Found problems: 175
$(MON 3)$ 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 $X$.
Show that for any finite set $S$ of distinct positive integers, we can find a set $T \supseteq S$ such that every member of $T$ divides the sum of all the members of $T$.
[b]Original Statement:[/b]
A finite set of (distinct) positive integers is called a [b]DS-set[/b] if each of the integers divides the sum of them all. Prove that every finite set of positive integers is a subset of some [b]DS-set[/b].
There are $20$ people in a certain community. $10$ of them speak English, $10$ of them speak German, and $10$ of them speak French. We call a [i]committee[/i] to a $3$-subset of this community if there is at least one who speaks English, at least one who speaks German, and at least one who speaks French in this subset. At most how many commitees are there in this community?
$
\textbf{(A)}\ 120
\qquad\textbf{(B)}\ 380
\qquad\textbf{(C)}\ 570
\qquad\textbf{(D)}\ 1020
\qquad\textbf{(E)}\ 1140
$
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.
Consider the set $S = \{1,2,3, ...,1441\}$.
1. Nora counts thoses subsets of $S$ having exactly two elements, tbe sum of which is even. Rania counts those subsets of $S$ having exactly two elements, the sum of which is odd. Determine the numbers counted by Nora and Rania.
2. Let $t$ be the number of subsets of $S$ which have at least two elements and the product of the elements is even. Determine the greatest power of $2$ which divides $t$.
3. Ahmad counts the subsets of $S$ having $77$ elements such that in each subset the sum of the elements is even. Bushra counts the subsets of $S$ having $77$ elements such that in each subset the sum of the elements is odd. Whose number is bigger? Determine the difference between the numbers found by Ahmad and Bushra.
$(MON 3)$ 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 $X$.
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 $p$.
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)
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]
Suppose that we have $2n$ non-empty subset of $ \big\{0,1,2,...,2n-1\big\} $ that sum of the elements of these subsets is $ \binom{2n+1}{2}$ . Prove that we can choose one element from every subset that some of them is $ \binom{2n}{2}$
[i]Proposed by Morteza Saghafian and Afrouz Jabalameli [/i]
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
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.
For every nonempty subset $R$ of $S = \{1,2,...,10\}$, we define the alternating sum $A(R)$ as follows:
If $r_1,r_2,...,r_k$ are the elements of $R$ in the increasing order, then $A(R) = r_k -r_{k-1} +r_{k-2}- ... +(-1)^{k-1}r_1$.
(a) Is it possible to partition $S$ into two sets having the same alternating sum?
(b) Determine the sum $\sum_{R} A(R)$, where $R$ runs over all nonempty subsets of $S$.
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.
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?
There are $n$ students each having $r$ positive integers. Their $nr$ positive integers are all different. Prove that we can divide the students into $k$ classes satisfying the following conditions:
(a) $k \le 4r$
(b) If a student $A$ has the number $m$, then the student $B$ in the same class can't have a number $\ell$ such that $(m - 1)! < \ell < (m + 1)! + 1$
$k > 1$ is fixed. Show that for $n$ sufficiently large for every partition of $\{1,2,\dots,n\}$ into $k$ disjoint subsets we can find $a \neq b$ such that $a$ and $b$ are in the same subset and $a+1$ and $b+1$ are in the same subset. What is the smallest $n$ for which this is true?
Let $ Z $ be a set of $ n $ elements. Find the number of such pairs of sets $ (A, B) $ such that $ A $ is contained in $ B $ and $ B $ is contained in $ Z $. We assume that every set also contains itself and the empty set.
Suppose that each set $X$ of points in the plane has an associated set $\overline{X}$ of points called its cover. Suppose further that (1) $\overline{X\cup Y} \supset \overline{\overline{X}} \cup \overline{Y} \cup Y$ for all sets $X,Y$ . Show that i) $\overline{X} \supset X$, ii) $\overline{\overline{X}}=\overline{X}$ and iii) $X\supset Y \Rightarrow \overline{X} \supset \overline{Y}.$ Prove also that these three statements imply (1).
$ P_1, P_2, \ldots, P_n $ are different two-element subsets of $ \{1,2,\ldots,n\} $. The sets $ P_i $, $ P_j $ for $ i\neq j $ have a common element if and only if the set $ \{i,j\} $ is one of the sets $ P_1, P_2, \ldots, P_n $. Prove that each of the numbers $ 1,2,\ldots,n $ is a common element of exactly two sets from $ P_1, P_2, \ldots, P_n $.
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 $x$
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) $n=2018$
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 $B \subseteq A$?
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.
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. $T_3\subseteq T_1 \cup T_2$.