Found problems: 233
2007 Dutch Mathematical Olympiad, 2
Is it possible to partition the set $A = \{1, 2, 3, ... , 32, 33\}$ into eleven subsets that contain three integers each, such that for every one of these eleven subsets, one of the integers is equal to the sum of the other two? If so, give such a partition, if not, prove that such a partition cannot exist.
2011 JBMO Shortlist, 5
A set $S$ of natural numbers is called [i]good[/i], if for each element $x \in S, x$ does not divide the sum of the remaining numbers in $S$. Find the maximal possible number of elements of a [i]good [/i]set which is a subset of the set $A = \{1,2, 3, ...,63\}$.
2008 Bulgarian Autumn Math Competition, Problem 12.4
Veni writes down finitely many real numbers (possibly one), squares them, and then subtracts 1 from each of them and gets the same set of numbers as in the beginning. What were the starting numbers?
2018 Israel National Olympiad, 7
A [i]uniform covering[/i] of the integers $1,2,...,n$ is a finite multiset of subsets of $\{1,2,...,n\}$, so that each number lies in the same amount of sets from the covering. A covering may contain the same subset multiple times, it must contain at least one subset, and it may contain the empty subset. For example, $(\{1\},\{1\},\{2,3\},\{3,4\},\{2,4\})$ is a uniform covering of $1,2,3,4$ (every number occurs in two sets). The covering containing only the empty set is also uniform (every number occurs in zero sets).
Given two uniform coverings, we define a new uniform covering, their [i]sum[/i] (denoted by $\oplus$), by adding the sets from both coverings. For example:
$(\{1\},\{1\},\{2,3\},\{3,4\},\{2,4\})\oplus(\{1\},\{2\},\{3\},\{4\})=$
$(\{1\},\{1\},\{1\},\{2\},\{3\},\{4\},\{2,3\},\{3,4\},\{2,4\})$
A uniform covering is called [i]non-composite[/i] if it's not a sum of two uniform coverings.
Prove that for any $n\geq1$, there are only finitely many non-composite uniform coverings of $1,2,...,n$.
1997 Bosnia and Herzegovina Team Selection Test, 6
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 $m$
2020 Thailand TSTST, 6
A nonempty set $S$ is called [i]Bally[/i] if for every $m\in S$, there are fewer than $\frac{1}{2}m$ elements of $S$ which are less than $m$. Determine the number of Bally subsets of $\{1, 2, . . . , 2020\}$.
Russian TST 2019, P2
Given any set $S$ of positive integers, show that at least one of the following two assertions holds:
(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$;
(2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.
2021 Romania Team Selection Test, 2
Consider the set $M=\{1,2,3,...,2020\}.$ Find the smallest positive integer $k$ such that for any subset $A$ of $M$ with $k$ elements, there exist $3$ distinct numbers $a,b,c$ from $M$ such that $a+b, b+c$ and $c+a$ are all in $A.$