Found problems: 175
1988 Tournament Of Towns, (177) 3
The set of all $10$-digit numbers may be represented as a union of two subsets: the subset $M$ consisting of all $10$-digit numbers, each of which may be represented as a product of two $5$-digit numbers, and the subset $N$ , containing the remaining $10$-digit numbers . Which of the sets $M$ and $N$ contains more elements?
(S. Fomin , Leningrad)
1961 Putnam, A5
Let $\Omega$ be a set of $n$ points, where $n>2$. Let $\Sigma$ be a nonempty subcollection of the $2^n$ subsets of $\Omega$ that is closed with respect to the unions, intersections and complements. If $k$ is the number of elements of $\Sigma,$ what are the possible values of $k?$
1969 Vietnam National Olympiad, 1
A graph $G$ has $n + k$ vertices. Let $A$ be a subset of $n$ vertices of the graph $G$, and $B$ be a subset of other $k$ vertices. Each vertex of $A$ is joined to at least $k - p$ vertices of $B$. Prove that if $np < k$ then there is a vertex in $B$ that can be joined to all vertices of $A$.
2022 New Zealand MO, 3
Let $S$ be a set of $10$ positive integers. Prove that one can find two disjoint subsets $A =\{a_1, ..., a_k\}$ and $B = \{b_1, ... , b_k\}$ of $S$ with $|A| = |B|$ such that the sums $x =\frac{1}{a_1}+ ... +\frac{1}{a_k}$ and $y =\frac{1}{b_1}+ ... +\frac{1}{b_k}$ differ by less than $0.01$, i.e., $|x - y| < 1/100$.
2022 Bulgarian Spring Math Competition, Problem 11.4
Let $n \geq 2$ be a positive integer. The set $M$ consists of $2n^2-3n+2$ positive rational numbers. Prove that there exists a subset $A$ of $M$ with $n$ elements with the following property: $\forall$ $2 \leq k \leq n$ the sum of any $k$ (not necessarily distinct) numbers from $A$ is not in $A$.
2014 India PRMO, 20
What is the number of ordered pairs $(A,B)$ where $A$ and $B$ are subsets of $\{1,2,..., 5\}$ such that neither $A \subseteq B$ nor $B \subseteq A$?
2020 Malaysia IMONST 1, 19
A set $S$ has $7$ elements. Several $3$-elements subsets of $S$ are listed, such
that any $2$ listed subsets have exactly $1$ common element. What is the maximum number of subsets that can be listed?
1986 Czech And Slovak Olympiad IIIA, 6
Assume that $M \subset N$ has the property that every two numbers $m,n$ of $M$ satisfy $|m-n| \ge mn/25$.
Prove that the set $M$ contains no more than $9$ elements.
Decide whether there exists such set M.
1999 Abels Math Contest (Norwegian MO), 4
For every nonempty subset $R$ of $S = \{1,2,...,10\}$, we define the alternating sum $A(R)$ as follows:
If $r_1,r_2,...,r_k$ are the elements of $R$ in the increasing order, then $A(R) = r_k -r_{k-1} +r_{k-2}- ... +(-1)^{k-1}r_1$.
(a) Is it possible to partition $S$ into two sets having the same alternating sum?
(b) Determine the sum $\sum_{R} A(R)$, where $R$ runs over all nonempty subsets of $S$.
2016 Saudi Arabia IMO TST, 3
Given two positive integers $r > s$, and let $F$ be an infinite family of sets, each of size $r$, no two of which share fewer than $s$ elements. Prove that there exists a set of size $r -1$ that shares at least $s$ elements with each set in $F$.
1972 Czech and Slovak Olympiad III A, 5
Determine how many unordered pairs $\{A,B\}$ is there such that $A,B\subseteq\{1,\ldots,n\}$ and $A\cap B=\emptyset.$
2008 IMAC Arhimede, 6
Consider the set of natural numbers $ U = \{1,2,3, ..., 6024 \} $ Prove that for any partition of the $ U $ in three subsets with $ 2008 $ elements each, we can choose a number in each subset so that one of the numbers is the sum of the other two numbers.
2010 Benelux, 1
A finite set of integers is called [i]bad[/i] if its elements add up to $2010$. A finite set of integers is a [i]Benelux-set[/i] if none of its subsets is bad. Determine the smallest positive integer $n$ such that the set $\{502, 503, 504, . . . , 2009\}$ can be partitioned into $n$ Benelux-sets.
(A partition of a set $S$ into $n$ subsets is a collection of $n$ pairwise disjoint subsets of $S$, the union of which equals $S$.)
[i](2nd Benelux Mathematical Olympiad 2010, Problem 1)[/i]
2009 Ukraine Team Selection Test, 3
Let $S$ be a set consisting of $n$ elements, $F$ a set of subsets of $S$ consisting of $2^{n-1}$ subsets such that every three such subsets have a non-empty intersection.
a) Show that the intersection of all subsets of $F$ is not empty.
b) If you replace the number of sets from $2^{n-1}$ with $2^{n-1}-1$, will the previous answer change?
2008 Indonesia TST, 1
Let $A$ be the subset of $\{1, 2, ..., 16\}$ that has $6$ elements. Prove that there exist $2$ subsets of $A$ that are disjoint, and the sum of their elements are the same.
2023 Iran Team Selection Test, 6
Suppose that we have $2n$ non-empty subset of $ \big\{0,1,2,...,2n-1\big\} $ that sum of the elements of these subsets is $ \binom{2n+1}{2}$ . Prove that we can choose one element from every subset that some of them is $ \binom{2n}{2}$
[i]Proposed by Morteza Saghafian and Afrouz Jabalameli [/i]
2016 Saudi Arabia GMO TST, 2
Let $n \ge 1$ be a fixed positive integer. We consider all the sets $S$ which consist of sub-sequences of the sequence $0, 1,2, ..., n$ satisfying the following conditions:
i) If $(a_i)_{i=0}^k$ belongs to $S$, then $a_0 = 0$, $a_k = n$ and $a_{i+1} - a_i \le 2$ for all $0 \le i \le k - 1$.
ii) If $(a_i)_{i=0}^k$ and $(b_j)^h_{j=0}$ both belong to $S$, then there exist $0 \le i_0 \le k - 1$ and $0 \le j_0 \le h - 1$ such that $a_{i_0} = b_{j_0}$ and $a_{i_0+1} = b_{j_0+1}$.
Find the maximum value of $|S|$ (among all the above-mentioned sets $S$).
2018 India PRMO, 22
A positive integer $k$ is said to be [i]good [/i] if there exists a partition of $ \{1, 2, 3,..., 20\}$ into disjoint proper subsets such that the sum of the numbers in each subset of the partition is $k$. How many [i]good [/i] numbers are there?
1993 IMO Shortlist, 4
Show that for any finite set $S$ of distinct positive integers, we can find a set $T \supseteq S$ such that every member of $T$ divides the sum of all the members of $T$.
[b]Original Statement:[/b]
A finite set of (distinct) positive integers is called a [b]DS-set[/b] if each of the integers divides the sum of them all. Prove that every finite set of positive integers is a subset of some [b]DS-set[/b].
2017 China Team Selection Test, 3
Suppose $S=\{1,2,3,...,2017\}$,for every subset $A$ of $S$,define a real number $f(A)\geq 0$ such that:
$(1)$ For any $A,B\subset S$,$f(A\cup B)+f(A\cap B)\leq f(A)+f(B)$;
$(2)$ For any $A\subset B\subset S$, $f(A)\leq f(B)$;
$(3)$ For any $k,j\in S$,$$f(\{1,2,\ldots,k+1\})\geq f(\{1,2,\ldots,k\}\cup \{j\});$$
$(4)$ For the empty set $\varnothing$, $f(\varnothing)=0$.
Confirm that for any three-element subset $T$ of $S$,the inequality $$f(T)\leq \frac{27}{19}f(\{1,2,3\})$$ holds.
1969 IMO Longlists, 42
$(MON 3)$ Let $A_k (1 \le k \le h)$ be $n-$element sets such that each two of them have a nonempty intersection. Let $A$ be the union of all the sets $A_k,$ and let $B$ be a subset of $A$ such that for each $k (1\le k \le h)$ the intersection of $A_k$ and $B$ consists of exactly two different elements $a_k$ and $b_k$. Find all subsets $X$ of the set $A$ with $r$ elements satisfying the condition that for at least one index $k,$ both elements $a_k$ and $b_k$ belong to $X$.
1993 Romania Team Selection Test, 3
Show that the set $\{1,2,....,2^n\}$ can be partitioned in two classes, none of which contains an arithmetic progression of length $2n$.
2011 Korea Junior Math Olympiad, 8
There are $n$ students each having $r$ positive integers. Their $nr$ positive integers are all different. Prove that we can divide the students into $k$ classes satisfying the following conditions:
(a) $k \le 4r$
(b) If a student $A$ has the number $m$, then the student $B$ in the same class can't have a number $\ell$ such that $(m - 1)! < \ell < (m + 1)! + 1$
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 $m$
2008 Indonesia TST, 1
Let $A$ be the subset of $\{1, 2, ..., 16\}$ that has $6$ elements. Prove that there exist $2$ subsets of $A$ that are disjoint, and the sum of their elements are the same.