Found problems: 175
2024 Indonesia TST, C
Let $A$ be a set with $1000$ members and $\mathcal F =${$A_1,A_2,\ldots,A_n$} a family of subsets of A such that
(a) Each element in $\mathcal F$ consists of 3 members
(b) For every five elements in $\mathcal F$, the union of them all will have at least $12$ members
Find the largest value of $n$
1994 Korea National Olympiad, Problem 2
Given a set $S \subset N$ and a positive integer n, let $S\oplus \{n\} = \{s+n / s \in S\}$. The sequence $S_k$ of sets is defined inductively as follows: $S_1 = {1}$, $S_k=(S_{k-1} \oplus \{k\}) \cup \{2k-1\}$ for $k = 2,3,4, ...$
(a) Determine $N - \cup _{k=1}^{\infty} S_k$.
(b) Find all $n$ for which $1994 \in S_n$.
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?
1993 Romania Team Selection Test, 4
Let $Y$ be the family of all subsets of $X = \{1,2,...,n\}$ ($n > 1$) and let $f : Y \to X$ be an arbitrary mapping. Prove that there exist distinct subsets $A,B$ of $X$ such that $f(A) = f(B) = max A\triangle B$, where $A\triangle B = (A-B)\cup(B-A)$.
2002 Switzerland Team Selection Test, 10
Given an integer $m\ge 2$, find the smallest integer $k > m$ such that for any partition of the set $\{m,m + 1,..,k\}$ into two classes $A$ and $B$ at least one of the classes contains three numbers $a,b,c$ (not necessarily distinct) such that $a^b = c$.
2010 Miklós Schweitzer, 3
Let $ A_i,i=1,2,\dots,t$ be distinct subsets of the base set $\{1,2,\dots,n\}$ complying to the following condition
$$ \displaystyle A_ {i} \cap A_ {k} \subseteq A_ {j}$$for any $1 \leq i <j <k \leq t.$ Find the maximum value of $t.$
Thanks @dgrozev
2006 Lithuania National Olympiad, 4
Find the maximal cardinality $|S|$ of the subset $S \subset A=\{1, 2, 3, \dots, 9\}$ given that no two sums $a+b | a, b \in S, a \neq b$ are equal.
2002 USAMO, 1
Let $S$ be a set with 2002 elements, and let $N$ be an integer with $0 \leq N \leq 2^{2002}$. Prove that it is possible to color every subset of $S$ either black or white so that the following conditions hold:
(a) the union of any two white subsets is white;
(b) the union of any two black subsets is black;
(c) there are exactly $N$ white subsets.
2007 Indonesia TST, 4
Given a collection of sets $X = \{A_1, A_2, ..., A_n\}$. A set $\{a_1, a_2, ..., a_n\}$ is called a single representation of $X$ if $a_i \in A_i$ for all i. Let $|S| = mn$, $S = A_1\cup A_2 \cup ... \cup A_n = B_1 \cup B_2 \cup ... \cup B_n$ with $|A_i| = |B_i| = m$ for all $i$. Prove that $S = C_1 \cup C_2 \cup ... \cup C_n$ where for every $i, C_i $ is a single represenation for $\{A_j\}_{j=1}^n $and $\{B_j\}_{j=1}^n$.
2018 Danube Mathematical Competition, 4
Let $M$ be the set of positive odd integers.
For every positive integer $n$, denote $A(n)$ the number of the subsets of $M$ whose sum of elements equals $n$.
For instance, $A(9) = 2$, because there are exactly two subsets of $M$ with the sum of their elements equal to $9$: $\{9\}$ and $\{1, 3, 5\}$.
a) Prove that $A(n) \le A(n + 1)$ for every integer $n \ge 2$.
b) Find all the integers $n \ge 2$ such that $A(n) = A(n + 1)$
2020 Iran MO (3rd Round), 3
find all $k$ distinct integers $a_1,a_2,...,a_k$ such that there exists an injective function $f$ from reals to themselves such that for each positive integer $n$ we have
$$\{f^n(x)-x| x \in \mathbb{R} \}=\{a_1+n,a_2+n,...,a_k+n\}$$.
2000 Iran MO (2nd round), 1
Find all positive integers $n$ such that we can divide the set $\{1,2,3,\ldots,n\}$ into three sets with the same sum of members.
1990 Czech and Slovak Olympiad III A, 6
Let $k\ge 1$ be an integer and $\mathsf S$ be a family of 2-element subsets of the index set $\{1,\ldots,2k\}$ with the following property: if $\mathsf M_1,\ldots,\mathsf M_{2k}$ are arbitrary sets such that \[\mathsf M_i\cap\mathsf M_j\neq\emptyset\quad\Leftrightarrow\quad\{i,j\}\in\mathsf S,\] then the union $\mathsf M_1\cup\ldots\cup\mathsf M_{2k}$ contains at least $k^2$ elements. Show that there is a suitable family $\mathsf S$ for any integer $k\ge1.$
2010 Belarus Team Selection Test, 4.1
Find all finite sets $M \subset R, |M| \ge 2$, satisfying the following condition:
[i]for all $a, b \in M, a \ne b$, the number $a^3 - \frac{4}{9}b$ also belongs to $M$.
[/i]
(I. Voronovich)
2012 India Regional Mathematical Olympiad, 4
Let $X=\{1,2,3,...,10\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{2,3,5,7\}$.
2012 Belarus Team Selection Test, 3
Define $M_n = \{1,2,...,n\}$, for any $n\in N$. A collection of $3$-element subsets of $M_n$ is said to be [i]fine [/i] if for any coloring of elements of $M_n$ in two colors there is a subset of the collection all three elements of which are of the same color.
For any $n\ge 5$ find the minimal possible number of the $3$-element subsets of $M_n$ in the fine collection.
(E. Barabanov, S. Mazanik, I. Voronovich)
2022 SEEMOUS, 4
Let $\mathcal{F}$ be the family of all nonempty finite subsets of $\mathbb{N} \cup \{0\}.$ Find all real numbers $a$ for which the series
$$\sum_{A \in \mathcal{F}} \frac{1}{\sum_{k \in A}a^k}$$
is convergent.
2008 Indonesia TST, 2
Let $S = \{1, 2, 3, ..., 100\}$ and $P$ is the collection of all subset $T$ of $S$ that have $49$ elements, or in other words: $$P = \{T \subset S : |T| = 49\}.$$ Every element of $P$ is labelled by the element of $S$ randomly (the labels may be the same). Show that there exist subset $M$ of $S$ that has $50$ members such that for every $x \in M$, the label of $M -\{x\}$ is not equal to $x$
2024 OMpD, 1
We say that a subset \( T \) of \(\{1, 2, \dots, 2024\}\) is [b]kawaii[/b] if \( T \) has the following properties:
1. \( T \) has at least two distinct elements;
2. For any two distinct elements \( x \) and \( y \) of \( T \), \( x - y \) does not divide \( x + y \).
For example, the subset \( T = \{31, 71, 2024\} \) is [b]kawaii[/b], but \( T = \{5, 15, 75\} \) is not [b]kawaii[/b] because \( 15 - 5 = 10 \) divides \( 15 + 5 = 20 \).
What is the largest possible number of elements that a [b]kawaii [/b]subset can have?
2019 Junior Balkan Team Selection Tests - Romania, 4
Consider two disjoint finite sets of positive integers, $A$ and $B$, have $n$ and $m$ elements, respectively. It is knows that all $k$ belonging to $A \cup B$ satisfies at least one of the conditions $k + 17 \in A$ and $k - 31 \in B$.
Prove that $17n = 31m$.
2024 European Mathematical Cup, 4
Let $\mathcal{F}$ be a family of (distinct) subsets of the set $\{1,2,\dots,n\}$ such that for all $A$, $B\in \mathcal{F}$,we have that $A^C\cup B\in \mathcal{F}$, where $A^C$ is the set of all members of ${1,2,\dots,n}$ that are not in $A$.
Prove that every $k\in {1,2,\dots,n}$ appears in at least half of the sets in $\mathcal{F}$.
[i]Stijn Cambie, Mohammad Javad Moghaddas Mehr[/i]
2006 MOP Homework, 6
Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that
$$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$
2023 Bulgaria JBMO TST, 4
Given is a set of $n\ge5$ people and $m$ commissions with $3$ persons in each. Let all the commissions be [i]nice[/i] if there are no two commissions $A$ and $B$, such that $\mid A\cap B\mid=1$. Find the biggest possible $m$ (as a function of $n$).
2017 China Team Selection Test, 3
Suppose $S=\{1,2,3,...,2017\}$,for every subset $A$ of $S$,define a real number $f(A)\geq 0$ such that:
$(1)$ For any $A,B\subset S$,$f(A\cup B)+f(A\cap B)\leq f(A)+f(B)$;
$(2)$ For any $A\subset B\subset S$, $f(A)\leq f(B)$;
$(3)$ For any $k,j\in S$,$$f(\{1,2,\ldots,k+1\})\geq f(\{1,2,\ldots,k\}\cup \{j\});$$
$(4)$ For the empty set $\varnothing$, $f(\varnothing)=0$.
Confirm that for any three-element subset $T$ of $S$,the inequality $$f(T)\leq \frac{27}{19}f(\{1,2,3\})$$ holds.
2003 Olympic Revenge, 7
Let $X$ be a subset of $R_{+}^{*}$ with $m$ elements.
Find $X$ such that the number of subsets with the same sum is maximum.