EGMO 2023 | Problem 4 - 17 April 2023

Problem 4.
Turbo the snail sits on a point on a circle with circumference $1$ . Given an infinite sequence of positive real numbers $c_1, c_2, c_3, \dots$ , Turbo successively crawls distances $c_1, c_2, c_3, \dots$ 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 $c_1, c_2, c_3, \dots$ with $c_i < C$ 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.