IMO 2014 Shortlist N6
Let a1 ă a2 ă ¨¨¨ ă an be pairwise coprime positive integers with a1 being prime and a1 ě n 2. On the segment I “ r0,a1a...
Category: Number Theory
Problem
Let a1 ă a2 ă ¨¨¨ ă an be pairwise coprime positive integers with a1 being prime and a1 ě n ` 2. On the segment I “ r0,a1a2 ¨¨¨ans of the real line, mark all integers that are divisible by at least one of the numbers a1,...,an. 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 a1. (Serbia)