2002 IMO 第 2 题

N2 (ROM) ${ }^{\mathrm{IMO} 4}$ Let $n \geq 2$ be a positive integer, with divisors $1=d_{1}<$ $d_{2}<\cdots<d_{k}=n$. Prove that $d_{1} d_{2}+d_{2} d_{3}+\cdots+d_{k-1} d_{k}$ is always less than $n^{2}$, and determine when it is a divisor of $n^{2}$.

N2 (ROM) ${ }^{\mathrm{IMO} 4}$ 设 $n \geq 2$ 为正整数，除数为 $1=d_{1}<$ $d_{2}<\cdots<d_{k}=n$。证明 $d_{1} d_{2}+d_{2} d_{3}+\cdots+d_{k-1} d_{k}$ 始终小于 $n^{2}$，并确定何时它是 $n^{2}$ 的约数。