MathDB

Problem 7

Part of 2024 Ukraine National Mathematical Olympiad

Problems(4)

Nobody solved this NT

Source: Ukrainian Mathematical Olympiad 2024. Day 2, Problem 8.7

3/20/2024
Prove that there exist infinitely many positive integers that can't be represented in form abcāˆ’bada^{bc} - b^{ad}, where a,b,c,da, b, c, d are positive integers and a,b>1a, b>1.
Proposed by Anton Trygub, Oleksii Masalitin
number theory
Number theory without tex

Source: Ukrainian Mathematical Olympiad 2024. Day 2, Problem 9.7

3/20/2024
Find all composite odd positive integers, all divisors of which can be divided into pairs so that the sum of the numbers in each pair is a power of two, and each divisor belongs to exactly one such pair.
Proposed by Anton Trygub
number theory
Amazing number theory about polynomials

Source: Ukrainian Mathematical Olympiad 2024. Day 2, Problem 10.7

3/20/2024
Find all polynomials P(x)P(x) with integer coefficients, such that for any positive integer nn number P(n)P(n) is a positive integer and a divisor of n!n!.
Proposed by Mykyta Kharin
polynomialnumber theory
My (almost) first combgeo

Source: Ukrainian Mathematical Olympiad 2024. Day 2, Problem 11.7

3/20/2024
You are given 20242024 yellow and 20242024 blue points on the plane, and no three of the points are on the same line. We call a pair of nonnegative integers (a,b)(a, b) good if there exists a half-plane with exactly aa yellow and bb blue points. Find the smallest possible number of good pairs. The points that lie on the line that is the boundary of the half-plane are considered to be outside the half-plane.
Proposed by Anton Trygub
combinatorial geometrycombinatorics