ILL 1986 Problem 45

Given $n$ real numbers $a_1 \leq a_2 \leq \cdots \leq a_n$ , define $$M_1=\frac 1n \sum_{i=1}^{n} a_i , M_2=\frac{2}{n(n-1)} \sum_{1 \leq i<j \leq n} a_ia_j, Q=\sqrt{M_1^2-M_2}$$ Prove that $$a_1 \leq M_1 - Q \leq M_1 + Q \leq a_n$$ and that equality holds if and only if $a_1 = a_2 = \cdots = a_n.$