Found problems: 233
2021 Junior Balkаn Mathematical Olympiad, 2
For any set $A = \{x_1, x_2, x_3, x_4, x_5\}$ of five distinct positive integers denote by $S_A$ the sum of its elements, and denote by $T_A$ the number of triples $(i, j, k)$ with $1 \le i < j < k \le 5$ for which $x_i + x_j + x_k$ divides $S_A$.
Find the largest possible value of $T_A$.
1962 Putnam, A6
Let $S$ be a set of rational numbers such that whenever $a$ and $b$ are members of $S$, so are $ab$ and $a+b$, and having the property that for every rational number $r$ exactly one of the following three statements is true:
$$r\in S,\;\; -r\in S,\;\;r =0.$$
Prove that $S$ is the set of all positive rational numbers.
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 $|a - b|\le 3$.
2021 Romania Team Selection Test, 1
Let $k>1$ be a positive integer. A set $S{}$ is called [i]good[/i] if there exists a colouring of the positive integers with $k{}$ colours, such that no element from $S{}$ can be written as the sum of two distinct positive integers having the same colour. Find the greatest positive integer $t{}$ (in terms of $k{}$) for which the set \[S=\{a+1,a+2,\ldots,a+t\}\]is good, for any positive integer $a{}$.
2021 Science ON all problems, 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]
2016 Indonesia TST, 3
Let $\{E_1, E_2, \dots, E_m\}$ be a collection of sets such that $E_i \subseteq X = \{1, 2, \dots, 100\}$, $E_i \neq X$, $i = 1, 2, \dots, m$. It is known that every two elements of $X$ is contained together in exactly one $E_i$ for some $i$. Determine the minimum value of $m$.
1979 VTRMC, 2
Let $S$ be a set which is closed under the binary operation $\circ$, with the following properties:
(i) there is an element $e \in S$ such that $a \circ e = e \circ a = a$, for each $a \in S$.
(ii) $(a \circ b) \circ (c \circ d)=(a \circ c) \circ (b \circ d)$, for all $a,b, c,d \in S$.
Prove or disprove:
(a) $\circ$ is associative on S
(b) $\circ$ is commutative on S
2015 Ukraine Team Selection Test, 7
Let $A$ and $B$ be two sets of real numbers. Suppose that the elements of the set $AB = \{ab: a\in A, b\in B\}$ form a finite arithmetic progression. Prove that one of these sets contains no more than three elements
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.
2021 EGMO, 1
The number 2021 is fantabulous. For any positive integer $m$, if any element of the set $\{m, 2m+1, 3m\}$ is fantabulous, then all the elements are fantabulous. Does it follow that the number $2021^{2021}$ is fantabulous?
2011 Philippine MO, 1
Find all nonempty finite sets $X$ of real numbers such that for all $x\in X$, $x+|x| \in X$.
2021 Korea - Final Round, P4
There are $n$($\ge 2$) clubs $A_1,A_2,...A_n$ in Korean Mathematical Society. Prove that there exist $n-1$ sets $B_1,B_2,...B_{n-1}$ that satisfy the condition below.
(1) $A_1\cup A_2\cup \cdots A_n=B_1\cup B_2\cup \cdots B_{n-1}$
(2) for any $1\le i<j\le n-1$, $B_i\cap B_j=\emptyset, -1\le\left\vert B_i \right\vert -\left\vert B_j \right\vert\le 1$
(3) for any $1\le i \le n-1$, there exist $A_k,A_j $($1\le k\le j\le n$)such that $B_i\subseteq A_k\cup A_j$
OMMC POTM, 2022 8
The positive integers are partitioned into two infinite sets so that the sum of any $2023$ distinct integers in one set is also in that set. Prove that one set contains all the odd positive integers, and one set contains all the even positive integers.
[i]Proposed by Evan Chang (squareman), USA[/i]
2025 Kosovo National Mathematical Olympiad`, P3
A subset $S$ of the natural numbers is called [i]dense [/i] for every $7$ consecutive natural numbers, at least $5$ of them are in $S$. Show that there exists a dense subset for which the equation $a^2+b^2=c^2$ has no solution for $a,b,c \in S$.
2011 German National Olympiad, 4
There are two points $A$ and $B$ in the plane.
a) Determine the set $M$ of all points $C$ in the plane for which $|AC|^2 +|BC|^2 = 2\cdot|AB|^2.$
b) Decide whether there is a point $C\in M$ such that $\angle ACB$ is maximal and if so, determine this angle.
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 $A$ ?
2017 China Team Selection Test, 3
Let $X$ be a set of $100$ elements. Find the smallest possible $n$ satisfying the following condition: Given a sequence of $n$ subsets of $X$, $A_1,A_2,\ldots,A_n$, there exists $1 \leq i < j < k \leq n$ such that
$$A_i \subseteq A_j \subseteq A_k \text{ or } A_i \supseteq A_j \supseteq A_k.$$
2015 Korea National Olympiad, 3
A positive integer $n$ is given. If there exists sets $F_1, F_2, \cdots F_m$ satisfying the following conditions, prove that $m \le n$. (For sets $A, B$, $|A|$ is the number of elements of $A$. $A-B$ is the set of elements that are in $A$ but not $B$. $\text{min}(x,y)$ is the number that is not larger than the other.)
(i): For all $1 \le i \le m$, $F_i \subseteq \{1,2,\cdots,n\}$
(ii): For all $1 \le i < j \le m$, $\text{min}(|F_i-F_j|,|F_j-F_i|) = 1$
2019 Brazil Team Selection Test, 2
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 $S$.
1996 Yugoslav Team Selection Test, Problem 1
Let $\mathfrak F=\{A_1,A_2,\ldots,A_n\}$ be a collection of subsets of the set $S=\{1,2,\ldots,n\}$ satisfying the following conditions:
(a) Any two distinct sets from $\mathfrak F$ have exactly one element in common;
(b) each element of $S$ is contained in exactly $k$ of the sets in $\mathfrak F$.
Can $n$ be equal to $1996$?
2019 Regional Olympiad of Mexico Southeast, 5
Let $n$ a natural number and $A=\{1, 2, 3, \cdots, 2^{n+1}-1\}$. Prove that if we choose $2n+1$ elements differents of the set $A$, then among them are three distinct number $a,b$ and $c$ such that
$$bc<2a^2<4bc$$
1959 AMC 12/AHSME, 14
Given the set $S$ whose elements are zero and the even integers, positive and negative. Of the five operations applied to any pair of elements: (1) addition (2) subtraction (3) multiplication (4) division (5) finding the arithmetic mean (average), those elements that only yield elements of $S$ are:
$ \textbf{(A)}\ \text{all} \qquad\textbf{(B)}\ 1,2,3,4\qquad\textbf{(C)}\ 1,2,3,5\qquad\textbf{(D)}\ 1,2,3\qquad\textbf{(E)}\ 1,3,5 $
2015 USAMO, 6
Consider $0<\lambda<1$, and let $A$ be a multiset of positive integers. Let $A_n=\{a\in A: a\leq n\}$. Assume that for every $n\in\mathbb{N}$, the set $A_n$ contains at most $n\lambda$ numbers. Show that there are infinitely many $n\in\mathbb{N}$ for which the sum of the elements in $A_n$ is at most $\frac{n(n+1)}{2}\lambda$. (A multiset is a set-like collection of elements in which order is ignored, but repetition of elements is allowed and multiplicity of elements is significant. For example, multisets $\{1, 2, 3\}$ and $\{2, 1, 3\}$ are equivalent, but $\{1, 1, 2, 3\}$ and $\{1, 2, 3\}$ differ.)
2024 Assara - South Russian Girl's MO, 8
Given a set $S$ of $2024$ natural numbers satisfying the following condition: if you select any $10$ (different) numbers from $S$, then you can select another number from $S$ so that the sum of all $11$ selected numbers is divisible by $10$. Prove that one of the numbers can be thrown out of $S$ so that the resulting set $S'$ of $2023$ numbers satisfies the condition: if you choose any $9$ (different) numbers from $S'$, then you can choose another number from $S'$ so that the sum of all $10$ selected numbers is divisible by $10$.
[i]K.A.Sukhov[/i]
2023 Germany Team Selection Test, 3
Let $A$ be a non-empty set of integers with the following property: For each $a \in A$, there exist not necessarily distinct integers $b,c \in A$ so that $a=b+c$.
(a) Proof that there are examples of sets $A$ fulfilling above property that do not contain $0$ as element.
(b) Proof that there exist $a_1,\ldots,a_r \in A$ with $r \ge 1$ and $a_1+\cdots+a_r=0$.
(c) Proof that there exist pairwise distinct $a_1,\ldots,a_r$ with $r \ge 1$ and $a_1+\cdots+a_r=0$.