Found problems: 175
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
$$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$
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?
2003 Greece JBMO TST, 3
Consider the set $M=\{1,2,3,...,2003\}$. How many subsets of $M$ with even number of elements exist?
2007 Postal Coaching, 4
Let $A_1,A_2,...,A_n$ be $n$ finite subsets of a set $X, n \ge 2$, such that
(i) $|A_i| \ge 2, 1 \le i \le n$,
(ii) $ |A_i \cap A_j | \ne 1, j \le i < j \le n$.
Prove that the elements of $A_1 \cup A_2 \cup ... \cup A_n$ may be colored with $2$ colors so that no $A_i$ is colored by the same color.
2014 Czech-Polish-Slovak Match, 6
Let $n \ge 6$ be an integer and $F$ be the system of the $3$-element subsets of the set $\{1, 2,...,n \}$ satisfying the following condition:
for every $1 \le i < j \le n$ there is at least $ \lfloor \frac{1}{3} n \rfloor -1$ subsets $A\in F$ such that $i, j \in A$.
Prove that for some integer $m \ge 1$ exist the mutually disjoint subsets $A_1, A_2 , ... , A_m \in F $ also, that $|A_1\cup A_2 \cup ... \cup A_m |\ge n-5 $
(Poland)
PS. just in case my translation does not make sense,
I leave the original in Slovak, in case someone understands something else
1986 Czech And Slovak Olympiad IIIA, 1
Given $n \in N$, let $A$ be a family of subsets of $\{1,2,...,n\}$. If for every two sets $B,C \in A$ the set $(B \cup C) -(B \cap C)$ has an even number of elements, find the largest possible number of elements of $A$ .
2009 Postal Coaching, 1
Let $n \ge 1$ be an integer. Prove that there exists a set $S$ of $n$ positive integers with the following property:
if $A$ and $B$ are any two distinct non-empty subsets of $S$, then the averages $\frac{P_{x\in A} x}{|A|}$ and $\frac{P_{x\in B} x}{|B|}$ are two relatively prime composite integers.
2021 Junior Balkan Team Selection Tests - Romania, P2
For any non-empty subset $X$ of $M=\{1,2,3,...,2021\}$, let $a_X$ be the sum of the greatest and smallest elements of $X$. Determine the arithmetic mean of all the values of $a_X$, as $X$ covers all the non-empty subsets of $M$.
2007 Thailand Mathematical Olympiad, 13
Let $S = \{1, 2,..., 8\}$. How many ways are there to select two disjoint subsets of $S$?
1988 Mexico National Olympiad, 7
Two disjoint subsets of the set $\{1,2, ... ,m\}$ have the same sums of elements. Prove that each of the subsets $A,B$ has less than $m / \sqrt2$ elements.
2021 China Girls Math Olympiad, 6
Given a finite set $S$, $P(S)$ denotes the set of all the subsets of $S$. For any $f:P(S)\rightarrow \mathbb{R}$ ,prove the following inequality:$$\sum_{A\in P(S)}\sum_{B\in P(S)}f(A)f(B)2^{\left| A\cap B \right|}\geq 0.$$
2019 Purple Comet Problems, 16
Find the number of ordered triples of sets $(T_1, T_2, T_3)$ such that
1. each of $T_1, T_2$, and $T_3$ is a subset of $\{1, 2, 3, 4\}$,
2. $T_1 \subseteq T_2 \cup T_3$,
3. $T_2 \subseteq T_1 \cup T_3$, and
4. $T_3\subseteq T_1 \cup T_2$.
2025 Philippine MO, P1
The set $S$ is a subset of $\{1, 2, \dots, 2025\}$ such that no two elements of $S$ differ by $2$ or by $7$. What is the largest number of elements that $S$ can have?
1999 Greece Junior Math Olympiad, 4
Defi ne alternate sum of a set of real numbers $A =\{a_1,a_2,...,a_k\}$ with $a_1 < a_2 <...< a_k$, the number
$S(A) = a_k - a_{k-1} + a_{k-2} - ... + (-1)^{k-1}a_1$ (for example if $A = \{1,2,5, 7\}$ then $S(A) = 7 - 5 + 2 - 1$)
Consider the alternate sums, of every subsets of $A = \{1, 2, 3, 4, 5, 6, 7, 8,9, 10\}$ and sum them.
What is the last digit of the sum obtained?
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 $a^b = c$.
2021 Indonesia TST, C
Let $p$ be an odd prime. Determine the number of nonempty subsets from $\{1, 2, \dots, p - 1\}$ for which the sum of its elements is divisible by $p$.
2014 Contests, 4
Givan the set $S = \{1,2,3,....,n\}$. We want to partition the set $S$ into three subsets $A,B,C$ disjoint (to each other) with $A\cup B\cup C=S$ , such that the sums of their elements $S_{A} S_{B} S_{C}$ to be equal .Examine if this is possible when:
a) $n=2014$
b) $n=2015 $
c) $n=2018$
2022 Korea Winter Program Practice Test, 4
For a finite set $A$ of positive integers and its subset $B$, call $B$ a [i]half subset[/i] of $A$ when it satisfies the equation $\sum_{a\in A}a=2\sum_{b\in B}b$. For example, if $A=\{1,2,3\}$, then $\{1,2\}$ and $\{3\}$ are half subset of $A$. Determine all positive integers $n$ such that there exists a finite set $A$ which has exactly $n$ half subsets.
2015 Moldova Team Selection Test, 4
Consider a positive integer $n$ and $A = \{ 1,2,...,n \}$. Call a subset $X \subseteq A$ [i][b]perfect[/b][/i] if $|X| \in X$. Call a perfect subset $X$ [i][b]minimal[/b][/i] if it doesn't contain another perfect subset. Find the number of minimal subsets of $A$.
1991 All Soviet Union Mathematical Olympiad, 556
$X$ is a set with $100$ members. What is the smallest number of subsets of $X$ such that every pair of elements belongs to at least one subset and no subset has more than $50$ members? What is the smallest number if we also require that the union of any two subsets has at most $80$ members?
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 $17n = 31m$.
1990 Romania Team Selection Test, 6
Prove that there are infinitely many n’s for which there exists a partition of $\{1,2,...,3n\}$ into subsets $\{a_1,...,a_n\}, \{b_1,...,b_n\}, \{c_1,...,c_n\}$ such that $a_i +b_i = c_i$ for all $i$, and prove that there are infinitely many $n$’s for which there is no such partition.
2008 Indonesia TST, 1
Let $A$ be the subset of $\{1, 2, ..., 16\}$ that has $6$ elements. Prove that there exist $2$ subsets of $A$ that are disjoint, and the sum of their elements are the same.
2013 Serbia Additional Team Selection Test, 3
Let $p > 3$ be a given prime number. For a set $S \subseteq \mathbb{Z}$ and $a \in \mathbb{N}$ , define
$S_a = \{ x \in \{ 0,1, 2,...,p-1 \}$ | $(\exists_s \in S) x \equiv_p a \cdot s \}$ .
$(a)$ How many sets $S \subseteq \{ 1, 2,...,p-1 \} $ are there for which the sequence
$S_1 , S_2 , ..., S_{p-1}$ contains exactly two distinct terms?
$(b)$ Determine all numbers $k \in \mathbb{N}$ for which there is a set $ S \subseteq \{ 1, 2,...,p-1 \} $ such
that the sequence $S_1 , S_2 , ..., S_{p-1} $ contains exactly $k$ distinct terms.
[i]Proposed by Milan Basic and Milos Milosavljevic[/i]