MathDB
Board numbers can be eventually all equal

Source: Iberoamerican MO 2024 Day 2 P5

September 23, 2024
combinatoricsnumber theory

Problem Statement

Let n≥2n \ge 2 be an integer and let a1,a2,⋯ana_1, a_2, \cdots a_n be fixed positive integers (not necessarily all distinct) in such a way that gcd⁡(a1,a2⋯an)=1\gcd(a_1, a_2 \cdots a_n)=1. In a board the numbers a1,a2⋯ana_1, a_2 \cdots a_n are all written along with a positive integer xx. A move consists of choosing two numbers a>ba>b from the n+1n+1 numbers in the board and replace them with a−b,2ba-b,2b. Find all possible values of xx, with respect of the values of a1,a2⋯ana_1, a_2 \cdots a_n, for which it is possible to achieve a finite sequence of moves (possibly none) such that eventually all numbers written in the board are equal.
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