Difference between revisions of "2021 IMO Shortlist Problems/C2"

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==Problem==
 
==Problem==
Let <math>n\ge 3</math> be an integer. An integer <math>m\ge n-1</math> is called <math>n</math>-colourful if, given infinitely
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Let <math>n\ge 3</math> be an integer. An integer <math>m\ge n+1</math> is called <math>n</math>-colourful if, given infinitely
 
many marbles in each of <math>n</math> colours <math>C_1, C_2,\dots, C_n</math>, it is possible to place <math>m</math> of them around a circle so that in any group of <math>n+1</math> consecutive marbles there is at least one marble of colour <math>C_i</math> for each <math>i = 1,\dots, n</math>. Prove that there are only finitely many positive integers which are not <math>n</math>-colourful. Find the largest among them.
 
many marbles in each of <math>n</math> colours <math>C_1, C_2,\dots, C_n</math>, it is possible to place <math>m</math> of them around a circle so that in any group of <math>n+1</math> consecutive marbles there is at least one marble of colour <math>C_i</math> for each <math>i = 1,\dots, n</math>. Prove that there are only finitely many positive integers which are not <math>n</math>-colourful. Find the largest among them.
  
 
==Solution==
 
==Solution==
 
https://youtu.be/Uw0G8uLDc2I
 
https://youtu.be/Uw0G8uLDc2I

Latest revision as of 08:24, 23 June 2023

Problem

Let $n\ge 3$ be an integer. An integer $m\ge n+1$ is called $n$-colourful if, given infinitely many marbles in each of $n$ colours $C_1, C_2,\dots, C_n$, 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 $i = 1,\dots, n$. Prove that there are only finitely many positive integers which are not $n$-colourful. Find the largest among them.

Solution

https://youtu.be/Uw0G8uLDc2I