MathDB

Problem 1

Part of 2024 Ukraine National Mathematical Olympiad

Problems(3)

Pairwise Sums

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 8.1, 9.1

3/19/2024
Oleksiy wrote several distinct positive integers on the board and calculated all their pairwise sums. It turned out that all digits from 00 to 99 appear among the last digits of these sums. What could be the smallest number of integers that Oleksiy wrote?
Proposed by Oleksiy Masalitin
number theoryDigits
Make equal. Fast!

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 10.1

3/19/2024
Solomiya wrote the numbers 1,2,…,20241, 2, \ldots, 2024 on the board. In one move, she can erase any two numbers a,ba, b from the board and write the sum a+ba+b instead of each of them. After some time, all the numbers on the board became equal. What is the minimum number of moves Solomiya could make to achieve this?
Proposed by Oleksiy Masalitin
combinatorics
NT equations make a huge comeback

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 11.1

3/19/2024
Find all pairs a,ba, b of positive integers, for which
(a,b)+3[a,b]=a3−b3(a, b) + 3[a, b] = a^3 - b^3
Here (a,b)(a, b) denotes the greatest common divisor of a,ba, b, and [a,b][a, b] denotes the least common multiple of a,ba, b.
Proposed by Oleksiy Masalitin
greatest common divisorleast common multiplenumber theory