Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2021 IMO Shortlist Problems/C2

Revision as of 09:23, 23 June 2023 by Renrenthehamster (talk | contribs) (Created page with "==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 many marbles in each of <math>n</ma...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2021_IMO_Shortlist_Problems/C2&oldid=194708"