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

Tags were heavily modified to better represent problems.

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Found problems: 27

1999 Tournament Of Towns, 3
Tags: partition , Sum , combinatorics , Subset
(a) The numbers $1, 2,... , 100$ are divided into two groups so that the sum of all numbers in one group is equal to that in the other. Prove that one can remove two numbers from each group so that the sums of all numbers in each group are still the same. (b) The numbers $1, 2 , ... , n$ are divided into two groups so that the sum of all numbers in one group is equal to that in the other . Is it true that for every such$ n > 4$ one can remove two numbers from each group so that the sums of all numbers in each group are still the same? (A Shapovalov) [(a) for Juniors, (a)+(b) for Seniors]

1997 Bosnia and Herzegovina Team Selection Test, 6
Tags: Sets , Subset , number theory
Let $k$, $m$ and $n$ be integers such that $1<n \leq m-1 \leq k$. Find maximum size of subset $S$ of set $\{1,2,...,k\}$ such that sum of any $n$ different elements from $S$ is not: $a)$ equal to $m$, $b)$ exceeding $m$