Found problems: 175
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.
Let $k$, $m$ and $n$ be integers such that $1<n \leq m-1 \leq k$. Find maximum size of subset $S$ of set $\{1,2,...,k\}$ such that sum of any $n$ different elements from $S$ is not:
$a)$ equal to $m$,
$b)$ exceeding $m$
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 $m \le 2^{n-1}$ -
Let $n \geq 2$ be a positive integer. The set $M$ consists of $2n^2-3n+2$ positive rational numbers. Prove that there exists a subset $A$ of $M$ with $n$ elements with the following property: $\forall$ $2 \leq k \leq n$ the sum of any $k$ (not necessarily distinct) numbers from $A$ is not in $A$.
Let $A_1, A_2, A_3$ be nonempty sets of integers such that for $\{i, j, k\} = \{1, 2, 3\}$ holds
$$(x \in A_i, y\in A_j) \Rightarrow (x + y \in A_k, x - y \in A_k).$$
Prove that at least two of the sets $A_1, A_2, A_3$ are equal. Can any of these sets be disjoint?
Determine the greatest possible value of n that satisfies the following condition:
for any choice of $n$ subsets $M_1, ...,M_n$ of the set $M = \{1,2,...,n\}$ satisfying the conditions
i) $i \in M_i$ and
ii) $i \in M_j \Leftrightarrow j \notin M_i$ for all $i \ne j$,
there exist $M_k$ and $M_l$ such that $M_k \cup M_l = M$.
(Moscow regional olympiad,adopted)
For a given natural $n$, we consider the set $A\subset \{1,2, ..., n\}$, which consists of at least $\left[\frac{n+1}{2}\right]$ items. Prove that for $n \ge 2015$ the set $A$ contains a three-element arithmetic sequence.
Let $S$ be the set of all n-tuples $(X_1,...,X_n)$ of subsets of the set $\{1,2,..,1000\}$, not necessarily different and not necessarily nonempty. For $a = (X_1,...,X_n)$ denote by $E(a)$ the number of elements of $X_1\cup ... \cup X_n$. Find an explicit formula for the sum $\sum_{a\in S} E(a)$
The set $N$ is partitioned into three subsets $A_1,A_2,A_3$.
Prove that at least one of them has the following property: There exists a positive number $m$ such that for any $k$ one can find numbers $a_1 < a_2 < ... < a_k$ in that subset satisfying $a_{j+1} -a_j \le m$ for $j = 1,...,k -1$.
Let $C$ be a unit circle and $n \ge 1$ be a fixed integer. For any set $A$ of $n$ points $P_1,..., P_n$ on $C$ define $D(A) = \underset{d}{max}\, \underset{i}{min}\delta (P_i, d)$, where $d$ goes over all diameters of $C$ and $\delta (P, \ell)$ denotes the distance from point $P$ to line $\ell$. Let $F_n$ be the family of all such sets $A$. Determine $D_n = \underset{A\in F_n}{min} D(A)$ and describe all sets $A$ with $D(A) = D_n$.
Let $X$ be a set of $n$ elements and $k$ be a positive integer.
Consider the family $S_k$ of all $k$-tuples $(E_1,...,E_k)$ with $E_i \subseteq X$ for each $i$.
Evaluate the sums $\sum_{(E_1,...,E_k) \in S_k }|E_1 \cap ... \cap E_k|$ and $\sum_{(E_1,...,E_k) \in S_k }|E_1 \cup ... \cup E_k|$
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 $n = 1000$.
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)
Divide the set of twelve numbers $A = \{3, 4, 5, ...,13, 14\}$ into two sets $ B$ and $C$ 'of six numbers each according to this condition: for any two different numbers with $ B$ their sum does not belong to $ B$ and for any two different numbers from $C$, the sum does not belong to $C$.
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 $A$ .
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 $n$
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.
Show that there is a subset of $A$ from $\{1,2, 3,... , 2014\}$ such that :
(i) $|A| = 12$
(ii) for each coloring number in $A$ with red or white , we can always find some numbers colored in $A$ whose sum is $2015$.
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]
Consider the set $A = \{1, 2, ..., n\}$, where $n \in N, n \ge 6$. Show that $A$ is the union of three pairwise disjoint sets, with the same cardinality and the same sum of their elements, if and only if $n$ is a multiple of $3$.
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 $F$.
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?
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.
Let $S$ be the subset of rational numbers that can be written in the form $a/b$, where $a$ is any integer and $b$ is an odd integer. Does the sum of two of its elements belong to the $S$ ? And the product? Are there elements in $S$ whose inverse belongs to $S$ ?
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 $f(32).$