Let $ X$ be a set containing $ 2k$ elements, $ F$ is a set of subsets of $ X$ consisting of certain $ k$ elements such that any one subset of $ X$ consisting of $ k \minus{} 1$ elements is exactly contained in an element of $ F.$ Show that $ k \plus{} 1$ is a prime number.

Let $ABCD$ be a rectangle consisting of unit squares. All vertices of these unit squares inside the rectangle and on its sides have been colored in four colors. Additionally, it is known that: [list] [*] every vertex that lies on the side $AB$ has been colored in either the $1.$ or $2.$ color; [*] every vertex that lies on the side $BC$ has been colored in either the $2.$ or $3.$ color; [*] every vertex that lies on the side $CD$ has been colored in either the $3.$ or $4.$ color; [*] every vertex that lies on the side $DA$ has been colored in either the $4.$ or $1.$ color; [*] no two neighboring vertices have been colored in $1.$ and $3.$ color; [*] no two neighboring vertices have been colored in $2.$ and $4.$ color. [/list] Notice that the constraints imply that vertex $A$ has been colored in $1.$ color etc. Prove that there exists a unit square that has all vertices in different colors (in other words it has one vertex of each color).

In a convex hexagon $ABCDEF$ the triangles $BDF, ACE$ are equilateral and congruent. Prove that the three lines connecting the midpoints of opposite sides are concurrent.

Suppose all of the 200 integers lying in between (and including) 1 and 200 are written on a blackboard. Suppose we choose exactly 100 of these numbers and circle each one of them. By the [i]score[/i] of such a choice, we mean the square of the difference between the sum of the circled numbers and the sum of the non-circled numbers. What is the average scores over all possible choices for 100 numbers?

Show that, if $m$ and $n$ are non-zero integers of like parity, and $n^2 -1$ is divisible by $m^2 - n^2 + 1$, then $m^2 - n^2 + 1$ is the square of an integer. Amer. Math. Monthly

$3n$ points are given ($n\ge 1$) in the plane, each $3$ of them are not collinear. Prove that there are $n$ distinct triangles with the vertices those points.

Prove that for every positive integer $n$ holds inequality $\{n\sqrt{7}\}>\frac{3\sqrt{7}}{14n}$, where $\{x\}$ is fractional part of $x$.

In a rectangular prism with volume $24$, the sum of the lengths of its $12$ edges is $60$, and the length of each space diagonal is $\sqrt{109}$. Let the dimensions of the prism be $a\times b\times c$, such that $a>b>c$. Given that $a$ can be written as $\frac{p+\sqrt{q}}{r}$ where $p$, $q$, and $r$ are integers and $q$ is square-free, find $p+q+r$.

Suppose that $T,U:V\longrightarrow V$ are two linear transformations on the vector space $V$ such that $T+U$ is an invertible transformation. Prove that $TU=UT=0 \Leftrightarrow \operatorname{rank} T+\operatorname{rank} U=n$.

There are 365 cards with 365 different numbers. Each step, we can choose 3 cards $a_{i},a_{j},a_{k}$ and we know the order of them (examble: $a_{i}<a_{j}<a_{k}$). With 2000 steps, can we order 365 cards from smallest to biggest??

Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells.

Let $ABCDE$ be a convex pentagon such that $AE\parallel BC$ and $\angle ADE = \angle BDC$. The diagonals $AC$ and $BE$ intersect at point $F$. Prove that $\angle CBD= \angle ADF$.

Find all positive real solutions $(a, b, c)$ to the following system: $$ \begin{aligned} a^2 + \frac{b}{a} &= 8, \\ ab + c^2 &= 18, \\ 3a + b + c &= 9\sqrt{3}. \end{aligned} $$

How many rectangles are in the following figure? [asy] size(80); draw((0,0)--(3,0)--(3,4)--(0,4)--cycle); draw((0,2)--(3,2)); draw((0.75,2)--(0.75,0)); draw((2.25,2)--(2.25,0)); [/asy] $\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

Arne and Berit are playing a game. They have chosen positive integers $m$ and $n$ with $n\geq 4$ and $m \leq 2n + 1$. Arne begins by choosing a number from the set $\{1, 2, \dots , n \}$, and writes it on a blackboard. Then Berit picks another number from the same set, and writes it on the board. They continue alternating turns, always choosing numbers that are not already on the blackboard. When the sum of all the numbers on the board exceeds or equals $m$, the game is over, and whoever wrote the last number has won. For which combinations of $m$ and $n$ does Arne have a winning strategy?

A student is playing computer. Computer shows randomly 2002 positive numbers. Game's rules let do the following operations - to take 2 numbers from these, to double first one, to add the second one and to save the sum. - to take another 2 numbers from the remainder numbers, to double the first one, to add the second one, to multiply this sum with previous and to save the result. - to repeat this procedure, until all the 2002 numbers won't be used. Student wins the game if final product is maximum possible. Find the winning strategy and prove it.

How many equilateral hexagons of side length $\sqrt{13}$ have one vertex at $(0,0)$ and the other five vertices at lattice points? (A lattice point is a point whose Cartesian coordinates are both integers. A hexagon may be concave but not self-intersecting.)

What condition should the angles of triangle $ ABC $ satisfy so that the bisector of angle $ A $, the median drawn from vertex $ B $ and the altitude drawn from vertex $ C $ intersect at one point?

Let $f:[a,b]\to[a,b]$ be a continuous function. It is known that there exist $\alpha,\beta\in (a,b)$ such that $f(\alpha)=a$ and $f(\beta)=b$. Prove that the function $f\circ f$ has at least three fixed points.

Given an acute angle triangle $ABC$ with circumcircle $\Gamma$ and circumcenter $O$. A point $P$ is taken on the line $BC$ but not on $[BC]$. Let $K$ be the reflection of the second intersection of the line $AP$ and $\Gamma$ with respect to $OP$. If $M$ is the intersection of the lines $AK$ and $OP$, prove that $\angle OMB+\angle OMC=180^{\circ}$.

Let $\triangle ABC$ be an acute triangle, and let $D,E,F$ respectively be three points on sides $BC,CA,AB$ such that $AEDF$ is a cyclic quadrilateral. Let $O_B$ and $O_C$ be the circumcenters of $\triangle BDF$ and $\triangle CDE$, respectively. Finally, let $D'$ be a point on segment $BC$ such that $BD'=CD$. Prove that $\triangle BD'O_B$ and $\triangle CD'O_C$ have the same surface.

Given a positive integer $c$, we construct a sequence of fractions $a_1, a_2, a_3,...$ as follows: $\bullet$ $a_1 =\frac{c}{c+1} $ $\bullet$ to get $a_n$, we take $a_{n-1}$ (in its most simplified form, with both the numerator and denominator chosen to be positive) and we add $2$ to the numerator and $3$ to the denominator. Then we simplify the result again as much as possible, with positive numerator and denominator. For example, if we take $c = 20$, then $a_1 =\frac{20}{21}$ and $a_2 =\frac{22}{24} = \frac{11}{12}$ . Then we find that $a_3 =\frac{13}{15}$ (which is already simplified) and $a_4 =\frac{15}{18} =\frac{5}{6}$. (a) Let $c = 10$, hence $a_1 =\frac{10}{11}$ . Determine the largest $n$ for which a simplification is needed in the construction of $a_n$. (b) Let $c = 99$, hence $a_1 =\frac{99}{100}$ . Determine whether a simplification is needed somewhere in the sequence. (c) Find two values of $c$ for which in the first step of the construction of $a_5$ (before simplification) the numerator and denominator are divisible by $5$.

Let $k$ be a positive integer. Find the smallest positive integer $n$ for which there exists $k$ nonzero vectors $v_1,v_2,…,v_k$ in $\mathbb{R}^n$ such that for every pair $i,j$ of indices with $|i-j|>1$ the vectors $v_i$ and $v_j$ are orthogonal. [i]Proposed by Alexey Balitskiy, Moscow Institute of Physics and Technology and M.I.T.[/i]

Given an integer $n > 2$, give an example of a set of $n$ mutually different numbers $a_1,...,a_n$ for which the set of their pairwise sums $a_i + a_j$ ($i \ne j$) contains as few different numbers as possible; also give an example of a set of n different numbers $b_1,...,b_n$ for which the set of their pairwise sums $b_i+b_j$ ($i \ne j$) contains as many different numbers as possible;

The leader of the Gnome country wants to print banknotes in $12$ different denominations (each with an integer number) in such a way that it is possible to pay an arbitrary amount from $1$ to $6543$ with these banknotes without a balance, using a maximum of $8$ banknotes. (Several bills with the same denomination can be used during payment.) Can the leader of the land of Gnomes do it?