IMO 2020 Shortlist C8
Players A and B play a game on a blackboard that initially contains 2020 copies of the number 1. In every round, player ...
Category: Combinatorics
Problem
Players A and B play a game on a blackboard that initially contains 2020 copies of the number 1. In every round, player A erases two numbers x and y from the blackboard, and then player B writes one of the numbers x ` y and |x ´ y| on the blackboard. The game terminates as soon as, at the end of some round, one of the following holds: (1) one of the numbers on the blackboard is larger than the sum of all other numbers; (2) there are only zeros on the blackboard. Player B must then give as many cookies to player A as there are numbers on the blackboard. Player A wants to get as many cookies as possible, whereas player B wants to give as few as possible. Determine the number of cookies that A receives if both players play optimally. (Austria) 8 Saint-Petersburg — Russia, 18th–28th September 2020