MathNet 2010 — Part 04 Question 164

Let $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ be five real numbers of zero sum, such that $|a_i - a_j| \le 1$, for all $i, j \in \{1, 2, 3, 4, 5\}$. Prove that $a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2 \le \frac{6}{5}$.