2024 CMO 第 4 题

Let $a_1, a_2, \ldots, a_{2023}$ be nonnegative real numbers such that $a_1 + a_2 + \ldots + a_{2023} = 100$ . Let $A = \left \{ (i,j) \mid 1 \leq i \leq j \leq 2023, \, a_ia_j \geq 1 \right\}$ . Prove that $|A| \leq 5050$ and determine when the equality holds.

*Proposed by Yunhao Fu*

令 $a_1, a_2, \ldots, a_{2023}$ 为非负实数，使得 $a_1 + a_2 + \ldots + a_{2023} = 100$ 。设 $A = \left \{ (i,j) \mid 1 \leq i \leq j \leq 2023, \, a_ia_j \geq 1 \right\}$ 。证明 $|A| \leq 5050$ 并确定何时等式成立。

*由付云浩提议*