Found problems: 175
1975 Czech and Slovak Olympiad III A, 6
Let $\mathbf M\subseteq\mathbb R^2$ be a set with the following properties:
1) there is a pair $(a,b)\in\mathbf M$ such that $ab(a-b)\neq0,$
2) if $\left(x_1,y_1\right),\left(x_2,y_2\right)\in\mathbf M$ and $c\in\mathbb R$ then also \[\left(cx_1,cy_1\right),\left(x_1+x_2,y_1+y_2\right),\left(x_1x_2,y_1y_2\right)\in\mathbf M.\]
Show that in fact \[\mathbf M=\mathbb R^2.\]
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 $L$.
2015 Ukraine Team Selection Test, 12
For a given natural $n$, we consider the set $A\subset \{1,2, ..., n\}$, which consists of at least $\left[\frac{n+1}{2}\right]$ items. Prove that for $n \ge 2015$ the set $A$ contains a three-element arithmetic sequence.
1998 Czech And Slovak Olympiad IIIA, 2
Given any set of $14$ (different) natural numbers, prove that for some $k$ ($1 \le k \le 7$) there exist two disjoint $k$-element subsets $\{a_1,...,a_k\}$ and $\{b_1,...,b_k\}$ such that $A =\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}$ and $B =\frac{1}{b_1}+\frac{1}{b_2}+...+\frac{1}{b_k}$ differ by less than $0.001$, i.e. $|A-B| < 0.001$
1978 Polish MO Finals, 4
Let $X$ be a set of $n$ elements. Prove that the sum of the numbers of elements of sets $A\cap B$, where $A$ and $B$ run over all subsets of $X$, is equal to $n4^{n-1}$.
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 $A$ ?
1991 Czech And Slovak Olympiad IIIA, 6
The set $N$ is partitioned into three subsets $A_1,A_2,A_3$.
Prove that at least one of them has the following property: There exists a positive number $m$ such that for any $k$ one can find numbers $a_1 < a_2 < ... < a_k$ in that subset satisfying $a_{j+1} -a_j \le m$ for $j = 1,...,k -1$.
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 $\{1,2,... ,n\}$.
2014 Czech-Polish-Slovak Junior Match, 1
The set of $\{1,2,3,...,63\}$ was divided into three non-empty disjoint sets $A,B$. Let $a,b,c$ be the product of all numbers in each set $A,B,C$ respectively and finally we have determined the greatest common divisor of these three products. What was the biggest result we could get?
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 $A_{k+2} = A_{k}.$
1999 Greece Junior Math Olympiad, 4
Defi ne alternate sum of a set of real numbers $A =\{a_1,a_2,...,a_k\}$ with $a_1 < a_2 <...< a_k$, the number
$S(A) = a_k - a_{k-1} + a_{k-2} - ... + (-1)^{k-1}a_1$ (for example if $A = \{1,2,5, 7\}$ then $S(A) = 7 - 5 + 2 - 1$)
Consider the alternate sums, of every subsets of $A = \{1, 2, 3, 4, 5, 6, 7, 8,9, 10\}$ and sum them.
What is the last digit of the sum obtained?
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
$$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$
2017 Federal Competition For Advanced Students, P2, 6
Let $S = \{1,2,..., 2017\}$.
Find the maximal $n$ with the property that there exist $n$ distinct subsets of $S$ such that for no two subsets their union equals $S$.
Proposed by Gerhard Woeginger
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.
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.
2001 Croatia Team Selection Test, 1
Consider $A = \{1, 2, ..., 16\}$. A partition of $A$ into nonempty sets $A_1, A_2,..., A_n$ is said to be good if none of the Ai contains elements $a, b, c$ (not necessarily distinct) such that $a = b + c$.
(a) Find a good partition $\{A_1, A_2, A_3, A_4\}$ of $A$.
(b) Prove that no partition $\{A_1, A_2, A_3\}$ of $A$ is good
1990 Romania Team Selection Test, 6
Prove that there are infinitely many n’s for which there exists a partition of $\{1,2,...,3n\}$ into subsets $\{a_1,...,a_n\}, \{b_1,...,b_n\}, \{c_1,...,c_n\}$ such that $a_i +b_i = c_i$ for all $i$, and prove that there are infinitely many $n$’s for which there is no such partition.
2015 Indonesia MO Shortlist, N6
Defined as $N_0$ as the set of all non-negative integers. Set $S \subset N_0$ with not so many elements is called beautiful if for every $a, b \in S$ with $a \ge b$ ($a$ and $b$ do not have to be different), exactly one of $a + b$ or $a - b$ is in $S$. Set $T \subset N_0$ with not so many elements is called charming if the largest number $k$ such that up to 3$^k | a$ is the same for each element $a \in T$. Prove that each beautiful set must be charming.
1991 Poland - Second Round, 5
$ P_1, P_2, \ldots, P_n $ are different two-element subsets of $ \{1,2,\ldots,n\} $. The sets $ P_i $, $ P_j $ for $ i\neq j $ have a common element if and only if the set $ \{i,j\} $ is one of the sets $ P_1, P_2, \ldots, P_n $. Prove that each of the numbers $ 1,2,\ldots,n $ is a common element of exactly two sets from $ P_1, P_2, \ldots, P_n $.
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 $p$.
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.$
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.
1987 Austrian-Polish Competition, 6
Let $C$ be a unit circle and $n \ge 1$ be a fixed integer. For any set $A$ of $n$ points $P_1,..., P_n$ on $C$ define $D(A) = \underset{d}{max}\, \underset{i}{min}\delta (P_i, d)$, where $d$ goes over all diameters of $C$ and $\delta (P, \ell)$ denotes the distance from point $P$ to line $\ell$. Let $F_n$ be the family of all such sets $A$. Determine $D_n = \underset{A\in F_n}{min} D(A)$ and describe all sets $A$ with $D(A) = D_n$.
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]
2012 India Regional Mathematical Olympiad, 6
Let $S$ be the set $\{1, 2, ..., 10\}$. Let $A$ be a subset of $S$.
We arrange the elements of $A$ in increasing order, that is, $A = \{a_1, a_2, ...., a_k\}$ with $a_1 < a_2 < ... < a_k$.
Define [i]WSUM [/i] for this subset as $3(a_1 + a_3 +..) + 2(a_2 + a_4 +...)$ where the first term contains the odd numbered terms and the second the even numbered terms.
(For example, if $A = \{2, 5, 7, 8\}$, [i]WSUM [/i] is $3(2 + 7) + 2(5 + 8)$.)
Find the sum of [i]WSUMs[/i] over all the subsets of S.
(Assume that WSUM for the null set is $0$.)