Prove that for any natural $n>1$ there are infinitely many natural numbers $m$ such that for any nonnegative integers $k_1$,$k_2$, $...$,$k_m$, $$m \ne k_1^n+ k_2^n+... k_n^n,$$

Let $F$ be a set of filters on X so that if $ \sigma, \tau \in F$ , $\forall S \in\sigma$ , $\forall T\in\tau$ , we have $S \cap T\neq\emptyset$ , then $\sigma \cap \tau \in F$. We say that $F$ is compatible with a topology on X when $x \in X$ is a contact point of $A\subset X$ , if and only if , there is $\sigma \in F$ such that $x \in S$ and $S \cap A \neq\emptyset$ for all $S \in\sigma$ . When is there an $F$ compatible with the topology on X in which finite subsets of X and X are closed ? contact point is also known as adherent point.

A trapezoid $ABCD$ with bases $BC = a$ and $AD = 2a$ is drawn on the plane. Using only with a ruler, construct a triangle whose area is equal to the area of the trapezoid. With the help of a ruler you can draw straight lines through two known points. (Rozhkova Maria)

Real numbers $a,b,c,d$ satisfy the equality $$\frac{1-ab}{a+b}=\frac{bc-1}{b+c}=\frac{1-cd}{c+d}=\sqrt{3}$$ Find all possible values of $ad$.

Find $ f(x): (0,\plus{}\infty) \to \mathbb R$ such that \[ f(x)\cdot f(y) \plus{} f(\frac{2008}{x})\cdot f(\frac{2008}{y})\equal{}2f(x\cdot y)\] and $ f(2008)\equal{}1$ for $ \forall x \in (0,\plus{}\infty)$.

Frieda the frog begins a sequence of hops on a $3 \times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around'' and jumps to the opposite edge. For example if Frieda begins in the center square and makes two hops "up'', the first hop would place her in the top row middle square, and the second hop would cause Frieda to jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the center square, makes at most four hops at random, and stops hopping if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops? $\textbf{(A) }\frac{9}{16}\qquad\textbf{(B) }\frac{5}{8}\qquad\textbf{(C) }\frac{3}{4}\qquad\textbf{(D) }\frac{25}{32}\qquad\textbf{(E) }\frac{13}{16}$

Let $a$ and $b$ be real numbers such that $\max_{0\le x\le 1} |x^3 - ax - b|$ is as small as possible. Find $a + b$ in simplest radical form. (Hint: If $f(x) = x^3 - cx - d$, then the maximum (or minimum) of $f(x)$ either occurs when $x = 0$ and/or $x = 1$ and/or when x satisfies $3x^2 - c = 0$).

In the space $n \geq 3$ points are given. Every pair of points determines some distance. Suppose all distances are different. Connect every point with the nearest point. Prove that it is impossible to obtain (closed) polygonal line in such a way.

There are a thousand non-intersecting arcs on a circle, and on each of them contains two natural numbers. Sum of numbers of each arc is divided by the product of the numbers of the arc following it clockwise arrow. What is the largest possible value of the largest number written?

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that \[ f(x) f(y) = |x - y| \cdot f \left( \frac{xy + 1}{x - y} \right) \] Holds for all $x \not= y \in \mathbb{R}$

Three acute triangles are inscribed in the same circle with their vertices being nine distinct points. Show that one can choose a vertex from each triangle so that the three chosen points determine a triangle each of whose angles is at most $90^\circ$.

Let $ABCD$ be a quadrilateral such that $\angle ABC = \angle ADC = 90º$ and $\angle BCD$ > $90º$. Let $P$ be a point inside of the $ABCD$ such that $BCDP$ is parallelogram, the line $AP$ intersects $BC$ in $M$. If $BM = 2, MC = 5, CD = 3$. Find the length of $AM$.

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

There are 3 heaps with $100,101,102$ stones. Ilya and Kostya play next game. Every step they take one stone from some heap, but not from same, that was on previous step. They make his steps in turn, Ilya make first step. Player loses if can not make step. Who has winning strategy?

A circle having common centre with the circumcircle of triangle $ABC$ meets the sides of the triangle at six points forming convex hexagon $A_{1}A_{2}B_{1}B_{2}C_{1}C_{2}$ ($A_{1}$ and $A_{2}$ lie on $BC$, $B_{1}$ and $B_{2}$ lie on $AC$, $C_{1}$ and $C_{2}$ lie on $AB$). If $A_{1}B_{1}$ is parallel to the bisector of angle $B$, prove that $A_{2}C_{2}$ is parallel to the bisector of angle $C$. [i]Proposed by S. Berlov[/i]

Show that there exists a convex polyhedron with all its vertices on the surface of a sphere and with all its faces congruent isosceles triangles whose ratio of sides are $\sqrt{3} :\sqrt{3} :2$.

Prove that for any real $ x $ and $ y $ the inequality $x^2 \sqrt {1+2y^2} + y^2 \sqrt {1+2x^2} \geq xy (x+y+\sqrt{2})$ .

Let $ a_1, a_2,\ldots, a_n$ be non-negative real numbers. Prove that $\frac{1}{1+ a_1}+\frac{ a_1}{(1+ a_1)(1+ a_2)}+\frac{ a_1 a_2}{(1+ a_1)(1+ a_2)(1+ a_3)}+$ $\cdots+\frac{ a_1 a_2\cdots a_{n-1}}{(1+ a_1)(1+ a_2)\cdots (1+ a_n)} \le 1.$

Let $f: \mathbb{R}^{2} \to \mathbb{R}^{+}$such that for every rectangle $A B C D$ one has $$ f(A)+f(C)=f(B)+f(D). $$ Let $K L M N$ be a quadrangle in the plane such that $f(K)+f(M)=f(L)+f(N)$, for each such function. Prove that $K L M N$ is a rectangle. [i]Proposed by Navid.[/i]

Let $ n \geq 3$ be an odd integer. Determine the maximum value of \[ \sqrt{|x_{1}\minus{}x_{2}|}\plus{}\sqrt{|x_{2}\minus{}x_{3}|}\plus{}\ldots\plus{}\sqrt{|x_{n\minus{}1}\minus{}x_{n}|}\plus{}\sqrt{|x_{n}\minus{}x_{1}|},\] where $ x_{i}$ are positive real numbers from the interval $ [0,1]$.

We will say that two sets of distinct numbers are $\textit{linked}$ to each other if between any two numbers of each set lies at least one number of the other set. Is it possible to fill the cells of a $100 \times 200$ rectangle with distinct numbers so that any two rows of the rectangle are linked to one another, and any two columns of the rectangle are linked to one another?

12. What is the sum of all possible $\left(\begin{array}{l}i \\ j\end{array}\right)$ subject to the restrictions that $i \geq 10, j \geq 0$, and $i+j \leq 20$ ? Count different $i, j$ that yield the same value separately - for example, count both $\left(\begin{array}{c}10 \\ 1\end{array}\right)$ and $\left(\begin{array}{c}10 \\ 9\end{array}\right)$.

Let us consider a simle graph with vertex set $ V$. All ordered pair $ (a,b)$ of integers with $ gcd(a,b) \equal{} 1$, are elements of V. $ (a,b)$ is connected to $ (a,b \plus{} kab)$ by an edge and to $ (a \plus{} kab,b)$ by another edge for all integer k. Prove that for all $ (a,b)\in V$, there exists a path fromm $ (1,1)$ to $ (a,b)$.

Each side of an acute triangle is multiplied by the cosine of the opposite angle. a) Prove that a triangle can be formed from the resulting segments. 6) Find the radius of the circle circumscribed around the resulting triangle if the radius of the circle circumscribed around the original triangle is equal to $R$.

How many integer pairs $(x,y)$ satisfy $x^2+4y^2-2xy-2x-4y-8=0$?