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OlympiadBench / 1887 / Players A and B play a game on a blackboard that initially contains 2020…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

answer type

Numerical

is multiple answer

false

language

English

question

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.

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:

(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.

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:

(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.

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 1887
task_id: 270c94a9-394f-5f77-bc95-a0ccae247439
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--1887
task_revision_id: 1
config / split: OE_TO_maths_en_COMP / train

Official source

initial import