MathDB

Problem 5

Part of 2023 Kyiv City MO Round 1

Problems(3)

Convex quadrilaterals don't exist??

Source: Kyiv City MO 2023 Round 1, Problem 8.5

12/16/2023
You are given a square n×nn \times n. The centers of some of some mm of its 1×11\times 1 cells are marked. It turned out that there is no convex quadrilateral with vertices at these marked points. For each positive integer n≥3n \geq 3, find the largest value of mm for which it is possible.
Proposed by Oleksiy Masalitin, Fedir Yudin
combinatoricscombinatorial geometrysquare grid
Integer convex equilateral 2023-gon

Source: Kyiv City MO 2023 Round 1, Problem 9.5

12/16/2023
Does there exist on the Cartesian plane a convex 20232023-gon with vertices at integer points, such that the lengths of all its sides are equal?
Proposed by Anton Trygub
geometrynumber theory
Graph wars

Source: Kyiv City MO 2023 Round 1, Problem 11.5

12/16/2023
In a galaxy far, far away there are 225225 inhabited planets. Between some pairs of inhabited planets there is a bidirectional space connection, and it is possible to reach any planet from any other (possibly with several transfers). The influence of a planet is the number of other planets with which this planet has a direct connection. It is known that if two planets are not connected by a direct space flight, they have different influences. What is the smallest number of connections possible under these conditions?
Proposed by Arsenii Nikolaev, Bogdan Rublov
combinatoricsgraph theory