + a_{4} + \cdots + a_{n}) + \cdots + a_{n-1} a_{n}
$$
Prove that the square of at least one of the numbers $a_{1}, a_{2}, \ldots, a_{n}$ does not exceed $2M / n(n-1)$.