Problem

Source: ELMO Shortlist 2023 C8

Tags: Elmo, combinatorics



Let \(n\ge3\) be a fixed integer, and let \(\alpha\) be a fixed positive real number. There are \(n\) numbers written around a circle such that there is exactly one \(1\) and the rest are \(0\)'s. An operation consists of picking a number \(a\) in the circle, subtracting some positive real \(x\le a\) from it, and adding \(\alpha x\) to each of its neighbors. Find all pairs \((n,\alpha)\) such that all the numbers in the circle can be made equal after a finite number of operations. Proposed by Anthony Wang