P7
Part of 2022/2023 Tournament of Towns
Problems(2)
It is important to divide pizza equally
Source: 44th International Tournament of Towns, Senior A-Level P7, Fall 2022
2/16/2023
There are friends and a round pizza. It is allowed to make no more than straight cuts without shifting the slices until all cuts are done; then the resulting slices are distributed among all the friends so that each of them gets a share off pizza having the same total area. Is there a cutting which gives the above result if a) and b) ?
combinatoricsgeometryTournament of Towns
Biting chameleons
Source: 44th International Tournament of Towns, Senior A-Level P7, Spring 2023
4/4/2023
Chameleons of five colors live on the island. When one chameleon bites another, the color of bitten chameleon changes to one of these five colors according to some rule, and the new color depends only on the color of the bitten and the color of the bitting. It is known that red chameleons can agree on a sequence of bites between
themselves, after which they will all turn blue.
What is the smallest that can guarantee that red chameleons, biting only each other, can turn blue?
(For example, the rules might be: if a red chameleon bites a green one, the bitten one changes color to blue; if a green one bites a red one, the bitten one remains red, that is, "changes color to red"; if red bites red, the bitten one changes color to yellow, etc. The rules for changing colors may be different.)
combinatorics