Found problems: 233
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.
2013 Junior Balkan Team Selection Tests - Moldova, 2
Determine the elements of the sets $A = \{x \in N | x \ne 4a + 7b, a, b \in N\}$, $B = \{x \in N | x\ne 3a + 11b, a, b \in N\}$.
1991 French Mathematical Olympiad, Problem 4
Let $p$ be a nonnegative integer and let $n=2^p$. Consider all subsets $A$ of the set $\{1,2,\ldots,n\}$ with the property that, whenever $x\in A$, $2x\notin A$. Find the maximum number of elements that such a set $A$ can have.
2006 Lithuania National Olympiad, 4
Find the maximal cardinality $|S|$ of the subset $S \subset A=\{1, 2, 3, \dots, 9\}$ given that no two sums $a+b | a, b \in S, a \neq b$ are equal.
MathLinks Contest 6th, 4.1
Let $F$ be a family of n subsets of a set $K$ with $5$ elements, such that any two subsets in $F$ have a common element. Find the minimal value of $n$ such that no matter how we choose $F$ with the properties above, there exists exactly one element of $K$ which belongs to all the sets in $F$.
1994 Italy TST, 4
Let $X$ be a set of $n$ elements and $k$ be a positive integer.
Consider the family $S_k$ of all $k$-tuples $(E_1,...,E_k)$ with $E_i \subseteq X$ for each $i$.
Evaluate the sums $\sum_{(E_1,...,E_k) \in S_k }|E_1 \cap ... \cap E_k|$ and $\sum_{(E_1,...,E_k) \in S_k }|E_1 \cup ... \cup E_k|$
2021 Science ON grade VII, 4
Take $k\in \mathbb{Z}_{\ge 1}$ and the sets $A_1,A_2,\dots, A_k$ consisting of $x_1,x_2,\dots ,x_k$ positive integers, respectively. For any two sets $A$ and $B$, define $A+B=\{a+b~|~a\in A,~b\in B\}$.
Find the least and greatest number of elements the set $A_1+A_2+\dots +A_k$ may have.
[i] (Andrei Bâra)[/i]
2013 Danube Mathematical Competition, 4
Show that there exists a proper non-empty subset $S$ of the set of real numbers such that, for every real number $x$, the set $\{nx + S : n \in N\}$ is finite, where $nx + S =\{nx + s : s \in S\}$
1984 Putnam, B3
Prove or disprove the following statement: If $F$ is a finite set with two or more elements, then there exists a binary operation $*$ on $F$ such that for all $x,y,z$ in $F$,
$(\text i)$ $x*z=y*z$ implies $x=y$
$(\text{ii})$ $x*(y*z)\ne(x*y)*z$
2024 Israel TST, P3
For a set $S$ of at least $3$ points in the plane, let $d_{\text{min}}$ denote the minimal distance between two different points in $S$ and $d_{\text{max}}$ the maximal distance between two different points in $S$.
For a real $c>0$, a set $S$ will be called $c$-[i]balanced[/i] if
\[\frac{d_{\text{max}}}{d_{\text{min}}}\leq c|S|\]
Prove that there exists a real $c>0$ so that for every $c$-balanced set of points $S$, there exists a triangle with vertices in $S$ that contains at least $\sqrt{|S|}$ elements of $S$ in its interior or on its boundary.
2021 Science ON all problems, 1
Supoose $A$ is a set of integers which contains all integers that can be written as $2^a-2^b$, $a,b\in \mathbb{Z}_{\ge 1}$ and also has the property that $a+b\in A$ whenever $a,b\in A$. Prove that if $A$ contains at least an odd number, then $A=\mathbb{Z}$.
[i] (Andrei Bâra)[/i]
2018 IMO Shortlist, A3
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 $S$.
Russian TST 2016, P2
A family of sets $F$ is called perfect if the following condition holds: For every triple of sets $X_1, X_2, X_3\in F$, at least one of the sets $$ (X_1\setminus X_2)\cap X_3,$$ $$(X_2\setminus X_1)\cap X_3$$ is empty. Show that if $F$ is a perfect family consisting of some subsets of a given finite set $U$, then $\left\lvert F\right\rvert\le\left\lvert U\right\rvert+1$.
[i]Proposed by Michał Pilipczuk[/i]
2020 Taiwan APMO Preliminary, P5
Let $S$ is the set of permutation of {1,2,3,4,5,6,7,8}
(1)For all $\sigma=\sigma_1\sigma_2...\sigma_8\in S$
Evaluate the sum of S=$\sigma_1\sigma_2+\sigma_3\sigma_4+\sigma_5\sigma_6+\sigma_7\sigma_8$. Then for all elements in $S$,what is the arithmetic mean of S?
(Notice $S$ and S are different.)
(2)In $S$, how many permutations are there which satisfies "For all $k=1,2,...,7$,the digit after k is [b]not[/b] (k+1)"?
2007 QEDMO 4th, 11
Let $S_{1},$ $S_{2},$ $...,$ $S_{n}$ be finitely many subsets of $\mathbb{N}$ such that $S_{1}\cup S_{2}\cup...\cup S_{n}=\mathbb{N}.$ Prove that there exists some $k\in\left\{ 1,2,...,n\right\} $ such that for each positive integer $m,$ the set $S_{k}$ contains infinitely many multiples of $m.$
2017 Costa Rica - Final Round, LR2
There is a set of $17$ consecutive positive integers. Let $m$ be the smallest of these numbers. Determine for which values of $m$ the set can be divided into three subsets disjoint, such that the sum of the elements of each subset is the same.
2023 Israel National Olympiad, P6
Determine if there exists a set $S$ of $5783$ different real numbers with the following property:
For every $a,b\in S$ (not necessarily distinct) there are $c\neq d$ in $S$ so that $a\cdot b=c+d$.
2024 Brazil Cono Sur TST, 3
For a pair of integers $a$ and $b$, with $0<a<b<1000$, a set $S\subset \begin{Bmatrix}1,2,3,...,2024\end{Bmatrix}$ $escapes$ the pair $(a,b)$ if for any elements $s_1,s_2\in S$ we have $\left|s_1-s_2\right| \notin \begin{Bmatrix}a,b\end{Bmatrix}$. Let $f(a,b)$ be the greatest possible number of elements of a set that escapes the pair $(a,b)$. Find the maximum and minimum values of $f$.
2018 Brazil Undergrad MO, 4
Consider the property that each a element of a group $G$ satisfies $a ^ 2 = e$, where e is the identity element of the group. Which of the following statements is not always valid for a
group $G$ with this property?
(a) $G$ is commutative
(b) $G$ has infinite or even order
(c) $G$ is Noetherian
(d) $G$ is vector space over $\mathbb{Z}_2$
2019 India PRMO, 30
Let $E$ denote the set of all natural numbers $n$ such that $3 < n < 100$ and the set $\{ 1, 2, 3, \ldots , n\}$ can be partitioned in to $3$ subsets with equal sums. Find the number of elements of $E$.
2014 Rioplatense Mathematical Olympiad, Level 3, 1
Let $n \ge 3$ be a positive integer. Determine, in terms of $n$, how many triples of sets $(A,B,C)$ satisfy the conditions:
$\bullet$ $A, B$ and $C$ are pairwise disjoint , that is, $A \cap B = A \cap C= B \cap C= \emptyset$.
$\bullet$ $A \cup B \cup C= \{ 1 , 2 , ... , n \}$.
$\bullet$ The sum of the elements of $A$, the sum of the elements of $B$ and the sum of the elements of $C$ leave the same remainder when divided by $3$.
Note: One or more of the sets may be empty.
2007 Rioplatense Mathematical Olympiad, Level 3, 6
Let $n > 2$ be a natural number. A subset $A$ of $R$ is said $n$-[i]small [/i]if there exist $n$ real numbers $t_1 , t_2 , ..., t_n$ such that the sets $t_1 + A , t_2 + A ,... , t_n + A$ are different . Show that $R$ can not be represented as a union of $ n - 1$ $n$-[i]small [/i] sets .
Notation : if $r \in R$ and $B \subset R$ , then $r + B = \{ r + b | b \in B\}$ .
2014 Contests, 4
Givan the set $S = \{1,2,3,....,n\}$. We want to partition the set $S$ into three subsets $A,B,C$ disjoint (to each other) with $A\cup B\cup C=S$ , such that the sums of their elements $S_{A} S_{B} S_{C}$ to be equal .Examine if this is possible when:
a) $n=2014$
b) $n=2015 $
c) $n=2018$
2015 Romania Team Selection Tests, 3
Given a positive real number $t$ , determine the sets $A$ of real numbers containing $t$ , for which there exists a set $B$ of real numbers depending on $A$ , $|B| \geq 4$ , such that the elements of the set $AB =\{ ab \mid a\in A , b \in B \}$ form a finite arithmetic progression .
2018 IMAR Test, 3
Let $S$ be a finite set and let $\mathcal{P}(S)$ be its power set, i.e., the set of all subsets of $S$, the empty set and $S$, inclusive. If $\mathcal{A}$ and $\mathcal{B}$ are non-empty subsets of $\mathcal{P}(S),$ let \[\mathcal{A}\vee \mathcal{B}=\{X:X\subseteq A\cup B,A\in\mathcal{A},B\in\mathcal{B}\}.\] Given a non-negative integer $n\leqslant |S|,$ determine the minimal size $\mathcal{A}\vee \mathcal{B}$ may have, where $\mathcal{A}$ and $\mathcal{B}$ are non-empty subsets of $\mathcal{P}(S)$ such that $|\mathcal{A}|+|\mathcal{B}|>2^n$.
[i]Amer. Math. Monthly[/i]