Found problems: 295
2008 Korea Junior Math Olympiad, 8
There are $12$ members in a club. The members created some small groups, which satisfy the following:
- The small group consists of $3$ or $4$ people.
- Also, for two arbitrary members, there exists exactly one small group that has both members.
Prove that all members are in the same number of small groups.
2001 VJIMC, Problem 1
Let $A$ be a set of positive integers such that for any $x,y\in A$,
$$x>y\implies x-y\ge\frac{xy}{25}.$$Find the maximal possible number of elements of the set $A$.
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]
2015 Bosnia Herzegovina Team Selection Test, 5
Let $N$ be a positive integer. It is given set of weights which satisfies following conditions:
$i)$ Every weight from set has some weight from $1,2,...,N$;
$ii)$ For every $i\in {1,2,...,N}$ in given set there exists weight $i$;
$iii)$ Sum of all weights from given set is even positive integer.
Prove that set can be partitioned into two disjoint sets which have equal weight
2011 BAMO, 3
Let $S$ be a finite, nonempty set of real numbers such that the distance between any two distinct points in $S$ is an element of $S$. In other words, $|x-y|$ is in $S$ whenever $x \ne y$ and $x$ and $y$ are both in $S$.
Prove that the elements of $S$ may be arranged in an arithmetic progression.
This means that there are numbers $a$ and $d$ such that $S = \{a, a+d, a+2d, a+3d, ..., a+kd, ...\}$.
2009 Jozsef Wildt International Math Competition, W. 10
Let consider the following function set $$F=\{f\ |\ f:\{1,\ 2,\ \cdots,\ n\}\to \{1,\ 2,\ \cdots,\ n\} \}$$
[list=1]
[*] Find $|F|$
[*] For $n=2k$ prove that $|F|< e{(4k)}^{k}$
[*] Find $n$, if $|F|=540$ and $n=2k$
[/list]
2011 IMO Shortlist, 1
Given any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers, we denote the sum $a_1 +a_2 +a_3 +a_4$ by $s_A$. Let $n_A$ denote the number of pairs $(i, j)$ with $1 \leq i < j \leq 4$ for which $a_i +a_j$ divides $s_A$. Find all sets $A$ of four distinct positive integers which achieve the largest possible value of $n_A$.
[i]Proposed by Fernando Campos, Mexico[/i]
2015 Bosnia And Herzegovina - Regional Olympiad, 4
It is given set $A=\{1,2,3,...,2n-1\}$. From set $A$, at least $n-1$ numbers are expelled such that:
$a)$ if number $a \in A$ is expelled, and if $2a \in A$ then $2a$ must be expelled
$b)$ if $a,b \in A$ are expelled, and $a+b \in A$ then $a+b$ must be also expelled
Which numbers must be expelled such that sum of numbers remaining in set stays minimal
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.
2012 IFYM, Sozopol, 3
Let $A$ be a set of natural numbers, for which for $\forall n\in \mathbb{N}$ exactly one of the numbers $n$, $2n$, and $3n$ is an element of $A$. If $2\in A$, show whether $13824\in A$.
2014 IFYM, Sozopol, 6
Let $A$ and $B$ be two non-infinite sets of natural numbers, each of which contains at least 3 elements. Two numbers $a\in A$ and $b\in B$ are called [i]"harmonious"[/i], if they are not coprime. It is known that each element from $A$ is not [i]harmonious[/i] with at least one element from $B$ and each element from $B$ is harmonious with at least one from $A$. Prove that there exist $a_1,a_2\in A$ and $b_1,b_2\in B$ such that $(a_1,b_1)$ and $(a_2,b_2)$ are [i]harmonious[/i] but $(a_1,b_2)$ and $(a_2,b_1)$ are not.
2000 Mexico National Olympiad, 3
Given a set $A$ of positive integers, the set $A'$ is composed from the elements of $A$ and all positive integers that can be obtained in the following way:
Write down some elements of $A$ one after another without repeating, write a sign $+ $ or $-$ before each of them, and evaluate the obtained expression. The result is included in $A'$.
For example, if $A = \{2,8,13,20\}$, numbers $8$ and $14 = 20-2+8$ are elements of $A'$.
Set $A''$ is constructed from $A'$ in the same manner.
Find the smallest possible number of elements of $A$, if $A''$ contains all the integers from $1$ to $40$.
2016 Dutch IMO TST, 4
Determine the number of sets $A = \{a_1,a_2,...,a_{1000}\}$ of positive integers satisfying $a_1 < a_2 <...< a_{1000} \le 2014$, for which we have that the set
$S = \{a_i + a_j | 1 \le i, j \le 1000$ with $i + j \in A\}$ is a subset of $A$.
2019 Brazil Team Selection Test, 4
Let $p \geq 7$ be a prime number and $$S = \bigg\{jp+1 : 1 \leq j \leq \frac{p-5}{2}\bigg\}.$$ Prove that at least one element of $S$ can be written as $x^2+y^2$, where $x, y$ are integers.
1998 Korea Junior Math Olympiad, 8
$T$ is a set of all the positive integers of the form $2^k 3^l$, where $k, l$ are some non-negetive integers. Show that there exists $1998$ different elements of $T$ that satisfy the following condition.
[b]Condition[/b]
The sum of the $1998$ elements is again an element of $T$.
2009 Jozsef Wildt International Math Competition, W. 11
Find all real numbers $m$ such that $$\frac{1-m}{2m} \in \{x\ |\ m^2x^4+3mx^3+2x^2+x=1\ \forall \ x\in \mathbb{R} \}$$
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?
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
2022 Turkey EGMO TST, 2
We are given some three element subsets of $\{1,2, \dots ,n\}$ for which any two of them have at most one common element. We call a subset of $\{1,2, \dots ,n\}$ [i]nice [/i] if it doesn't include any of the given subsets. If no matter how the three element subsets are selected in the beginning, we can add one more element to every 29-element [i]nice [/i] subset while keeping it nice, find the minimum value of $n$.
2020 Vietnam National Olympiad, 7
Given a positive integer $n>1$. Denote $T$ a set that contains all ordered sets $(x;y;z)$ such that $x,y,z$ are all distinct positive integers and $1\leq x,y,z\leq 2n$. Also, a set $A$ containing ordered sets $(u;v)$ is called [i]"connected"[/i] with $T$ if for every $(x;y;z)\in T$ then $\{(x;y),(x;z),(y;z)\} \cap A \neq \varnothing$.
a) Find the number of elements of set $T$.
b) Prove that there exists a set "connected" with $T$ that has exactly $2n(n-1)$ elements.
c) Prove that every set "connected" with $T$ has at least $2n(n-1)$ elements.
2006 Singapore MO Open, 4
Let $n$ be positive integer. Let $S_1,S_2,\cdots,S_k$ be a collection of $2n$-element subsets of $\{1,2,3,4,...,4n-1,4n\}$ so that $S_{i}\cap S_{j}$ contains at most $n$ elements for all $1\leq i<j\leq k$. Show that $$k\leq 6^{(n+1)/2}$$
1974 Poland - Second Round, 1
Let $ Z $ be a set of $ n $ elements. Find the number of such pairs of sets $ (A, B) $ such that $ A $ is contained in $ B $ and $ B $ is contained in $ Z $. We assume that every set also contains itself and the empty set.
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]
2013 Junior Balkan Team Selection Tests - Romania, 5
a) Prove that for every positive integer n, there exist $a, b \in R - Z$ such that
the set $A_n = \{a - b, a^2 - b^2, a^3 - b^3,...,a^n - b^n\}$ contains only positive integers.
b) Let $a$ and $b$ be two real numbers such that the set $A = \{a^k - b^k | k \in N*\}$ contains only positive integers.
Prove that $a$ and $b$ are integers.
2014 India PRMO, 8
Let $S$ be a set of real numbers with mean $M$. If the means of the sets $S\cup \{15\}$ and $S\cup \{15,1\}$ are $M + 2$ and $M + 1$, respectively, then how many elements does $S$ have?