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Omni-MATH / 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 …

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problem

Players AA and BB play a game on a blackboard that initially contains 2020 copies of the number 1 . In every round, player AA erases two numbers xx and yy from the blackboard, and then player BB writes one of the numbers x+yx+y and |x−y||x-y| on the blackboard. The game terminates as soon as, at the end of some round, one of the following holds: [list] [*] (1)(1) one of the numbers on the blackboard is larger than the sum of all other numbers; [*] (2)(2) there are only zeros on the blackboard. [/list] Player BB must then give as many cookies to player AA as there are numbers on the blackboard. Player AA wants to get as many cookies as possible, whereas player BB wants to give as few as possible. Determine the number of cookies that AA receives if both players play optimally.
Plain-text mathematical notation (without MathML)
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:
[list]
[*] (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.
[/list]
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.
Original LaTeX notation
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:
[list]
[*] $(1)$ one of the numbers on the blackboard is larger than the sum of all other numbers;
[*] $(2)$ there are only zeros on the blackboard.
[/list]
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.

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Source and history

task_id
5e117f4b-7b14-56d9-8127-098bf970cda1
task_key
test--5e117f4b-7b14-56d9-8127-098bf970cda1
task_revision_id
1
config / split
/ test

Official source

initial import