Found problems: 295
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.
2023 Brazil EGMO Team Selection Test, 2
Let $A$ be a finite set made up of prime numbers. Determine if there exists an infinite set $B$ that satisfies the following conditions:
$(i)$ the prime factors of any element of $B$ are in $A$;
$(ii)$ no term of $B$ divides another element of this set.
2019 Peru IMO TST, 6
Let $p$ and $q$ two positive integers. Determine the greatest value of $n$ for which there exists sets $A_1,\ A_2,\ldots,\ A_n$ and $B_1,\ B_2,\ldots,\ B_n$ such that:
[LIST]
[*] The sets $A_1,\ A_2,\ldots,\ A_n$ have $p$ elements each one. [/*]
[*] The sets $B_1,\ B_2,\ldots,\ B_n$ have $q$ elements each one. [/*]
[*] For all $1\leq i,\ j \leq n$, sets $A_i$ and $B_j$ are disjoint if and only if $i=j$.
[/LIST]
2024 AMC 10, 20
Let $S$ be a subset of $\{1, 2, 3, \dots, 2024\}$ such that the following two conditions hold:
- If $x$ and $y$ are distinct elements of $S$, then $|x-y| > 2$
- If $x$ and $y$ are distinct odd elements of $S$, then $|x-y| > 6$.
What is the maximum possible number of elements in $S$?
$
\textbf{(A) }436 \qquad
\textbf{(B) }506 \qquad
\textbf{(C) }608 \qquad
\textbf{(D) }654 \qquad
\textbf{(E) }675 \qquad
$
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.
1996 Italy TST, 2
2. Let $A_1,A_2,...,A_n$be distinct subsets of an n-element set $ X$ ($n \geq 2$). Show that there exists an element $x$ of $X$ such that the sets $A_1\setminus \{x\}$ ,:......., $A_n\setminus \{x\}$ are all distinct.
2013 IFYM, Sozopol, 7
Let $T$ be a set of natural numbers, each of which is greater than 1. A subset $S$ of $T$ is called “good”, if for each $t\in T$ there exists $s\in S$, for which $gcd(t,s)>1$. Prove that the number of "good" subsets of $T$ is odd.
2024 239 Open Mathematical Olympiad, 1
We will say that two sets of distinct numbers are $\textit{linked}$ to each other if between any two numbers of each set lies at least one number of the other set. Is it possible to fill the cells of a $100 \times 200$ rectangle with distinct numbers so that any two rows of the rectangle are linked to one another, and any two columns of the rectangle are linked to one another?
2015 NIMO Summer Contest, 6
Let $S_0 = \varnothing$ denote the empty set, and define $S_n = \{ S_0, S_1, \dots, S_{n-1} \}$ for every positive integer $n$. Find the number of elements in the set
\[ (S_{10} \cap S_{20}) \cup (S_{30} \cap S_{40}). \]
[i] Proposed by Evan Chen [/i]
2005 National Olympiad First Round, 24
There are $20$ people in a certain community. $10$ of them speak English, $10$ of them speak German, and $10$ of them speak French. We call a [i]committee[/i] to a $3$-subset of this community if there is at least one who speaks English, at least one who speaks German, and at least one who speaks French in this subset. At most how many commitees are there in this community?
$
\textbf{(A)}\ 120
\qquad\textbf{(B)}\ 380
\qquad\textbf{(C)}\ 570
\qquad\textbf{(D)}\ 1020
\qquad\textbf{(E)}\ 1140
$
1998 Belarus Team Selection Test, 2
Find all finite sets $M \subset R$ containing at least two elements such that $(2a/3 -b^2) \in M$ for any two different elements $a,b \in M$.
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.
2023 Belarusian National Olympiad, 11.1
On a set $G$ we are given an operation $*: G \times G \to G$, that for every pair $(x,y)$ of elements of $G$ gives back $x*y \in G$, and for every elements $x,y,z \in G$ the equation $(x*y)*z=x*(y*z)$ holds. $G$ is partitioned into three non-empty sets $A,B$ and $C$.
Can it be that for every three elements $a \in A, b \in B, c \in C$ we have $a*b \in C, b*c \in A, c*a \in B$
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 $S=\{n,n+1,n+2,...,5n-1\}$
2007 German National Olympiad, 2
Let $A$ be the set of odd integers $\leq 2n-1.$ For a positive integer $m$, let $B=\{a+m\,|\, a\in A \}.$ Determine for which positive integers $n$ there exists a positive integer $m$ such that the product of all elements in $A$ and $B$ is a square.
2014 Rioplatense Mathematical Olympiad, Level 3, 6
Let $n \in N$ such that $1 + 2 + ... + n$ is divisible by $3$. Integers $a_1\ge a_2\ge a_3\ge 2$ have sum $n$ and they satisfy $1 + 2 + ... + a_1\le \frac{1}{3}( 1 + 2 + ... + n ) $ and $1 + 2 + ... + (a_1+ a_2) \le \frac{2}{3}( 1 + 2 + ... + n )$.
Prove that there is a partition of $\{ 1 , 2 , ... , n\}$ in three subsets $A_1, A_2, A_3$ with cardinals $| A_i| = a_i, i = 1 , 2 , 3$, and with equal sums of their elements .
2018 Bosnia And Herzegovina - Regional Olympiad, 2
Determine all triplets $(a,b,c)$ of real numbers such that sets $\{a^2-4c, b^2-2a, c^2-2b \}$ and $\{a-c,b-4c,a+b\}$ are equal and $2a+2b+6=5c$. In every set all elements are pairwise distinct
2021 Saudi Arabia Training Tests, 31
Let $n$ be a positive integer. What is the smallest value of $m$ with $m > n$ such that the set $M = \{n, n + 1, ..., m\}$ can be partitioned into subsets so that in each subset, there is a number which equals to the sum of all other numbers of this subset?
2020 LIMIT Category 2, 8
Let $S$ be a finite set of size $s\geq 1$ defined with a uniform probability $\mathbb{P}$( i.e. for any subset $X\subset S$ of size $x$, $\mathbb{P}(x)=\frac{x}{s}$). Suppose $A$ and $B$ are subsets of $S$. They are said to be independent iff $\mathbb{P}(A)\mathbb{P}(B)=\mathbb{P}(A\cap B)$. Which if these is sufficient for independence?
(A)$|A\cup B|=|A|+|B|$
(B)$|A\cap B|=|A|+|B|$
(C)$|A\cup B|=|A|\cdot |B|$
(D)$|A\cap B|=|A|\cdot |B|$
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?
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
$$bc<2a^2<4bc$$
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 $\sum_{(E_1,...,E_k) \in S_k }|E_1 \cup ... \cup E_k|$
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 $40$.
2011 IFYM, Sozopol, 4
Prove that the set $\{1,2,…,12001\}$ can be partitioned into 5 groups so that none of them contains an arithmetic progression with length 11.
2023 AMC 12/AHSME, 24
Let $K$ be the number of sequences $A_1$, $A_2$, $\dots$, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of $\{1, 2, 3, \dots, 10\}$, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, $\{\}$, $\{5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 6, 7, 9\}$ is one such sequence, with $n = 5$. What is the remainder when $K$ is divided by $10$?
$\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$