Iranian Combinatorics Olympiad 2021 - Advanced level - Problem 1 - Frogs on the stone
- Anindya Biswas
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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
— John von Neumann
- Mehrab4226
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Re: Iranian Combinatorics Olympiad 2021 - Advanced level - Problem 1 - Frogs on the stone
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré
-Henri Poincaré