This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 28

2024/2025 TOURNAMENT OF TOWNS, P7
Tags: combinatorics , ToT
The hostess takes a piece of meat from the fridge; kittens gather around her. Each minute, the hostess cuts a part from the piece and feeds it to one of the kittens (on her choice). Each time, the cut part is in the same proportion to the current piece. At some moment, the hostess puts the rest of the meat into the fridge. Can the hostess give the same amount of meat in total to each kitten if a) the number of kittens equals two; (3 marks) b) the number of kittens equals three? (7 marks)

2024/2025 TOURNAMENT OF TOWNS, P5
Tags: ToT , combinatorics
A rectangular checkered board is painted black and white as a chessboard, and is tiled by dominoes $1 \times 2$. If a horizontal and a vertical dominoes have common segment, it has a door which has the color of the adjoining cell of the domino adjacent by a short side. Is it necessarily true that the number of white doors equals the number of black doors?

2024/2025 TOURNAMENT OF TOWNS, P2
Tags: ToT , combinatorics
There are $N$ pupils in a school class, and there are several communities among them. Sociability of a pupil will mean the number of pupils in the largest community to which the pupil belongs (if the pupil belongs to none then the sociability equals $1$). It occurred that all girls in the class have different sociabilities. What is the maximum possible number of girls in the class?