Found problems: 175
2004 Korea Junior Math Olympiad, 2
For $n\geq3$ define $S_n=\{1, 2, ..., n\}$. $A_1, A_{2}, ..., A_{n}$ are given subsets of $S_n$, each having an even number of elements. Prove that there exists a set $\{i_1, i_2, ..., i_t\}$, a nonempty subset of $S_n$ such that
$$A_{i_1} \Delta A_{i_2} \Delta \ldots \Delta A_{i_t}=\emptyset$$
(For two sets $A, B$, we define $\Delta$ as $A \Delta B=(A\cup B)-(A\cap B)$)
2012 Ukraine Team Selection Test, 12
We shall call the triplet of numbers $a, b, c$ of the interval $[-1,1]$ [i]qualitative [/i] if these numbers satisfy the inequality $1 + 2abc\ge a^2 + b^2 + c^2$. Prove that when the triples $a, b, c$, and $x, y, z$ are qualitative, then $ax, by, cz$ is also qualitative.
1994 Bulgaria National Olympiad, 6
Let $n$ be a positive integer and $A$ be a family of subsets of the set $\{1,2,...,n\},$ none of which contains another subset from A . Find the largest possible cardinality of $A$ .
1999 Tournament Of Towns, 3
(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]
2024 Indonesia TST, C
Let $A$ be a set with $1000$ members and $\mathcal F =${$A_1,A_2,\ldots,A_n$} a family of subsets of A such that
(a) Each element in $\mathcal F$ consists of 3 members
(b) For every five elements in $\mathcal F$, the union of them all will have at least $12$ members
Find the largest value of $n$
2007 Thailand Mathematical Olympiad, 13
Let $S = \{1, 2,..., 8\}$. How many ways are there to select two disjoint subsets of $S$?
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 $m$
2018 JBMO Shortlist, A7
Let $A$ be a set of positive integers satisfying the following :
$a.)$ If $n \in A$ , then $n \le 2018$.
$b.)$ If $S \subset A$ such that $|S|=3$, then there exists $m,n \in S$ such that $|n-m| \ge \sqrt{n}+\sqrt{m}$
What is the maximum cardinality of $A$ ?
1954 Moscow Mathematical Olympiad, 286
Consider the set of all $10$-digit numbers expressible with the help of figures $1$ and $2$ only. Divide it into two subsets so that the sum of any two numbers of the same subset is a number which is written with not less than two $3$’s.
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 $(m - 1)! < \ell < (m + 1)! + 1$
2015 Indonesia MO Shortlist, C7
Show that there is a subset of $A$ from $\{1,2, 3,... , 2014\}$ such that :
(i) $|A| = 12$
(ii) for each coloring number in $A$ with red or white , we can always find some numbers colored in $A$ whose sum is $2015$.
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.
2004 Junior Tuymaada Olympiad, 4
Given the disjoint finite sets of natural numbers $ A $ and $ B $, consisting of $ n $ and $ m $ elements, respectively. It is known that every natural number belonging to $ A $ or $ B $ satisfies at least one of the conditions $ k + 17 \in A $, $ k-31 \in B $. Prove that $ 17n = 31m $
1956 Putnam, B2
Suppose that each set $X$ of points in the plane has an associated set $\overline{X}$ of points called its cover. Suppose further that (1) $\overline{X\cup Y} \supset \overline{\overline{X}} \cup \overline{Y} \cup Y$ for all sets $X,Y$ . Show that i) $\overline{X} \supset X$, ii) $\overline{\overline{X}}=\overline{X}$ and iii) $X\supset Y \Rightarrow \overline{X} \supset \overline{Y}.$ Prove also that these three statements imply (1).
2016 India PRMO, 14
Find the minimum value of $m$ such that any $m$-element subset of the set of integers $\{1,2,...,2016\}$ contains at least two distinct numbers $a$ and $b$ which satisfy $|a - b|\le 3$.
2015 Saudi Arabia BMO TST, 2
Given $2015$ subsets $A_1, A_2,...,A_{2015}$ of the set $\{1, 2,..., 1000\}$ such that $|A_i| \ge 2$ for every $i \ge 1$ and $|A_i \cap A_j| \ge 1$ for every $1 \le i < j \le 2015$. Prove that $k = 3$ is the smallest number of colors such that we can always color the elements of the set $\{1, 2,..., 1000\}$ by $k$ colors with the property that the subset $A_i$ has at least two elements of different colors for every $i \ge 1$.
Lê Anh Vinh
2012 Danube Mathematical Competition, 4
Let $A$ be a subset with seven elements of the set $\{1,2,3, ...,26\}$.
Show that there are two distinct elements of $A$, having the same sum of their elements.
1991 All Soviet Union Mathematical Olympiad, 556
$X$ is a set with $100$ members. What is the smallest number of subsets of $X$ such that every pair of elements belongs to at least one subset and no subset has more than $50$ members? What is the smallest number if we also require that the union of any two subsets has at most $80$ members?
2018 Brazil Team Selection Test, 4
Given a set $S$ of positive real numbers, let $$\Sigma (S) = \Bigg\{ \sum_{x \in A} x : \emptyset \neq A \subset S \Bigg\}.$$
be the set of all the sums of elements of non-empty subsets of $S$. Find the least constant $L> 0$ with the following property: for every integer greater than $1$ and every set $S$ of $n$ positive real numbers, it is possible partition $\Sigma(S)$ into $n$ subsets $\Sigma_1,\ldots, \Sigma_n$ so that the ratio between the largest and smallest element of each $\Sigma_i$ is at most $L$.
2002 Switzerland Team Selection Test, 10
Given an integer $m\ge 2$, find the smallest integer $k > m$ such that for any partition of the set $\{m,m + 1,..,k\}$ into two classes $A$ and $B$ at least one of the classes contains three numbers $a,b,c$ (not necessarily distinct) such that $a^b = c$.
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 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 $A$.
1996 IMO Shortlist, 3
Let $ k,m,n$ be integers such that $ 1 < n \leq m \minus{} 1 \leq k.$ Determine the maximum size of a subset $ S$ of the set $ \{1,2,3, \ldots, k\minus{}1,k\}$ such that no $ n$ distinct elements of $ S$ add up to $ m.$
2025 Kyiv City MO Round 2, Problem 3
In a school, \( n \) different languages are taught. It is known that for any subset of these languages (including the empty set), there is exactly one student who knows these and only these languages (there are \( 2^n \) students in total). Each day, the students are divided into pairs and teach each other the languages that only one of them knows. If students are not allowed to be in the same pair twice, what is the minimum number of days the school administration needs to guarantee that all their students know all \( n \) languages?
[i]Proposed by Oleksii Masalitin[/i]
2001 Abels Math Contest (Norwegian MO), 2
Let $A$ be a set, and let $P (A)$ be the powerset of all non-empty subsets of $A$. (For example, $A = \{1,2,3\}$, then $P (A) = \{\{1\},\{2\} ,\{3\},\{1,2\}, \{1,3\},\{2,3\}, \{1,2,3\}\}$.)
A subset $F$ of P $(A)$ is called [i]strong [/i] if the following is true:
If $B_1$ and $B_2$ are elements of $F$, then $B_1 \cup B_2$ is also an element of $F$.
Suppose that $F$ and $G$ are strong subsets of $P (A)$.
a) Is the union $F \cup G$ necessarily strong?
b) Is the intersection $F \cap G$ necessarily strong?