MathDB

P7

Part of 2020/2021 Tournament of Towns

Problems(2)

Chessboard Combinatorics

Source: ToT 2020/2021 Senior A-Level p7, Fall

3/7/2021
A white bug sits in one corner square of a 10001000 × nn chessboard, where nn is an odd positive integer and n>2020n > 2020. In the two nearest corner squares there are two black chess bishops. On each move, the bug either steps into a square adjacent by side or moves as a chess knight. The bug wishes to reach the opposite corner square by never visiting a square occupied or attacked by a bishop, and visiting every other square exactly once. Show that the number of ways for the bug to attain its goal does not depend on nn.
combinatorics
Drawing disjoint arcs on the unit sphere

Source: 42nd International Tournament of Towns, Senior A-Level P7, Spring 2021

2/18/2023
An integer n>2n > 2 is given. Peter wants to draw nn{} arcs of length α\alpha{} of great circles on a unit sphere so that they do not intersect each other. Prove that
[*]for all α<π+2π/n\alpha<\pi+2\pi/n it is possible; [*]for all α>π+2π/n\alpha>\pi+2\pi/n it is impossible;
Ilya Bogdanov
combinatoricscombinatorial geometryTournament of Towns