Found problems: 233
2022 3rd Memorial "Aleksandar Blazhevski-Cane", P4
Find all positive integers $n$ such that the set $S=\{1,2,3, \dots 2n\}$ can be divided into $2$ disjoint subsets $S_1$ and $S_2$, i.e. $S_1 \cap S_2 = \emptyset$ and $S_1 \cup S_2 = S$, such that each one of them has $n$ elements, and the sum of the elements of $S_1$ is divisible by the sum of the elements in $S_2$.
[i]Proposed by Viktor Simjanoski[/i]
2009 Junior Balkan Team Selection Tests - Romania, 3
Let $A$ be a finite set of positive real numbers satisfying the property:
[i]For any real numbers a > 0, the sets $\{x \in A | x > a\}$ and $\{x \in A | x < \frac{1}{a}\}$ have the cardinals of the same parity.[/i]
Show that the product of all elements in $A$ is equal to $1$.
2019 Belarusian National Olympiad, 9.4
The sum of several (not necessarily different) positive integers not exceeding $10$ is equal to $S$.
Find all possible values of $S$ such that these numbers can always be partitioned into two groups with the sum of the numbers in each group not exceeding $70$.
[i](I. Voronovich)[/i]
2022 Belarus - Iran Friendly Competition, 3
Let $n > k$ be positive integers and let $F$ be a family of finite sets with the following
properties:
i. $F$ contains at least $\binom{n}{k}+ 1$ distinct sets containing exactly $k$ elements;
ii. For any two sets $A, B \in F$ their union, i.e., $A \cup B$ also belongs to $F$.
Prove that $F$ contains at least three sets with at least $n$ elements.
2001 Korea Junior Math Olympiad, 5
$A$ is a set satisfying the following the condition. Show that $2001+\sqrt{2001}$ is an element of $A$.
[b]Condition[/b]
(1) $1 \in A$
(2) If $x \in A$, then $x^2 \in A$.
(3) If $(x-3)^2 \in A$, then $x \in A$.
1996 Korea National Olympiad, 6
Find the minimum value of $k$ such that there exists two sequence ${a_i},{b_i}$ for $i=1,2,\cdots ,k$ that satisfies the following conditions.
(i) For all $i=1,2,\cdots ,k,$ $a_i,b_i$ is the element of $S=\{1996^n|n=0,1,2,\cdots\}.$
(ii) For all $i=1,2,\cdots, k, a_i\ne b_i.$
(iii) For all $i=1,2,\cdots, k, a_i\le a_{i+1}$ and $b_i\le b_{i+1}.$
(iv) $\sum_{i=1}^{k} a_i=\sum_{i=1}^{k} b_i.$
2002 AIME Problems, 14
A set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that 1 belongs to $\mathcal{S}$ and that 2002 is the largest element of $\mathcal{S},$ what is the greatet number of elements that $\mathcal{S}$ can have?
2017 Kazakhstan National Olympiad, 5
Consider all possible sets of natural numbers $(x_1, x_2, ..., x_{100})$ such that $1\leq x_i \leq 2017$ for every $i = 1,2, ..., 100$. We say that the set $(y_1, y_2, ..., y_{100})$ is greater than the set $(z_1, z_2, ..., z_{100})$ if $y_i> z_i$ for every $i = 1,2, ..., 100$. What is the largest number of sets that can be written on the board, so that any set is not more than the other set?
2013 Israel National Olympiad, 2
Let $A=\{n\in\mathbb{Z}\mid 0<n<2013\}$. A subset $B\subseteq A$ is called [b]reduced[/b] if for any two numbers $x,y\in B$, we must have $x\cdot y \notin B$. For example, any subset containing the numbers $3,5,15$ cannot be reduced, and same for a subset containing $4,16$.
[list=a]
[*] Find the maximal size of a reduced subset of $A$.
[*] How many reduced subsets are there with that maximal size?
[/list]
1996 Italy TST, 4
4.4. Prove that there exists a set X of 1996 positive integers with the following properties:
(i) the elements of X are pairwise coprime;
(ii) all elements of X and all sums of two or more distinct elements of X are
composite numbers
2019 India PRMO, 20
Consider the set $E$ of all natural numbers $n$ such that whenn divided by $11, 12, 13$, respectively, the remainders, int that order, are distinct prime numbers in an arithmetic progression. If $N$ is the largest number in $E$, find the sum of digits of $N$.
2022 European Mathematical Cup, 4
A collection $F$ of distinct (not necessarily non-empty) subsets of $X = \{1,2,\ldots,300\}$ is [i]lovely[/i] if for any three (not necessarily distinct) sets $A$, $B$ and $C$ in $F$ at most three out of the following eight sets are non-empty
\begin{align*}A \cap B \cap C, \ \ \ \overline{A} \cap B \cap C, \ \ \ A \cap \overline{B} \cap C, \ \ \ A \cap B \cap \overline{C}, \\ \overline{A} \cap \overline{B} \cap C, \ \ \ \overline{A} \cap B \cap \overline {C}, \ \ \ A \cap \overline{B} \cap \overline{C}, \ \ \ \overline{A} \cap \overline{B} \cap \overline{C}
\end{align*}
where $\overline{S}$ denotes the set of all elements of $X$ which are not in $S$.
What is the greatest possible number of sets in a lovely collection?
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 $m$.
2017 Bosnia And Herzegovina - Regional Olympiad, 4
Let $S$ be a set of $n$ distinct real numbers, and $A_S$ set of arithemtic means of two distinct numbers from $S$. For given $n \geq 2$ find minimal number of elements in $A_S$
2016 Indonesia TST, 1
Let $k$ and $n$ be positive integers. Determine the smallest integer $N \ge k$ such that the following holds: If a set of $N$ integers contains a complete residue modulo $k$, then it has a non-empty subset whose sum of elements is divisible by $n$.
2018 Junior Balkan Team Selection Tests - Romania, 3
Let $A =\left\{a = q + \frac{1}{q }/ q \in Q^*,q > 0 \right\}$, $A + A = \{a + b |a,b \in A\}$,$A \cdot A =\{a \cdot b | a, b \in A\}$.
Prove that:
i) $A + A \ne A \cdot A$
ii) $(A + A) \cap N = (A \cdot A) \cap N$.
Vasile Pop
2018 Thailand TST, 2
For finite sets $A,M$ such that $A \subseteq M \subset \mathbb{Z}^+$, we define $$f_M(A)=\{x\in M \mid x\text{ is divisible by an odd number of elements of }A\}.$$ Given a positive integer $k$, we call $M$ [i]k-colorable[/i] if it is possible to color the subsets of $M$ with $k$ colors so that for any $A \subseteq M$, if $f_M(A)\neq A$ then $f_M(A)$ and $A$ have different colors.
Determine the least positive integer $k$ such that every finite set $M \subset\mathbb{Z}^+$ is k-colorable.
2009 Jozsef Wildt International Math Competition, W. 11
Find all real numbers $m$ such that $$\frac{1-m}{2m} \in \{x\ |\ m^2x^4+3mx^3+2x^2+x=1\ \forall \ x\in \mathbb{R} \}$$
1986 Czech And Slovak Olympiad IIIA, 4
Let $C_1,C_2$, and $C_3$ be points inside a bounded convex planar set $M$. Rays $l_1,l_2,l_3$ emanating from $C_1,C_2,C_3$ respectively partition the complement of the set $M \cup l_1 \cup l_2 \cup l_3$ into three regions $D_1,D_2,D_3$. Prove that if the convex sets $A$ and $B$ satisfy $A\cap l_j =\emptyset = B\cap l_j$ and $A\cap D_j \ne \emptyset \ne B\cap D_j$ for $j = 1,2,3$, then $A\cap B \ne \emptyset$
1990 Bulgaria National Olympiad, Problem 4
Suppose $M$ is an infinite set of natural numbers such that, whenever the sum of two natural numbers is in $M$, one of these two numbers is in $M$ as well. Prove that the elements of any finite set of natural numbers not belonging to $M$ have a common divisor greater than $1$.
2011 Junior Balkan Team Selection Tests - Romania, 2
Find all the finite sets $A$ of real positive numbers having at least two elements, with the property that $a^2 + b^2 \in A$ for every $a, b \in A$ with $a \ne b$
2019 IFYM, Sozopol, 2
There are some boys and girls that study in a school. A group of boys is called [i]sociable[/i], if each girl knows at least one of the boys in the group. A group of girls is called [i]sociable[/i], if each boy knows at least one of the girls in the group. If the number of [i]sociable[/i] groups of boys is odd, prove that the number of [i]sociable[/i] groups of girls is also odd.
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.
ICMC 4, 1
Let \(S\) be a set with 10 distinct elements. A set \(T\) of subsets of \(S\) (possibly containing the empty set) is called [i]union-closed[/i] if, for all \(A, B \in T\), it is true that \(A \cup B \in T\). Show that the number of union-closed sets \(T\) is less than \(2^{1023}\).
[i]Proposed by Tony Wang[/i]
2011 Bosnia And Herzegovina - Regional Olympiad, 4
Let $n$ be a positive integer and set $S=\{n,n+1,n+2,...,5n\}$
$a)$ If set $S$ is divided into two disjoint sets , prove that there exist three numbers $x$, $y$ and $z$(possibly equal) which belong to same subset of $S$ and $x+y=z$
$b)$ Does $a)$ hold for set $S=\{n,n+1,n+2,...,5n-1\}$