Question 96
Let $a_1, a_2, \dots, a_n$ be real numbers such that
$$
a_m + a_{m+1} + \dots + a_n \ge m + (m+1) + \dots + n
$$
for every $m = 1, 2, \dots, n$. Prove that
$$
a_1^2 + a_2^2 + \dots + a_n^2 \ge \frac{n(n+1)(2n+1)}{6}.
$$
$$
a_m + a_{m+1} + \dots + a_n \ge m + (m+1) + \dots + n
$$
for every $m = 1, 2, \dots, n$. Prove that
$$
a_1^2 + a_2^2 + \dots + a_n^2 \ge \frac{n(n+1)(2n+1)}{6}.
$$