MathDB

Problem 8

Part of 2024 Ukraine National Mathematical Olympiad

Problems(3)

Oleksii and Solomiya go pro gaming on a grid

Source: Ukrainian Mathematical Olympiad 2024. Day 2, Problem 8.8, 9.8

3/20/2024
Oleksii and Solomiya play the following game on a square 6n×6n6n\times 6n, where nn is a positive integer. Oleksii in his turn places a piece of type FF, consisting of three cells, on the board. Solomia, in turn, after each move of Oleksii, places the numbers 0,1,20, 1, 2 in the cells of the figure that Oleksii has just placed, using each of the numbers exactly once. If two of Oleksii's pieces intersect at any moment (have a common square), he immediately loses.
Once the square is completely filled with numbers, the game stops. In this case, if the sum of the numbers in each row and each column is divisible by 33, Solomiya wins, and otherwise Oleksii wins. Who can win this game if the figure of type FF is: a) a rectangle ; b) a corner of three cells?
Proposed by Oleksii Masalitin
combinatoricsgame
Almost Hamiltonian path

Source: Ukrainian Mathematical Olympiad 2024. Day 2, Problem 10.8

3/20/2024
There are 20242024 cities in a country, some pairs of which are connected by bidirectional flights. For any distinct cities A,B,C,X,Y,ZA, B, C, X, Y, Z, it is possible to fly directly from some of the cities A,B,CA, B, C to some of the cities X,Y,ZX, Y, Z. Prove that it is possible to plan a route T1→T2→…→T2022T_1\to T_2 \to \ldots \to T_{2022} that passes through 20222022 distinct cities.
Proposed by Lior Shayn
graph theoryHamiltonian pathcombinatorics
Generalization of 10.7

Source: Ukrainian Mathematical Olympiad 2024. Day 2, Problem 11.8

3/20/2024
Find all polynomials P(x)P(x) with integer coefficients, such that for each of them there exists a positive integer NN, such that for any positive integer n≥Nn\geq N, number P(n)P(n) is a positive integer and a divisor of n!n!.
Proposed by Mykyta Kharin
number theorypolynomial