MathNet 2009-part-01 — 175 problems (EN–VI) Question 131

Let $n \ge 3$ be a given integer, and $a_1, a_2, \dots, a_n$ be real numbers satisfying $\min_{1 \le i \lt j \le n} |a_i - a_j| = 1$. Find the minimum value of $\sum_{k=1}^n |a_k|^3$. (Posed by Zhu Huawei)