IMO 2023 Shortlist C1

Let m and n be positive integers greater than 1. In each unit square of an mˆn grid lies a coin with its tail-side up. A...

imomathematicsolympiadshortlistcombinatorics

IMO 2023 Shortlist C1

Category: Combinatorics

Problem

Let m and n be positive integers greater than 1. In each unit square of an mˆn grid lies a coin with its tail-side up. A move consists of the following steps:

  1. select a 2 ˆ 2 square in the grid;
  2. flip the coins in the top-left and bottom-right unit squares;
  3. flip the coin in either the top-right or bottom-left unit square. Determine all pairs pm,nq for which it is possible that every coin shows head-side up after a finite number of moves. (Thailand)