Found problems: 175
2005 National Olympiad First Round, 24
There are $20$ people in a certain community. $10$ of them speak English, $10$ of them speak German, and $10$ of them speak French. We call a [i]committee[/i] to a $3$-subset of this community if there is at least one who speaks English, at least one who speaks German, and at least one who speaks French in this subset. At most how many commitees are there in this community?
$
\textbf{(A)}\ 120
\qquad\textbf{(B)}\ 380
\qquad\textbf{(C)}\ 570
\qquad\textbf{(D)}\ 1020
\qquad\textbf{(E)}\ 1140
$
2021 Indonesia TST, C
Let $p$ be an odd prime. Determine the number of nonempty subsets from $\{1, 2, \dots, p - 1\}$ for which the sum of its elements is divisible by $p$.
1995 Austrian-Polish Competition, 2
Let $X= \{A_1, A_2, A_3, A_4\}$ be a set of four distinct points in the plane. Show that there exists a subset $Y$ of $X$ with the property that there is no (closed) disk $K$ such that $K\cap X = Y$.
1977 Chisinau City MO, 136
We represent the number line $R$ as the union of two non-empty sets $A, B$ different from $R$. Prove that one of the sets $A, B$ does not have the following property: the difference of any elements of the set belongs to the same set.
1990 Czech and Slovak Olympiad III A, 6
Let $k\ge 1$ be an integer and $\mathsf S$ be a family of 2-element subsets of the index set $\{1,\ldots,2k\}$ with the following property: if $\mathsf M_1,\ldots,\mathsf M_{2k}$ are arbitrary sets such that \[\mathsf M_i\cap\mathsf M_j\neq\emptyset\quad\Leftrightarrow\quad\{i,j\}\in\mathsf S,\] then the union $\mathsf M_1\cup\ldots\cup\mathsf M_{2k}$ contains at least $k^2$ elements. Show that there is a suitable family $\mathsf S$ for any integer $k\ge1.$
1993 IMO Shortlist, 4
Let $n \geq 2, n \in \mathbb{N}$ and $A_0 = (a_{01},a_{02}, \ldots, a_{0n})$ be any $n-$tuple of natural numbers, such that $0 \leq a_{0i} \leq i-1,$ for $i = 1, \ldots, n.$
$n-$tuples $A_1= (a_{11},a_{12}, \ldots, a_{1n}), A_2 = (a_{21},a_{22}, \ldots, a_{2n}), \ldots$ are defined by: $a_{i+1,j} = Card \{a_{i,l}| 1 \leq l \leq j-1, a_{i,l} \geq a_{i,j}\},$ for $i \in \mathbb{N}$ and $j = 1, \ldots, n.$ Prove that there exists $k \in \mathbb{N},$ such that $A_{k+2} = A_{k}.$
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?
2001 Bosnia and Herzegovina Team Selection Test, 3
Find maximal value of positive integer $n$ such that there exists subset of $S=\{1,2,...,2001\}$ with $n$ elements, such that equation $y=2x$ does not have solutions in set $S \times S$
2005 Czech And Slovak Olympiad III A, 2
Determine for which $m$ there exist exactly $2^{15}$ subsets $X$ of $\{1,2,...,47\}$ with the following property: $m$ is the smallest element of $X$, and for every $x \in X$, either $x+m \in X$ or $x+m > 47$.
2025 Romania EGMO TST, P2
Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that
$$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$
2012 India Regional Mathematical Olympiad, 4
Let $X=\{1,2,3,...,12\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{2,3,5,7,8\}$.
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.
1999 Tournament Of Towns, 4
Is it possible to divide the integers from $1$ to $100$ inclusive into $50$ pairs such that for $1\le k\le 50$, the difference between the two numbers in the $k$-th pair is $k$?
(V Proizvolov)
2020 Iran MO (3rd Round), 4
What is the maximum number of subsets of size $5$, taken from the set $A=\{1,2,3,...,20\}$ such that any $2$ of them share exactly $1$ element.
1998 Singapore Team Selection Test, 2
Let $n \ge 2$ be an integer. Let $S$ be a set of $n$ elements and let $A_i, 1 \le i \le m$, be distinct subsets of $S$ of size at least $2$ such that $A_i \cap A_j \ne \emptyset$, $A_i \cap A_k \ne \emptyset$, $A_j \cap A_k \ne \emptyset$ imply $A_i \cap A_j \cap A_k \ne \emptyset$. Show that $m \le 2^{n-1}$ -
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].
1985 Czech And Slovak Olympiad IIIA, 2
Let $A_1, A_2, A_3$ be nonempty sets of integers such that for $\{i, j, k\} = \{1, 2, 3\}$ holds
$$(x \in A_i, y\in A_j) \Rightarrow (x + y \in A_k, x - y \in A_k).$$
Prove that at least two of the sets $A_1, A_2, A_3$ are equal. Can any of these sets be disjoint?
2018 Regional Competition For Advanced Students, 3
Let $n \ge 3$ be a natural number.
Determine the number $a_n$ of all subsets of $\{1, 2,...,n\}$ consisting of three elements such that one of them is the arithmetic mean of the other two.
[i]Proposed by Walther Janous[/i]
2001 Switzerland Team Selection Test, 10
Prove that every $1000$-element subset $M$ of the set $\{0,1,...,2001\}$ contains either a power of two or two distinct numbers whose sum is a power of two.
1992 IMO Longlists, 75
A sequence $\{an\}$ of positive integers is defined by
\[a_n=\left[ n +\sqrt n + \frac 12 \right] , \qquad \forall n \in \mathbb N\]
Determine the positive integers that occur in the sequence.
2020 Canadian Mathematical Olympiad Qualification, 2
Given a set $S$, of integers, an [i]optimal partition[/i] of S into sets T, U is a partition which minimizes the value $|t - u|$, where $t$ and $u$ are the sum of the elements of $T$ and U respectively.
Let $P$ be a set of distinct positive integers such that the sum of the elements of $P$ is $2k$ for a positive integer $k$, and no subset of $P$ sums to $k$.
Either show that there exists such a $P$ with at least $2020$ different optimal partitions, or show that such a $P$ does not exist.
1969 IMO Shortlist, 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 $X$.
2016 Latvia Baltic Way TST, 9
The numbers from$ 1$ to $2016$ are divided into three (disjoint) subsets $A, B$ and $C$, each one contains exactly $672$ numbers. Prove that you can find three numbers, each from a different subset, such that the sum of two of them is equal to the third.
[hide=original wording]Skaitļi no 1 līdz 2016 ir sadalīti trīs (nešķeļošās) apakškopās A, B un C, katranotām satur tieši 672 skaitļus. Pierādīt, ka var atrast trīs tādus skaitļus, katru no citas apakškopas, ka divu no tiem summa ir vienāda ar trešo.
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1985 Spain Mathematical Olympiad, 2
Determine if there exists a subset $E$ of $Z \times Z$ with the properties:
(i) $E$ is closed under addition,
(ii) $E$ contains $(0,0),$
(iii) For every $(a,b) \ne (0,0), E$ contains exactly one of $(a,b)$ and $-(a,b)$.
Remark: We define $(a,b)+(a',b') = (a+a',b+b')$ and $-(a,b) = (-a,-b)$.
2018 JBMO Shortlist, C1
A set $S$ is called [i]neighbouring [/i] if it has the following two properties:
a) $S$ has exactly four elements
b) for every element $x$ of $S$, at least one of the numbers $x - 1$ or $x+1$ belongs to $S$.
Find the number of all [i]neighbouring [/i] subsets of the set $\{1,2,... ,n\}$.