2023 Kyiv City MO Round 1

Part of 2023 Kyiv City MO

Subcontests

(5)

3

Convex quadrilaterals don't exist??

You are given a square

n \times n

. The centers of some of some

m

1\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 \geq 3

, find the largest value of

m

for which it is possible.
Proposed by Oleksiy Masalitin, Fedir Yudin

Integer convex equilateral 2023-gon

Does there exist on the Cartesian plane a convex

2023

-gon with vertices at integer points, such that the lengths of all its sides are equal?
Proposed by Anton Trygub

Graph wars

In a galaxy far, far away there are

225

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

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