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OlympiadBench / 1974 / Let n≥2 be an integer. Consider an n×n chessboard divided…

Problem

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

question

Let

n \geqslant 2

be an integer. Consider an

n \times n

chessboard divided into

n^{2}

unit squares. We call a configuration of

n

rooks on this board happy if every row and every column contains exactly one rook. Find the greatest positive integer

k

such that for every happy configuration of rooks, we can find a

k \times k

square without a rook on any of its

k^{2}

unit squares.

Plain-text mathematical notation (without MathML)

Let n≥2 be an integer. Consider an n×n chessboard divided into n² unit squares. We call a configuration of n rooks on this board happy if every row and every column contains exactly one rook. Find the greatest positive integer k such that for every happy configuration of rooks, we can find a k×k square without a rook on any of its k² unit squares.

Original LaTeX notation

Let $n \geqslant 2$ be an integer. Consider an $n \times n$ chessboard divided into $n^{2}$ unit squares. We call a configuration of $n$ rooks on this board happy if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that for every happy configuration of rooks, we can find a $k \times k$ square without a rook on any of its $k^{2}$ unit squares.

answer type

Expression

is multiple answer

false

language

English

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 1974
task_id: 9b8871ea-0861-59a2-8d69-5670a6ffa1f4
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--1974
task_revision_id: 3
config / split: OE_TO_maths_en_COMP / train

Official source

initial import