2025 CMO 第 6 题

Let $a_1, a_2, \ldots, a_n$ be real numbers such that $\sum_{i=1}^n a_i = n$ , $\sum_{i = 1}^n a_i^2 = 2n$ , $\sum_{i=1}^n a_i^3 = 3n$ .

(i) Find the largest constant $C$ , such that for all $n \geq 4$ , $$\max \left\{ a_1, a_2, \ldots, a_n \right\} - \min \left\{ a_1, a_2, \ldots, a_n \right\} \geq C.$$

(ii) Prove that there exists a positive constant $C_2$ , such that $$\max \left\{ a_1, a_2, \ldots, a_n \right\} - \min \left\{ a_1, a_2, \ldots, a_n \right\} \geq C + C_2 n^{-\frac 32},$$where $C$ is the constant determined in (i).

令 $a_1, a_2, \ldots, a_n$ 为实数，使得 $\sum_{i=1}^n a_i = n$ 、 $\sum_{i = 1}^n a_i^2 = 2n$ 、 $\sum_{i=1}^n a_i^3 = 3n$ 。

(i) 找到最大常数 $C$ ，使得对于所有 $n \geq 4$ ， $$\max \left\{ a_1, a_2, \ldots, a_n \right\} - \min \left\{ a_1, a_2, \ldots, a_n \right\} \geq C。$$

(ii) 证明存在正常数 $C_2$ ，使得 $$\max \left\{ a_1, a_2, \ldots, a_n \right\} - \min \left\{ a_1, a_2, \ldots, a_n \right\} \geq C + C_2 n^{-\frac 32},$$其中 $C$ 是 (i) 中确定的常数。