MathDB

P7

Part of 2021/2022 Tournament of Towns

Problems(3)

A 2x2 grid covered in triangles

Source: 43rd International Tournament of Towns, Senior A-Level P7, Fall 2021

2/18/2023
A checkered square of size 2×22\times2 is covered by two triangles. Is it necessarily true that:
[*]at least one of its four cells is fully covered by one of the triangles; [*]some square of size 1×11\times1 can be placed into one of these triangles?
Alexandr Shapovalov
combinatoricscombinatorial geometryTournament of Towns
2022 TTs SA7

Source: 2022 Tournament of towns Spring Senior A-level

5/14/2022
A starship is located in a halfspace at the distance aa from its boundary. The crew knows this but does not know which direction to move to reach the boundary plane. The starship may travel through the space by any path, may measure the way it has already travelled and has a sensor that signals when the boundary is reached. Is it possible to reach the boundary for sure, having passed no more than:
a)14aa)14a
b)13ab)13a?
3D geometrygeometry
2022TTsJA7

Source: 2022 Tournament of towns Spring Junior A level

5/18/2022
Let pp be a prime number and let MM be a convex polygon. Suppose that there are precisely pp ways to tile mm with equilateral triangles with side 11 and squares with side 11. Show there is some side of MM of length p−1p-1.
number theoryprime numbersgeometry