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 | + | 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 09:24, 23 June 2023
Problem
Let 
be an integer. An integer 
is called 
-colourful if, given infinitely
many marbles in each of colours 
, it is possible to place 
of them around a circle so that in any group of 
consecutive marbles there is at least one marble of colour 
for each 
. Prove that there are only finitely many positive integers which are not -colourful. Find the largest among them.
