MathDB

Problem 2

Part of 2024 Kyiv City MO Round 2

Problems(5)

Nondivisible power sums

Source: Kyiv City MO 2024 Round 2, Problem 9.2

2/4/2024
You are given a positive integer n>1n > 1. What is the largest possible number of integers that can be chosen from the set {1,2,3,…,2n}\{1, 2, 3, \ldots, 2^n\} so that for any two different chosen integers a,ba, b, the value ak+bka^k + b^k is not divisible by 2n2^n for any positive integer kk?
Proposed by Oleksii Masalitin
number theory
Anti-GCD problem

Source: Kyiv City MO 2024 Round 2, Problem 7.2/8.1

2/4/2024
You are given a positive integer nn. What is the largest possible number of numbers that can be chosen from the set {1,2,…,2n}\{1, 2, \ldots, 2n\} so that there are no two chosen numbers x>yx > y for which x−y=(x,y)x - y = (x, y)?
Here (x,y)(x, y) denotes the greatest common divisor of x,yx, y.
Proposed by Anton Trygub
GCDnumber theory
8-graders solving algebra from Shortlist (unintended)

Source: Kyiv City MO 2024 Round 2, Problem 8.2

2/4/2024
Find the smallest positive integer nn for which one can select nn distinct real numbers such that each of them is equal to the sum of some two other selected numbers.
Proposed by Anton Trygub
algebraconstruction
Comeback of inequalities (I'm sorry)

Source: Kyiv City MO 2024 Round 2, Problem 10.2

2/4/2024
For any positive real numbers a,b,c,da, b, c, d, prove the following inequality: (a2+b2)(b2+c2)(c2+d2)(d2+a2)≥64abcd∣(a−b)(b−c)(c−d)(d−a)∣(a^2+b^2)(b^2+c^2)(c^2+d^2)(d^2+a^2) \geq 64abcd|(a-b)(b-c)(c-d)(d-a)| Proposed by Anton Trygub
inequalitiesalgebra
Fantastic number theory

Source: Kyiv City MO 2024 Round 2, Problem 11.2

2/4/2024
Mykhailo wants to arrange all positive integers from 11 to 20242024 in a circle so that each number is used exactly once and for any three consecutive numbers a,b,ca, b, c the number a+ca + c is divisible by b+1b + 1. Can he do it?
Proposed by Fedir Yudin
number theoryDivisibility