IMO 2021 Shortlist C2
Let n ě3 be an integer. An integer m ěn1 is called n-colourful if, given infinitely many marbles in each of n colours C1...
Category: Combinatorics
Problem
Let n ě3 be an integer. An integer m ěn1 is called n-colourful if, given infinitely many marbles in each of n colours C1,C2,...,Cn, 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
Ci for each i “1,...,n.
Prove that there are only finitely many positive integers which are not n-colourful. Find
the largest among them.