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National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Kyiv City MO
2023 Kyiv City MO
2023 Kyiv City MO Round 1
2023 Kyiv City MO Round 1
Part of
2023 Kyiv City MO
Subcontests
(5)
Problem 5
3
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Convex quadrilaterals don't exist??
You are given a square
n
×
n
n \times n
n
×
n
. The centers of some of some
m
m
m
of its
1
×
1
1\times 1
1
×
1
cells are marked. It turned out that there is no convex quadrilateral with vertices at these marked points. For each positive integer
n
≥
3
n \geq 3
n
≥
3
, find the largest value of
m
m
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
2023
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
225
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
Problem 1
5
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Problem 4
4
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Problem 3
5
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Problem 2
5
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