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OlympiadBench / 2153 / Players A and B play a game with N≥2012 coins and 2012 boxes arranged…

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 with

N \geq 2012

coins and 2012 boxes arranged around a circle. Initially

A

distributes the coins among the boxes so that there is at least 1 coin in each box. Then the two of them make moves in the order

B, A, B, A, \ldots

by the following rules: - On every move of his

B

passes 1 coin from every box to an adjacent box. - On every move of hers

A

chooses several coins that were not involved in

B

's previous move and are in different boxes. She passes every chosen coin to an adjacent box. Player

A

's goal is to ensure at least 1 coin in each box after every move of hers, regardless of how

B

plays and how many moves are made. Find the least

N

that enables her to succeed.

Plain-text mathematical notation (without MathML)

Players A and B play a game with N≥2012 coins and 2012 boxes arranged around a circle. Initially A distributes the coins among the boxes so that there is at least 1 coin in each box. Then the two of them make moves in the order B,A,B,A,… by the following rules:

- On every move of his B passes 1 coin from every box to an adjacent box.
- On every move of hers A chooses several coins that were not involved in B 's previous move and are in different boxes. She passes every chosen coin to an adjacent box.

Player A 's goal is to ensure at least 1 coin in each box after every move of hers, regardless of how B plays and how many moves are made. Find the least N that enables her to succeed.

Original LaTeX notation

Players $A$ and $B$ play a game with $N \geq 2012$ coins and 2012 boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least 1 coin in each box. Then the two of them make moves in the order $B, A, B, A, \ldots$ by the following rules:

- On every move of his $B$ passes 1 coin from every box to an adjacent box.
- On every move of hers $A$ chooses several coins that were not involved in $B$ 's previous move and are in different boxes. She passes every chosen coin to an adjacent box.

Player $A$ 's goal is to ensure at least 1 coin in each box after every move of hers, regardless of how $B$ plays and how many moves are made. Find the least $N$ that enables her to succeed.

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 2153
task_id: 061e6849-1c3e-57f2-ba1d-e4e7aea2d8b3
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--2153
task_revision_id: 1
config / split: OE_TO_maths_en_COMP / train

Official source

initial import