This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 2

2002 German National Olympiad, 6
Tags: coin , distributions , combinatorics
Theo Travel, who has $5$ children, has already visited $8$ countries of the eurozone. From every country, he brought $5$ not necessarily distinct coins home. Moreover, among these $40$ coins there are exactly $5$ of every value ($1,2,5,10,20,$ and $50$ ct, $1$ and $2$ euro). He wants to give each child $8$ coins such that they are from different countries and that each child gets the same amount of money. Is this always possible?

2019-IMOC, C1
Tags: partition , distributions , parity , combinatorics
Given a natural number $n$, if the tuple $(x_1,x_2,\ldots,x_k)$ satisfies $$2\mid x_1,x_2,\ldots,x_k$$ $$x_1+x_2+\ldots+x_k=n$$ then we say that it's an [i]even partition[/i]. We define [i]odd partition[/i] in a similar way. Determine all $n$ such that the number of even partitions is equal to the number of odd partitions.