Found problems: 233
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]
2009 Ukraine Team Selection Test, 3
Let $S$ be a set consisting of $n$ elements, $F$ a set of subsets of $S$ consisting of $2^{n-1}$ subsets such that every three such subsets have a non-empty intersection.
a) Show that the intersection of all subsets of $F$ is not empty.
b) If you replace the number of sets from $2^{n-1}$ with $2^{n-1}-1$, will the previous answer change?
2015 Junior Regional Olympiad - FBH, 5
Prove that for every parititon of set $X=\{1,2,...,9\}$ on two disjoint sets at least one of them contains three elements such that sum of some two of them is equal to third
2000 Saint Petersburg Mathematical Olympiad, 11.7
It is known that for irrational numbers $\alpha$, $\beta$, $\gamma$, $\delta$ and for any positive integer $n$ the following is true:
$$[n\alpha]+[n\beta]=[n\gamma]+[n\delta]$$
Does this mean that sets $\{\alpha,\beta\}$ and $\{\gamma,\delta\}$ are equal? (As usual $[x]$ means the greatest integer not greater than $x$).
Oliforum Contest V 2017, 5
Find the smallest integer $n > 3$ such that, for each partition of $\{3, 4,..., n\}$ in two sets, at least one of these sets contains three (not necessarily distinct) numbers $ a, b, c$ for which $ab = c$.
(Alberto Alfarano)
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.
2010 BAMO, 1
We write $\{a,b,c\}$ for the set of three different positive integers $a, b$, and $c$. By choosing some or all of the numbers a, b and c, we can form seven nonempty subsets of $\{a,b,c\}$. We can then calculate the sum of the elements of each subset. For example, for the set $\{4,7,42\}$ we will find sums of $4, 7, 42,11, 46, 49$, and $53$ for its seven subsets. Since $7, 11$, and $53$ are prime, the set $\{4,7,42\}$ has exactly three subsets whose sums are prime. (Recall that prime numbers are numbers with exactly two different factors, $1$ and themselves. In particular, the number $1$ is not prime.)
What is the largest possible number of subsets with prime sums that a set of three different positive integers can have? Give an example of a set $\{a,b,c\}$ that has that number of subsets with prime sums, and explain why no other three-element set could have more.
2014 Korea Junior Math Olympiad, 8
Let there be $n$ students and $m$ clubs. The students joined the clubs so that the following is true:
- For all students $x$, you can choose some clubs such that $x$ is the only student who joined all of the chosen clubs.
Let the number of clubs each student joined be $a_1,a_2,...,a_m$. Prove that
$$a_1!(m - a_1)! + a_2!(m - a_2)! + ... + a_n!(m -a_n)! \le m!$$
2015 District Olympiad, 3
Consider the following sequence of sets: $ \{ 1,2\} ,\{ 3,4,5\}, \{ 6,7,8,9\} ,... $
[b]a)[/b] Find the samllest element of the $ 100\text{-th} $ term.
[b]b)[/b] Is $ 2015 $ the largest element of one of these sets?
2017 AIME Problems, 12
Call a set $S$ [i]product-free[/i] if there do not exist $a, b, c \in S$ (not necessarily distinct) such that $a b = c$. For example, the empty set and the set $\{16, 20\}$ are product-free, whereas the sets $\{4, 16\}$ and $\{2, 8, 16\}$ are not product-free. Find the number of product-free subsets of the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$.
2010 Bosnia And Herzegovina - Regional Olympiad, 4
It is given set with $n^2$ elements $(n \geq 2)$ and family $\mathbb{F}$ of subsets of set $A$, such that every one of them has $n$ elements. Assume that every two sets from $\mathbb{F}$ have at most one common element. Prove that
$i)$ Family $\mathbb{F}$ has at most $n^2+n$ elements
$ii)$ Upper bound can be reached for $n=3$
2019 Taiwan APMO Preliminary Test, P5
Find the minimum positive integer $n$ such that for any set $A$ with $n$ positive intergers has $15$ elements which sum is divisible by $15$.
2019 India IMO Training Camp, P1
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 $S$.
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?
2021 Science ON grade VII, 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]
1987 China Team Selection Test, 1
a.) For all positive integer $k$ find the smallest positive integer $f(k)$ such that $5$ sets $s_1,s_2, \ldots , s_5$ exist satisfying:
[b]i.[/b] each has $k$ elements;
[b]ii.[/b] $s_i$ and $s_{i+1}$ are disjoint for $i=1,2,...,5$ ($s_6=s_1$)
[b]iii.[/b] the union of the $5$ sets has exactly $f(k)$ elements.
b.) Generalisation: Consider $n \geq 3$ sets instead of $5$.
2018 Iran MO (1st Round), 16
A subset of the real numbers has the property that for any two distinct elements of it such as $x$ and $y$, we have $(x+y-1)^2 = xy+1$. What is the maximum number of elements in this set?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ \text{Infinity}$
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)$|A\cap B|=|A|\cdot |B|$
2023 OMpD, 3
Let $m$ and $n$ be positive integers integers such that $2m + 1 < n$, and let $S$ be the set of the $2^n$ subsets of $\{1,2,\ldots,n\}$. Prove that we can place the elements of $S$ on a circle, so that for any two adjacent elements $A$ and $B$, the set $A \Delta B$ has exactly $2m + 1$ elements.
[b]Note[/b]: $A \Delta B = (A \cup B) - (A \cap B)$ is the set of elements that are exclusively in $A$ or exclusively in $B$.
2009 Kyiv Mathematical Festival, 6
Let $\{a_1,...,a_n\}\subset \{-1,1\}$ and $a>0$ . Denote by $X$ and $Y$ the number of collections $\{\varepsilon_1,...,\varepsilon_n\}\subset \{-1,1\}$, such that $$max_{1\le k\le n}(\varepsilon_1a_1+...+\varepsilon_ka_k) >\alpha$$ and $$\varepsilon_1a_1+...+\varepsilon_na_n>a$$ respectively. Prove that $X\le 2Y$.
2013 IFYM, Sozopol, 8
The irrational numbers $\alpha ,\beta ,\gamma ,\delta$ are such that $\forall$ $n\in \mathbb{N}$ :
$[n\alpha ].[n\beta ]=[n\gamma ].[n\delta ]$.
Is it true that the sets $\{ \alpha ,\beta \}$ and $\{ \gamma ,\delta \}$ are equal?
India EGMO 2025 TST, 1
Let $n$ be a positive integer. Initially the sequence $0,0,\cdots,0$ ($n$ times) is written on the board. In each round, Ananya choses an integer $t$ and a subset of the numbers written on the board and adds $t$ to all of them. What is the minimum number of rounds in which Ananya can make the sequence on the board strictly increasing?
Proposed by Shantanu Nene
2014 India PRMO, 20
What is the number of ordered pairs $(A,B)$ where $A$ and $B$ are subsets of $\{1,2,..., 5\}$ such that neither $A \subseteq B$ nor $B \subseteq A$?
2023 Turkey MO (2nd round), 5
Is it possible that a set consisting of $23$ real numbers has a property that the number of the nonempty subsets whose product of the elements is rational number is exactly $2422$?
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.