$T$ is a set of all the positive integers of the form $2^k 3^l$, where $k, l$ are some non-negetive integers. Show that there exists $1998$ different elements of $T$ that satisfy the following condition. [b]Condition[/b] The sum of the $1998$ elements is again an element of $T$.

Determine the sets $S{}$ of positive integers satisfying the following two conditions: [list=a] [*]For any positive integers $a, b, c{}$, if $ab + bc + ca{}$ is in $S$, then so are $a + b + c{}$ and $abc$; and [*]The set $S{}$ contains an integer $N \geqslant 160$ such that $N-2$ is not divisible by $4$. [/list] [i]Bogdan Blaga, United Kingdom[/i]

Show that there is some rational number in the interval $(0,1)$ that can be expressed as a sum of $2021$ reciprocals of positive integers, but cannot be expressed as a sum of $2020$ reciprocals of positive integers.

Find the number of triples of sets $(A, B, C)$ such that $A \cup B \cup C = \{1, 2, 3, ... , 2549\}$

Given an $n$-gon ($n\ge 4$), consider a set $T$ of triangles formed by vertices of the polygon having the following property: Every two triangles in T have either two common vertices, or none. Prove that $T$ contains at most $n$ triangles.

Define $S_n$ as the set ${1,2,\cdots,n}$. A non-empty subset $T_n$ of $S_n$ is called $balanced$ if the average of the elements of $T_n$ is equal to the median of $T_n$. Prove that, for all $n$, the number of balanced subsets $T_n$ is odd.

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.

Prove that the numbers $1,2,...,1998$ cannot be separated into three classes whose sums of elements are divisible by $2000,3999$, and $5998$, respectively.

Suppose an ordered set of $ ({{a} _{1}}, \ {{a} _{2}},\ \ldots,\ {{a} _{n}}) $ real numbers, $n \ge 3 $. It is possible to replace the number $ {{a} _ {i}} $, $ i = \overline {2, \ n-1} $ by the number $ a_ {i} ^ {*} $ that $ {{a} _ {i}} + a_ {i} ^ {*} = {{a} _ {i-1}} + {{a} _ {i + 1}} $. Let $ ({{b} _ {1}},\ {{b} _ {2}}, \ \ldots, \ {{b} _ {n}}) $ be the set with the largest sum of numbers that can be obtained from this, and $ ({{c} _ {1}},\ {{c} _ {2}}, \ \ldots, \ {{c} _ {n}}) $ is a similar set with the least amount. For the odd $n \ge 3 $ and set $ (1,\ 3, \ \ldots, \ n, \ 2, \ 4, \ \ldots,\ n-1) $ find the values of the expressions $ {{b} _ {1}} + {{b} _ {2}} + \ldots + {{b} _ {n}} $ and $ {{c} _ {1}} + {{c} _ {2}} + \ldots + {{c} _ {n}} $.

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?

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 set $S$ of positive integers is called magic if for any two distinct members of $S, i$ and $j$, $\frac{i+ j}{GCD(i, j)}$is also a member of $S$. The $GCD$, or greatest common divisor, of two positive integers is the largest integer that divides evenly into both of them; for example, $GCD(36,80) = 4$. Find and describe all finite magic sets.

$M$ is an integer set with a finite number of elements. Among any three elements of this set, it is always possible to choose two such that the sum of these two numbers is an element of $M.$ How many elements can $M$ have at most?

Let $\mathbb N$ denote the set of all natural numbers. Show that there exists two nonempty subsets $A$ and $B$ of $\mathbb N$ such that [list=1] [*] $A\cap B=\{1\};$ [*] every number in $\mathbb N$ can be expressed as the product of a number in $A$ and a number in $B$; [*] each prime number is a divisor of some number in $A$ and also some number in $B$; [*] one of the sets $A$ and $B$ has the following property: if the numbers in this set are written as $x_1<x_2<x_3<\cdots$, then for any given positive integer $M$ there exists $k\in \mathbb N$ such that $x_{k+1}-x_k\ge M$. [*] Each set has infinitely many composite numbers. [/list]

Let $n > 2$ be a natural number. A subset $A$ of $R$ is said $n$-[i]small [/i]if there exist $n$ real numbers $t_1 , t_2 , ..., t_n$ such that the sets $t_1 + A , t_2 + A ,... , t_n + A$ are different . Show that $R$ can not be represented as a union of $ n - 1$ $n$-[i]small [/i] sets . Notation : if $r \in R$ and $B \subset R$ , then $r + B = \{ r + b | b \in B\}$ .

Two sets of intervals $A ,B$ on the line are given. The set $A$ contains $2m-1$ intervals, every two of which have an interior point in common. Moreover, every interval from $A$ contains at least two disjoint intervals from $B$. Show that there exists an interval in $B$ which belongs to at least $m$ intervals from $A$ .

For a positive integer $n$, ($n\ge 2$), find the number of sets with $2n + 1$ points $P_0, P_1,..., P_{2n}$ in the coordinate plane satisfying the following as its elements: - $P_0 = (0, 0),P_{2n}= (n, n)$ - For all $i = 1,2,..., 2n - 1$, line $P_iP_{i+1}$ is parallel to $x$-axis or $y$-axis and its length is $1$. - Out of $2n$ lines$P_0P_1, P_1P_2,..., P_{2n-1}P_{2n}$, there are exactly $4$ lines that are enclosed in the domain $y \le x$.

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]

Given is a set of $n\ge5$ people and $m$ commissions with $3$ persons in each. Let all the commissions be [i]nice[/i] if there are no two commissions $A$ and $B$, such that $\mid A\cap B\mid=1$. Find the biggest possible $m$ (as a function of $n$).

Consider the set $M=\{1,2,3,...,2020\}.$ Find the smallest positive integer $k$ such that for any subset $A$ of $M$ with $k$ elements, there exist $3$ distinct numbers $a,b,c$ from $M$ such that $a+b, b+c$ and $c+a$ are all in $A.$

Let $p,q$ be positive integers. For any $a,b\in\mathbb{R}$ define the sets $$P(a)=\bigg\{a_n=a \ + \ n \ \cdot \ \frac{1}{p} : n\in\mathbb{N}\bigg\}\text{ and }Q(b)=\bigg\{b_n=b \ + \ n \ \cdot \ \frac{1}{q} : n\in\mathbb{N}\bigg\}.$$ The [i]distance[/i] between $P(a)$ and $Q(b)$ is the minimum value of $|x-y|$ as $x\in P(a), y\in Q(b)$. Find the maximum value of the distance between $P(a)$ and $Q(b)$ as $a,b\in\mathbb{R}$.

Consider positive integers $a<b$ and the set $C\subset\{a,a+1,a+2,\dots ,b-2,b-1,b\}$. Suppose $C$ has more than $\frac{b-a+1}{2}$ elements. Prove that there are two elements $x,y\in C$ that satisfy $x+y=a+b$. [i] (From "Radu Păun" contest, Radu Miculescu)[/i]

Given $2n$ natural numbers, so that the average arithmetic of those $2n$ number is $2$. If all the number is not more than $2n$. Prove we can divide those $2n$ numbers into $2$ sets, so that the sum of each set to be the same.

In some primary school there were $94$ students in $7$th grade. Some students are involved in extracurricular activities: spanish and german language and sports. Spanish language studies $40$ students outside school program, german $27$ students and $60$ students do sports. Out of the students doing sports, $24$ of them also goes to spanish language. $10$ students who study spanish also study german. $12$ students who study german also do sports. Only $4$ students go to all three activities. How many of them does only one of the activities, and how much of them do not go to any activity?

There are $12$ members in a club. The members created some small groups, which satisfy the following: - The small group consists of $3$ or $4$ people. - Also, for two arbitrary members, there exists exactly one small group that has both members. Prove that all members are in the same number of small groups.