MathDB

Let

a_1 < a_2 < \cdots <a_n

be pairwise coprime positive integers with

a_1

being prime and

a_1 \ge n + 2

. On the segment

I = [0, a_1 a_2 \cdots a_n ]

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

.
Proposed by Serbia

Problem Statement