Found problems: 175
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.
2005 Czech And Slovak Olympiad III A, 2
Determine for which $m$ there exist exactly $2^{15}$ subsets $X$ of $\{1,2,...,47\}$ with the following property: $m$ is the smallest element of $X$, and for every $x \in X$, either $x+m \in X$ or $x+m > 47$.
2023 Korea - Final Round, 3
Let $p$ be an odd prime. Let $A(n)$ be the number of subsets of $\{1,2,...,n\}$ such that the sum of elements of the subset is a multiple of $p$. Prove that if $2^{p-1}-1$ is not a multiple of $p^2$, there exists infinitely many positive integer $m$ for any integer $k$ that satisfies the following. (The sum of elements of the empty set is 0.)
$$\frac{A(m)-k}{p}\in\mathbb{Z}$$
2009 Thailand Mathematical Olympiad, 2
Let $k$ and $n$ be positive integers with $k < n$. Find the number of subsets of $\{1, 2, . . . , n\}$ such that the difference between the largest and smallest elements in the subset is $k$.
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]
1986 Czech And Slovak Olympiad IIIA, 4
Let $C_1,C_2$, and $C_3$ be points inside a bounded convex planar set $M$. Rays $l_1,l_2,l_3$ emanating from $C_1,C_2,C_3$ respectively partition the complement of the set $M \cup l_1 \cup l_2 \cup l_3$ into three regions $D_1,D_2,D_3$. Prove that if the convex sets $A$ and $B$ satisfy $A\cap l_j =\emptyset = B\cap l_j$ and $A\cap D_j \ne \emptyset \ne B\cap D_j$ for $j = 1,2,3$, then $A\cap B \ne \emptyset$
1991 Spain Mathematical Olympiad, 2
Given two distinct elements $a,b \in \{-1,0,1\}$, consider the matrix $A$ .
Find a subset $S$ of the set of the rows of $A$, of minimum size, such that every other row of $A$ is a linear combination of the rows in $S$ with integer coefficients.
2014 Greece JBMO TST, 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) $n=2018$
1999 Switzerland Team Selection Test, 2
Can the set $\{1,2,...,33\}$ be partitioned into $11$ three-element sets, in each of which one element equals the sum of the other two?
2007 Dutch Mathematical Olympiad, 2
Is it possible to partition the set $A = \{1, 2, 3, ... , 32, 33\}$ into eleven subsets that contain three integers each, such that for every one of these eleven subsets, one of the integers is equal to the sum of the other two? If so, give such a partition, if not, prove that such a partition cannot exist.
2012 Ukraine Team Selection Test, 12
We shall call the triplet of numbers $a, b, c$ of the interval $[-1,1]$ [i]qualitative [/i] if these numbers satisfy the inequality $1 + 2abc\ge a^2 + b^2 + c^2$. Prove that when the triples $a, b, c$, and $x, y, z$ are qualitative, then $ax, by, cz$ is also qualitative.
2003 Greece JBMO TST, 3
Consider the set $M=\{1,2,3,...,2003\}$. How many subsets of $M$ with even number of elements exist?
1987 Bundeswettbewerb Mathematik, 2
Let $n$ be a positive integer and $M=\{1,2,\ldots, n\}.$ A subset $T\subset M$ is called [i]heavy[/i] if each of its elements is greater or equal than $|T|.$ Let $f(n)$ denote the number of heavy subsets of $M.$ Describe a method for finding $f(n)$ and use it to calculate $f(32).$
2015 Saudi Arabia BMO TST, 2
Given $2015$ subsets $A_1, A_2,...,A_{2015}$ of the set $\{1, 2,..., 1000\}$ such that $|A_i| \ge 2$ for every $i \ge 1$ and $|A_i \cap A_j| \ge 1$ for every $1 \le i < j \le 2015$. Prove that $k = 3$ is the smallest number of colors such that we can always color the elements of the set $\{1, 2,..., 1000\}$ by $k$ colors with the property that the subset $A_i$ has at least two elements of different colors for every $i \ge 1$.
Lê Anh Vinh
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 $p$.
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 $p = 5$.
1974 Putnam, B6
For a set with $n$ elements, how many subsets are there whose cardinality is respectively $\equiv 0$ (mod $3$), $\equiv 1$ (mod $3$), $ \equiv 2$ (mod $3$)? In other words, calculate
$$s_{i,n}= \sum_{k\equiv i \;(\text{mod} \;3)} \binom{n}{k}$$
for $i=0,1,2$. Your result should be strong enough to permit direct evaluation of the numbers $s_{i,n}$ and to show clearly the relationship of $s_{0,n}, s_{1,n}$ and $s_{2,n}$ to each other for all positive integers $n$. In particular, show the relationships among these three sums for $n = 1000$.
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. $T_3\subseteq T_1 \cup T_2$.
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) $n=2018$
1985 Spain Mathematical Olympiad, 2
Determine if there exists a subset $E$ of $Z \times Z$ with the properties:
(i) $E$ is closed under addition,
(ii) $E$ contains $(0,0),$
(iii) For every $(a,b) \ne (0,0), E$ contains exactly one of $(a,b)$ and $-(a,b)$.
Remark: We define $(a,b)+(a',b') = (a+a',b+b')$ and $-(a,b) = (-a,-b)$.
1989 Tournament Of Towns, (238) 2
Consider all the possible subsets of the set $\{1,2,..., N\}$ which do not contain any consecutive numbers. Prove that the sum of the squares of the products of the numbers in these subsets is $(N + 1)! - 1$.
(Based on idea of R.P. Stanley)
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 $A$.
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:$$\sum_{A\in P(S)}\sum_{B\in P(S)}f(A)f(B)2^{\left| A\cap B \right|}\geq 0.$$
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]
1998 Singapore Team Selection Test, 2
Let $n \ge 2$ be an integer. Let $S$ be a set of $n$ elements and let $A_i, 1 \le i \le m$, be distinct subsets of $S$ of size at least $2$ such that $A_i \cap A_j \ne \emptyset$, $A_i \cap A_k \ne \emptyset$, $A_j \cap A_k \ne \emptyset$ imply $A_i \cap A_j \cap A_k \ne \emptyset$. Show that $m \le 2^{n-1}$ -