MathDB

Problem 3

Part of 2024 Kyiv City MO Round 1

Problems(5)

Easy numbergame theory

Source: Kyiv City MO 2024 Round 1, Problem 7.3

1/28/2024
Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from 11 to 20242024 that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by 20242024 loses. Who wins if every player wants to win?
Proposed by Mykhailo Shtandenko
gamenumber theoryDivisibility
Congenial concyclicity

Source: Kyiv City MO 2024 Round 1, Problem 8.3

1/28/2024
The circle γ\gamma passing through the vertex AA of triangle ABCABC intersects its sides ABAB and ACAC for the second time at points XX and YY, respectively. Also, the circle γ\gamma intersects side BCBC at points DD and EE so that AD=AEAD = AE. Prove that the points B,X,Y,CB, X, Y, C lie on the same circle.
Proposed by Mykhailo Shtandenko
geometryConcyclic
Medium numbergame theory

Source: Kyiv City MO 2024 Round 1, Problem 9.3

1/28/2024
Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from 11 to 20232023 that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by 20232023 loses. Who wins if every player wants to win?
Proposed by Mykhailo Shtandenko
gamenumber theoryDivisibility
Hard numbergame theory

Source: Kyiv City MO 2024 Round 1, Problem 11.3

1/28/2024
Let n>1n>1 be a given positive integer. Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from 11 to nn that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by nn loses. Who wins if every player wants to win? Find answer for each n>1n>1.
Proposed by Mykhailo Shtandenko, Anton Trygub
gamenumber theorycombinatoricsDivisibility
Knights accuse each other in lying!

Source: Kyiv City MO 2024 Round 1, Problem 10.3

1/28/2024
There are 20252025 people living on the island, each of whom is either a knight, i.e. always tells the truth, or a liar, which means they always lie. Some of the inhabitants of the island know each other, and everyone has at least one acquaintance, but no more than three. Each inhabitant of the island claims that there are exactly two liars among his acquaintances.
a) What is the smallest possible number of knights among the inhabitants of the island? b) What is the largest possible number of knights among the inhabitants of the island?
Proposed by Oleksii Masalitin
combinatoricslying