Problem

Source: 2017 Argentina OMA Finals L3 p2

Tags: Sum, algebra, number theory, consecutive



In a row there are $51$ written positive integers. Their sum is $100$ . An integer is representable if it can be expressed as the sum of several consecutive numbers in a row of $51$ integers. Show that for every $k$ , with $1\le k \le 100$ , one of the numbers $k$ and $100-k$ is representable.