MathDB
sub-decks of cards, sum of cards divisible by n

Source: IMOC 2018 C6

August 16, 2021
combinatoricsnumber theory

Problem Statement

In a deck of cards, there are knkn cards numbered from 11 to nn and there are kk cards of each number. Now, divide this deck into kk sub-decks with equal sizes. Prove that if gcd(k,n)=1\gcd(k,n)=1, then one could always pick nn cards, one from each sub-deck, such that the sum of those cards is divisible by nn.