Problem

Source: Tournament of Towns, Fall 2002, Junior A Level, P3

Tags: combinatorics proposed, combinatorics



The vertices of a $50\text{-gon}$ divide a circumference into $50$ arcs, whose lengths are $1,2,\ldots 50$ in some order. It is known that any two opposite arcs (corresponding to opposite sides) differ by $25$. Prove that the polygon has two parallel sides.