Bariz - Problem

Source: JBMO Shortlist 2022

Tags: combinatorics, Junior, Balkan, shortlist

Lukaluce

27.06.2023 00:37

There are $200$ boxes on the table. In the beginning, each of the boxes contains a positive integer (the integers are not necessarily distinct). Every minute, Alice makes one move. A move consists of the following. First, she picks a box $X$ which contains a number $c$ such that $c = a + b$ for some numbers $a$ and $b$ which are contained in some other boxes. Then she picks a positive integer $k > 1$. Finally, she removes $c$ from $X$ and replaces it with $kc$. If she cannot make any mobes, she stops. Prove that no matter how Alice makes her moves, she won't be able to make infinitely many moves.

A bit silly Assume otherwise. Note that there is a nonzero number of boxes whose number changes infinitely many times. We may ignore any boxes whose number changes a finite number of times, as after a finite number of moves we may assume that all those boxes retain their number, and since the number in each of the other boxes increases, we may assume that the process happens between among these particular boxes. Hence, the number in each of the boxes changes infinitely may times. Consider the smallest number among the boxes. Note that this number cannot change, as it is smaller than all the remaining numbers, and the numbers on the other boxes keep increasing during the process, a contradiction. Hence, the process will eventually stop.