Take $k\in \mathbb{Z}_{\ge 1}$ and the sets $A_1,A_2,\dots, A_k$ consisting of $x_1,x_2,\dots ,x_k$ positive integers, respectively. For any two sets $A$ and $B$, define $A+B=\{a+b~|~a\in A,~b\in B\}$. Find the least and greatest number of elements the set $A_1+A_2+\dots +A_k$ may have. [i] (Andrei Bâra)[/i]

Let $A$ be a finite set of positive real numbers satisfying the property: [i]For any real numbers a > 0, the sets $\{x \in A | x > a\}$ and $\{x \in A | x < \frac{1}{a}\}$ have the cardinals of the same parity.[/i] Show that the product of all elements in $A$ is equal to $1$.

If $A=\{1,2,...,4s-1,4s\}$ and $S \subseteq A$ such that $\mid S \mid =2s+2$, prove that in $S$ we can find three distinct numbers $x$, $y$ and $z$ such that $x+y=2z$

Let $A$ be a set of $65$ integers with pairwise different remainders modulo $2016$. Prove that exists a subset $B=\{a,b,c,d\}$ of set $A$ such that $a+b-c-d$ is divisible with $2016$

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 $\{E_1, E_2, \dots, E_m\}$ be a collection of sets such that $E_i \subseteq X = \{1, 2, \dots, 100\}$, $E_i \neq X$, $i = 1, 2, \dots, m$. It is known that every two elements of $X$ is contained together in exactly one $E_i$ for some $i$. Determine the minimum value of $m$.

Yamin and Tamim are playing a game with subsets of $\{1, 2, \ldots, n\}$ where $n \geq 3$. [list] [*] Tamim starts the game with the empty set. [*] On Yamin's turn, he adds a proper non-empty subset of $\{1, 2, \ldots, n\}$ to his collection $F$ of blocked sets. [*] On Tamim's turn, he adds or removes a positive integer less than or equal to $n$ to or from their set but Tamim can never add or remove an element so that his set becomes one of the blocked sets in $F$. [/list] Tamim wins if he can make his set to be $\{1, 2, \ldots, n\}$. Yamin wins if he can stop Tamim from doing so. Yamin goes first and they alternate making their moves. Does Tamim have a winning strategy? [i]Proposed by Ahmed Ittihad Hasib[/i]

Consider all possible sets of natural numbers $(x_1, x_2, ..., x_{100})$ such that $1\leq x_i \leq 2017$ for every $i = 1,2, ..., 100$. We say that the set $(y_1, y_2, ..., y_{100})$ is greater than the set $(z_1, z_2, ..., z_{100})$ if $y_i> z_i$ for every $i = 1,2, ..., 100$. What is the largest number of sets that can be written on the board, so that any set is not more than the other set?

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 $S$.

Let $A$ be a set of positive integers satisfying the following : $a.)$ If $n \in A$ , then $n \le 2018$. $b.)$ If $S \subset A$ such that $|S|=3$, then there exists $m,n \in S$ such that $|n-m| \ge \sqrt{n}+\sqrt{m}$ What is the maximum cardinality of $A$ ?

Suppose that $S$ is a subset of $\{1, 2, 3,...,25\}$ such that the sum of any two (not necessarily distinct) elements of $S$ is never an element of $S$. What is the maximum number of elements $S$ may contain? $\textbf{(A) }12 \qquad \textbf{(B) }13 \qquad \textbf{(C) }14 \qquad \textbf{(D) }15 \qquad \textbf{(E) }16$

A subset $S$ of the natural numbers is called [i]dense [/i] for every $7$ consecutive natural numbers, at least $5$ of them are in $S$. Show that there exists a dense subset for which the equation $a^2+b^2=c^2$ has no solution for $a,b,c \in S$.

A set $S$ is called [i]neighbouring [/i] if it has the following two properties: a) $S$ has exactly four elements b) for every element $x$ of $S$, at least one of the numbers $x - 1$ or $x+1$ belongs to $S$. Find the number of all [i]neighbouring [/i] subsets of the set $\{1,2,... ,n\}$.

$a)$ Prove that for all positive integers $n \geq 3$ holds: $$\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n-1}=2^n-2$$ where $\binom{n}{k}$ , with integer $k$ such that $n \geq k \geq 0$, is binomial coefficent $b)$ Let $n \geq 3$ be an odd positive integer. Prove that set $A=\left\{ \binom{n}{1},\binom{n}{2},...,\binom{n}{\frac{n-1}{2}} \right\}$ has odd number of odd numbers

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 $S$.

It is known that subsets $A_1,A_2, \cdots , A_n$ of set $I=\{1,2,\cdots ,101\}$ satisfy the following condition $$\text{For any } i,j \text{ } (1 \leq i < j \leq n) \text{, there exists } a,b \in A_i \cap A_j \text{ so that } (a,b)=1$$ Determine the maximum positive integer $n$. *$(a,b)$ means $\gcd (a,b)$

Does there exist a set $S$ of positive integers satisfying the following conditions? $\text{(i)}$ $S$ contains $2020$ distinct elements; $\text{(ii)}$ the number of distinct primes in the set $\{\gcd(a, b) : a, b \in S, a \neq b\}$ is exactly $2019$; and $\text{(iii)}$ for any subset $A$ of $S$ containing at least two elements, $\sum\limits_{a,b\in A; a<b} ab$ is not a prime power.

There are two points $A$ and $B$ in the plane. a) Determine the set $M$ of all points $C$ in the plane for which $|AC|^2 +|BC|^2 = 2\cdot|AB|^2.$ b) Decide whether there is a point $C\in M$ such that $\angle ACB$ is maximal and if so, determine this angle.

Let $n\ge k$ be positive integers, and let $\mathcal{F}$ be a family of finite sets with the following properties: (i) $\mathcal{F}$ contains at least $\binom{n}{k}+1$ distinct sets containing exactly $k$ elements; (ii) for any two sets $A, B\in \mathcal{F}$, their union $A\cup B$ also belongs to $\mathcal{F}$. Prove that $\mathcal{F}$ contains at least three sets with at least $n$ elements. (Proposed by Fedor Petrov, St. Petersburg State University)

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 $202/100$ .

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.

Let $n \ge 1$ be an integer. For each subset $S \subset \{1, 2, \ldots , 3n\}$, let $f(S)$ be the sum of the elements of $S$, with $f(\emptyset) = 0$. Determine, as a function of $n$, the sum $$\sum_{\mathclap{\substack{S \subset \{1,2,\ldots,3n\}\\ 3 \mid f(S)}}} f(S)$$ where $S$ runs through all subsets of $\{1, 2,\ldots, 3n\}$ such that $f(S)$ is a multiple of $3$.

Let $F$ be a family of n subsets of a set $K$ with $5$ elements, such that any two subsets in $F$ have a common element. Find the minimal value of $n$ such that no matter how we choose $F$ with the properties above, there exists exactly one element of $K$ which belongs to all the sets in $F$.

Let $A = \{1,2,3, . . . ,11\}$. How many subsets $B$ of $A$ are there, such that for each $n\in \{1,2, . . . ,8\}$, if $n$ and $n+2$ are in $B$ then at least one of the numbers $ n+1$ and $n+3$ is also in $B$?

Numbers $1,2,3,...,2n$ are divided onto two equal groups. Let $a_1,a_2,...,a_n$ be the first group numbers in the increasing order, and $b_1,b_2,...,b_n$ -- the second group numbers in the decreasing order. Prove that $$|a_1 - b_1| + |a_2 - b_2| + ... + |a_n - b_n| = n^2$$