IMO 2017 Shortlist C7
For any nite sets X and Y of positive integers, denote by fXpkq the kth smallest positive integer not in X, and let X ˚...
Category: Combinatorics
Problem
For any nite sets X and Y of positive integers, denote by fXpkq the kth smallest positive integer not in X, and let X ˚ Y “ X Y tfXpyq: y P Y u. Let A be a set of a ą 0 positive integers, and let B be a set of b ą 0 positive integers. Prove that if A ˚ B “ B ˚ A, then A ˚ pA ˚ ¨¨¨ ˚ pA ˚ pA ˚ Aqq...q looooooooooooooooooomooooooooooooooooooon A appears b times “ B ˚ pB ˚ ¨¨¨ ˚ pB ˚ pB ˚ Bqq...q looooooooooooooooooomooooooooooooooooooon B appears a times . (U.S.A.)