Iranian Combinatorics Olympiad 2021 - Advanced level - Problem 1 - Frogs on the stone
Posted: Thu Aug 26, 2021 2:55 am
In the lake, there are $23$ stones arranged along a circle. There are $22$ frogs numbered $1,2,\dots,22$ (each number appears once). Initially, each frog randomly sits on a stone (several frogs might sit on the same stone). Every minute, all frogs jump at the same time as follows: the frog number $i$ jumps $i$ stones forward in the clockwise direction. (In particular, the frog number $22$ jumps $1$ stone in the counter-clockwise direction.) Prove that at some point, at least $6$ stones will be empty.