Question 23
Let $a_1, a_2, \dots, a_n$ ($n \ge 3$) be real numbers. Prove that
$$
\sum_{i=1}^{n} a_i^2 - \sum_{i=1}^{n} a_i a_{i+1} \le \left[ \frac{n}{2} \right] (M-m)^2,
$$
where $a_{n+1} = a_1$, $M = \max_{1 \le i \le n} a_i$, $m = \min_{1 \le i \le n} a_i$. $[x]$ is the largest integer not exceeding $x$.
$$
\sum_{i=1}^{n} a_i^2 - \sum_{i=1}^{n} a_i a_{i+1} \le \left[ \frac{n}{2} \right] (M-m)^2,
$$
where $a_{n+1} = a_1$, $M = \max_{1 \le i \le n} a_i$, $m = \min_{1 \le i \le n} a_i$. $[x]$ is the largest integer not exceeding $x$.