Found problems: 175
2017 Bosnia and Herzegovina Junior BMO TST, 2
Let $A$ be a set $A=\{1,2,3,...,2017\}$. Subset $S$ of set $A$ is [i]good [/i] if for all $x\in A$ sum of remaining elements of set $S$ has same last digit as $x$. Prove that [i]good[/i] subset with $405$ elements is not possible.
2004 Regional Olympiad - Republic of Srpska, 4
Set $S=\{1,2,...,n\}$ is firstly divided on $m$ disjoint nonempty subsets, and then on $m^2$ disjoint nonempty subsets. Prove that some $m$ elements of set $S$ were after first division in same set, and after the second division were in $m$ different sets
2018 India PRMO, 22
A positive integer $k$ is said to be [i]good [/i] if there exists a partition of $ \{1, 2, 3,..., 20\}$ into disjoint proper subsets such that the sum of the numbers in each subset of the partition is $k$. How many [i]good [/i] numbers are there?
1969 IMO Shortlist, 42
$(MON 3)$ Let $A_k (1 \le k \le h)$ be $n-$element sets such that each two of them have a nonempty intersection. Let $A$ be the union of all the sets $A_k,$ and let $B$ be a subset of $A$ such that for each $k (1\le k \le h)$ the intersection of $A_k$ and $B$ consists of exactly two different elements $a_k$ and $b_k$. Find all subsets $X$ of the set $A$ with $r$ elements satisfying the condition that for at least one index $k,$ both elements $a_k$ and $b_k$ belong to $X$.
2003 Greece JBMO TST, 3
Consider the set $M=\{1,2,3,...,2003\}$. How many subsets of $M$ with even number of elements exist?
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?
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.
2016 Saudi Arabia IMO TST, 3
Given two positive integers $r > s$, and let $F$ be an infinite family of sets, each of size $r$, no two of which share fewer than $s$ elements. Prove that there exists a set of size $r -1$ that shares at least $s$ elements with each set in $F$.
2015 Indonesia MO Shortlist, N6
Defined as $N_0$ as the set of all non-negative integers. Set $S \subset N_0$ with not so many elements is called beautiful if for every $a, b \in S$ with $a \ge b$ ($a$ and $b$ do not have to be different), exactly one of $a + b$ or $a - b$ is in $S$. Set $T \subset N_0$ with not so many elements is called charming if the largest number $k$ such that up to 3$^k | a$ is the same for each element $a \in T$. Prove that each beautiful set must be charming.
2010 Benelux, 1
A finite set of integers is called [i]bad[/i] if its elements add up to $2010$. A finite set of integers is a [i]Benelux-set[/i] if none of its subsets is bad. Determine the smallest positive integer $n$ such that the set $\{502, 503, 504, . . . , 2009\}$ can be partitioned into $n$ Benelux-sets.
(A partition of a set $S$ into $n$ subsets is a collection of $n$ pairwise disjoint subsets of $S$, the union of which equals $S$.)
[i](2nd Benelux Mathematical Olympiad 2010, Problem 1)[/i]
1999 IMO Shortlist, 7
Let $p >3$ be a prime number. For each nonempty subset $T$ of $\{0,1,2,3, \ldots , p-1\}$, let $E(T)$ be the set of all $(p-1)$-tuples $(x_1, \ldots ,x_{p-1} )$, where each $x_i \in T$ and $x_1+2x_2+ \ldots + (p-1)x_{p-1}$ is divisible by $p$ and let $|E(T)|$ denote the number of elements in $E(T)$. Prove that
\[|E(\{0,1,3\})| \geq |E(\{0,1,2\})|\]
with equality if and only if $p = 5$.
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 $A\triangle B = (A-B)\cup(B-A)$.
2020 Malaysia IMONST 1, 19
A set $S$ has $7$ elements. Several $3$-elements subsets of $S$ are listed, such
that any $2$ listed subsets have exactly $1$ common element. What is the maximum number of subsets that can be listed?
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.
2001 Croatia Team Selection Test, 1
Consider $A = \{1, 2, ..., 16\}$. A partition of $A$ into nonempty sets $A_1, A_2,..., A_n$ is said to be good if none of the Ai contains elements $a, b, c$ (not necessarily distinct) such that $a = b + c$.
(a) Find a good partition $\{A_1, A_2, A_3, A_4\}$ of $A$.
(b) Prove that no partition $\{A_1, A_2, A_3\}$ of $A$ is good
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.
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.
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?
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 $n$
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 $k\ge1.$
2015 NIMO Problems, 5
Compute the number of subsets $S$ of $\{0,1,\dots,14\}$ with the property that for each $n=0,1,\dots,
6$, either $n$ is in $S$ or both of $2n+1$ and $2n+2$ are in $S$.
[i]Proposed by Evan Chen[/i]
1961 Putnam, A5
Let $\Omega$ be a set of $n$ points, where $n>2$. Let $\Sigma$ be a nonempty subcollection of the $2^n$ subsets of $\Omega$ that is closed with respect to the unions, intersections and complements. If $k$ is the number of elements of $\Sigma,$ what are the possible values of $k?$
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 $A(n) = A(n + 1)$
1970 Poland - Second Round, 6
If $ A $ is a subset of $ X $, then we take $ A^1 = A $, $ A^{-1} = X - A $. The subsets $ A_1, A_2, \ldots, A_k $ are called mutually independent if the product $ A_1^{\varepsilon_1} \cap A_2^{\varepsilon_2} \ldots A_k^{\varepsilon_k} $ is nonempty for every system of numbers $ \varepsilon_1 , \varepsilon_2, \ldots, \varepsilon_k $, such that $ |\varepsilon_2| = $1 for $ i = 1, 2, \ldots, k $.
What is the maximum number of mutually independent subsets of a $2^n $-element set?