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OlympiadBench / 1782 / Let n≥3 be an integer. An integer m≥n+1 is called…

Problem

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

question

Let n≥3n \geqslant 3 be an integer. An integer m≥n+1m \geqslant n+1 is called nn-colourful if, given infinitely many marbles in each of nn colours C1,C2,…,CnC_{1}, C_{2}, \ldots, C_{n}, it is possible to place mm of them around a circle so that in any group of n+1n+1 consecutive marbles there is at least one marble of colour CiC_{i} for each i=1,…,ni=1, \ldots, n. Prove that there are only finitely many positive integers which are not nn-colourful. Find the largest among them.
Plain-text mathematical notation (without MathML)
Let n≥3 be an integer. An integer m≥n+1 is called n-colourful if, given infinitely many marbles in each of n colours C₁,C₂,…,C_(n), it is possible to place m of them around a circle so that in any group of n+1 consecutive marbles there is at least one marble of colour C_(i) for each i=1,…,n.

Prove that there are only finitely many positive integers which are not n-colourful. Find the largest among them.
Original LaTeX notation
Let $n \geqslant 3$ be an integer. An integer $m \geqslant n+1$ is called $n$-colourful if, given infinitely many marbles in each of $n$ colours $C_{1}, C_{2}, \ldots, C_{n}$, it is possible to place $m$ of them around a circle so that in any group of $n+1$ consecutive marbles there is at least one marble of colour $C_{i}$ for each $i=1, \ldots, n$.

Prove that there are only finitely many positive integers which are not $n$-colourful. Find the largest among them.

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
1782
task_id
8082a1f1-0d87-5800-9de9-2f73261b05d4
task_key
OE~5fTO~5fmaths~5fen~5fCOMP--train--1782
task_revision_id
4
config / split
OE_TO_maths_en_COMP / train

Official source

initial import