## Iranian Combinatorics Olympiad 2021 - Advanced level - Problem 1 - Frogs on the stone

For discussing Olympiad Level Combinatorics problems
Anindya Biswas
Posts:263
Joined:Fri Oct 02, 2020 8:51 pm
Contact:
Iranian Combinatorics Olympiad 2021 - Advanced level - Problem 1 - Frogs on the stone
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.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

Mehrab4226
Posts:230
Joined:Sat Jan 11, 2020 1:38 pm