Problem

Source: EGMO 2023/4

Tags: EGMO 2023, EGMO, combinatorics, game, ilostthegame



Turbo the snail sits on a point on a circle with circumference 1. Given an infinite sequence of positive real numbers c1,c2,c3,, Turbo successively crawls distances c1,c2,c3, around the circle, each time choosing to crawl either clockwise or counterclockwise. Determine the largest constant C>0 with the following property: for every sequence of positive real numbers c1,c2,c3, with ci<C for all i, Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.