2014 IMO Shortlist N6

Let $a_{1}<a_{2}<\cdots<a_{n}$ be pairwise coprime positive integers with $a_{1}$ being prime and $a_{1} \geq n+2$. On the segment $I=\left[0, a_{1} a_{2} \cdots a_{n}\right]$ of the real line, mark all integers that are divisible by at least one of the numbers $a_{1}, \ldots, a_{n}$. These points split $I$ into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by $a_{1}$. (Serbia)

设 $a_{1}<a_{2}<\cdots<a_{n}$ 为成对互质正整数，其中 $a_{1}$ 为素数，$a_{1} \geq n+2$。在实数线段 $I=\left[0, a_{1} a_{2} \cdots a_{n}\right]$ 上，标记所有至少能被 $a_{1}, \ldots, a_{n}$ 之一整除的整数。这些点将 $I$ 分成许多较小的部分。证明这些线段长度的平方和能被$a_{1}$整除。 (塞尔维亚)