Let $A$ be a given set with $n$ elements. Let $k<n$ be a given positive integer. Find the maximum value of $m$ for which it is possible to choose sets $B_i$ and $C_i$ for $i=1,2,\ldots,m$ satisfying the following conditions: [list=1] [*]$B_i\subset A,$ $|B_i|=k,$ [*]$C_i\subset B_i$ (there is no additional condition for the number of elements in $C_i$), and [*]$B_i\cap C_j\neq B_j\cap C_i$ for all $i\neq j.$ [/list]

Let $S$ be a set which is closed under the binary operation $\circ$, with the following properties: (i) there is an element $e \in S$ such that $a \circ e = e \circ a = a$, for each $a \in S$. (ii) $(a \circ b) \circ (c \circ d)=(a \circ c) \circ (b \circ d)$, for all $a,b, c,d \in S$. Prove or disprove: (a) $\circ$ is associative on S (b) $\circ$ is commutative on S

$(a)$ Is it possible to partition the segment $[0,1]$ into two sets $A$ and $B$ and to define a continuous function $f$ such that for every $x\in A \ f(x)$ is in $B$, and for every $x\in B \ f(x)$ is in $A$? $(b)$ The same question with $[0,1]$ replaced by $[0,1).$

Let $\{a_1,...,a_n\}\subset \{-1,1\}$ and $a>0$ . Denote by $X$ and $Y$ the number of collections $\{\varepsilon_1,...,\varepsilon_n\}\subset \{-1,1\}$, such that $$max_{1\le k\le n}(\varepsilon_1a_1+...+\varepsilon_ka_k) >\alpha$$ and $$\varepsilon_1a_1+...+\varepsilon_na_n>a$$ respectively. Prove that $X\le 2Y$.

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.

A [i]uniform covering[/i] of the integers $1,2,...,n$ is a finite multiset of subsets of $\{1,2,...,n\}$, so that each number lies in the same amount of sets from the covering. A covering may contain the same subset multiple times, it must contain at least one subset, and it may contain the empty subset. For example, $(\{1\},\{1\},\{2,3\},\{3,4\},\{2,4\})$ is a uniform covering of $1,2,3,4$ (every number occurs in two sets). The covering containing only the empty set is also uniform (every number occurs in zero sets). Given two uniform coverings, we define a new uniform covering, their [i]sum[/i] (denoted by $\oplus$), by adding the sets from both coverings. For example: $(\{1\},\{1\},\{2,3\},\{3,4\},\{2,4\})\oplus(\{1\},\{2\},\{3\},\{4\})=$ $(\{1\},\{1\},\{1\},\{2\},\{3\},\{4\},\{2,3\},\{3,4\},\{2,4\})$ A uniform covering is called [i]non-composite[/i] if it's not a sum of two uniform coverings. Prove that for any $n\geq1$, there are only finitely many non-composite uniform coverings of $1,2,...,n$.

The irrational numbers $\alpha ,\beta ,\gamma ,\delta$ are such that $\forall$ $n\in \mathbb{N}$ : $[n\alpha ].[n\beta ]=[n\gamma ].[n\delta ]$. Is it true that the sets $\{ \alpha ,\beta \}$ and $\{ \gamma ,\delta \}$ are equal?

Let $n > 1$ be an integer and $\Omega=\{1,2,...,2n-1,2n\}$ the set of all positive integers that are not larger than $2n$. A nonempty subset $S$ of $\Omega$ is called [i]sum-free[/i] if, for all elements $x, y$ belonging to $S, x + y$ does not belong to $S$. We allow $x = y$ in this condition. Prove that $\Omega$ has more than $2^n$ distinct [i]sum-free[/i] subsets.

Determine the elements of the sets $A = \{x \in N | x \ne 4a + 7b, a, b \in N\}$, $B = \{x \in N | x\ne 3a + 11b, a, b \in N\}$.

Given a set $S$, of integers, an [i]optimal partition[/i] of S into sets T, U is a partition which minimizes the value $|t - u|$, where $t$ and $u$ are the sum of the elements of $T$ and U respectively. Let $P$ be a set of distinct positive integers such that the sum of the elements of $P$ is $2k$ for a positive integer $k$, and no subset of $P$ sums to $k$. Either show that there exists such a $P$ with at least $2020$ different optimal partitions, or show that such a $P$ does not exist.

Prove or disprove the following statement: If $F$ is a finite set with two or more elements, then there exists a binary operation $*$ on $F$ such that for all $x,y,z$ in $F$, $(\text i)$ $x*z=y*z$ implies $x=y$ $(\text{ii})$ $x*(y*z)\ne(x*y)*z$

It is given set $A=\{1,2,3,...,2n-1\}$. From set $A$, at least $n-1$ numbers are expelled such that: $a)$ if number $a \in A$ is expelled, and if $2a \in A$ then $2a$ must be expelled $b)$ if $a,b \in A$ are expelled, and $a+b \in A$ then $a+b$ must be also expelled Which numbers must be expelled such that sum of numbers remaining in set stays minimal

Supoose $A$ is a set of integers which contains all integers that can be written as $2^a-2^b$, $a,b\in \mathbb{Z}_{\ge 1}$ and also has the property that $a+b\in A$ whenever $a,b\in A$. Prove that if $A$ contains at least an odd number, then $A=\mathbb{Z}$. [i] (Andrei Bâra)[/i]

Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of $S$? $\textbf{(A) } \text{a single point} \qquad \textbf{(B) } \text{two intersecting lines} \\ \\ \textbf{(C) } \text{three lines whose pairwise intersections are three distinct points} \\ \\ \textbf{(D) } \text{a triangle} \qquad \textbf{(E) } \text{three rays with a common endpoint}$

Let $X= \{A_1, A_2, A_3, A_4\}$ be a set of four distinct points in the plane. Show that there exists a subset $Y$ of $X$ with the property that there is no (closed) disk $K$ such that $K\cap X = Y$.

Let $S$ be a finite set and $P$ the set of all subsets of $S$. Show that one can label the elements of $P$ as $A_i$ such that (1) $A_1 =\emptyset$. (2) For each $n\geq1 $ we either have $A_{n-1}\subset A_{n}$ and $|A_{n} \setminus A_{n-1}|=1$ or $A_{n}\subset A_{n-1}$ and $|A_{n-1} \setminus A_{n}|=1.$

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 $B \subseteq A$?

Given a positive real number $t$ , determine the sets $A$ of real numbers containing $t$ , for which there exists a set $B$ of real numbers depending on $A$ , $|B| \geq 4$ , such that the elements of the set $AB =\{ ab \mid a\in A , b \in B \}$ form a finite arithmetic progression .

Let $\Bbb{N}$ be the set of positive integers. For each subset $\mathcal{X}$ of $\Bbb{N}$ we define the set $\Delta(\mathcal{X})$ as the set of all numbers $| m - n |,$ where $m$ and $n$ are elements of $\mathcal{X}$, ie: $$\Delta (\mathcal{X}) = \{ |m-n| \ | \ m, n \in \mathcal{X} \}$$ Let $\mathcal A$ and $\mathcal B$ be two infinite, disjoint sets whose union is $\Bbb{N.}$ a) Prove that the set $\Delta (\mathcal A) \cap \Delta (\mathcal B)$ has infinitely many elements. b) Prove that there exists an infinite subset $\mathcal C$ of $\Bbb{N}$ such that $\Delta (\mathcal C)$ is a subset of $\Delta (\mathcal A) \cap \Delta (\mathcal B).$

Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.

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) $n=2018$

Find all sets $S$ of positive integers that satisfy all of the following. $1.$ If $a,b$ are two not necessarily distinct elements in $S$, then $\gcd(a,b)$, $ab$ are also in $S$. $2.$ If $m,n$ are two positive integers with $n\nmid m$, then there exists an element $s$ in $S$ such that $m^2\mid s$ and $n^2\nmid s$. $3.$ For any odd prime $p$, the set formed by moduloing all elements in $S$ by $p$ has size exactly $\frac{p+1}2$.

In a class among 80 students number of boys is 40 and number of girls is 40. 50 of the students use spectacles. Which of the following is correct? [list=1] [*] Only 10 boys use spectacles [*] Only 20 girls use spectacles [*] At most 25 boys do not use spectacles [*] At most 30 girls do not use spectacles [/list]

Let $S$ be a set of positive integers such that $\lfloor \sqrt{x}\rfloor =\lfloor \sqrt{y}\rfloor $ for all $x, y \in S$. Show that the products $xy$, where $x, y \in S$, are pairwise distinct.

Define the distance between real numbers $x$ and $y$ by $d(x,y) =\sqrt{([x]-[y])^2+(\{x\}-\{y\})^2}$ . Determine (as a union of intervals) the set of real numbers whose distance from $3/2$ is less than $202/100$ .