2024 USAMO 第 1 题

Find all integers $n \geq 3$ such that the following property holds: if we list the divisors of $n !$ in increasing order as $1=d_1<d_2<\cdots<d_k=n!$, then we have

$$

d_2-d_1 \leq d_3-d_2 \leq \cdots \leq d_k-d_{k-1} .

$$

找到所有整数 $n \geq 3$ ，使得以下属性成立：如果我们按升序列出 $n !$ 的除数为 $1=d_1<d_2<\cdots<d_k=n!$，那么我们有

$$d_2-d_1 \leq d_3-d_2 \leq \cdots \leq d_k-d_{k-1} 。$$