Find all positive integer solutions of equation $n^3 - 2 = k! .$

There is convex $2009$-gon on the plane. a) Find the greatest number of vertices of $2009$-gon such that no two forms the side of the polygon. b) Find the greatest number of vertices of $2009$-gon such that among any three of them there is one that is not connected with other two by side.

On the party every boy gave $1$ candy to every girl and every girl gave $1$ candy to every boy. Then every boy ate $2$ candies and every girl ate $3$ candies. It is known that $\frac 14$ of all candies was eaten. Find the greatest possible number of children on the party.

In the triangle $ABC$ given that $\angle ABC = 120^\circ .$ The bisector of $\angle B$ meet $AC$ at $M$ and external bisector of $\angle BCA$ meet $AB$ at $P.$ Segments $MP$ and $BC$ intersects at $K$. Prove that $\angle AKM = \angle KPC .$

Let $a, b, c$ be integers satisfying $ab + bc + ca = 1.$ Prove that $(1+ a^2 )(1+ b^2 )(1+ c^2 )$ is a perfect square.

In acute-angled triangle $ABC,$ let $M$ be the midpoint of $BC$ and let $K$ be a point on side $AB.$ We know that $AM$ meet $CK$ at $F.$ Prove that if $AK = KF$ then $AB = CF .$

Given $2009 \times 4018$ rectangular board. Frame is a rectangle $n \times n$ or $n \times(n + 2)$ for $ ( n \geq 3 )$ without all cells which don’t have any common points with boundary of rectangle. Rectangles $1\times1,1\times 2,1\times 3$ and $ 2\times 4$ are also frames. Two players by turn paint all cells of some frame that has no painted cells yet. Player that can't make such move loses. Who has a winning strategy?

а) Prove that for any positive integer $n$ there exist a pair of positive integers $(m, k)$ such that \[{k + m^k + n^{m^k}} = 2009^n.\] b) Prove that there are infinitely many positive integers $n$ for which there is only one such pair.

Build the set of points $( x, y )$ on coordinate plane, that satisfies equality: \[ \sqrt{1-x^2}+\sqrt{1-y^2}=2-x^2-y^2.\]

On the party every boy gave $1$ candy to every girl and every girl gave $1$ candy to every boy. Then every boy ate $2$ candies and every girl ate $3$ candies. It is known that $\frac 14$ of all candies was eaten. Find the greatest possible number of children on the party.

In triangle $ABC$ points $M, N$ are midpoints of $BC, CA$ respectively. Point $P$ is inside $ABC$ such that $\angle BAP = \angle PCA = \angle MAC .$ Prove that $\angle PNA = \angle AMB .$

Let $x \leq y \leq z \leq t$ be real numbers such that $xy + xz + xt + yz + yt + zt = 1.$ a) Prove that $xt < \frac 13,$ b) Find the least constant $C$ for which inequality $xt < C$ holds for all possible values $x$ and $t.$

Pairwise distinct real numbers $a, b, c$ satisfies the equality \[a +\frac 1b =b + \frac 1c =c+\frac 1a.\] Find all possible values of $abc .$

Find all prime numbers $p$ and positive integers $m$ such that $2p^2 + p + 9 = m^2.$

Given $2009 \times 4018$ rectangular board. Frame is a rectangle $n \times n$ or $n \times(n + 2)$ for $ ( n \geq 3 )$ without all cells which don’t have any common points with boundary of rectangle. Rectangles $1\times1,1\times 2,1\times 3$ and $ 2\times 4$ are also frames. Two players by turn paint all cells of some frame that has no painted cells yet. Player that can't make such move loses. Who has a winning strategy?

In the trapezoid $ABCD$ we know that $CD \perp BC, $ and $CD \perp AD .$ Circle $w$ with diameter $AB$ intersects $AD$ in points $A$ and $P,$ tangent from $P$ to $w$ intersects $CD$ at $M.$ The second tangent from $M$ to $w$ touches $w$ at $Q.$ Prove that midpoint of $CD$ lies on $BQ.$

Compare the number of distinct prime divisors of $200^2 \cdot 201^2 \cdot ... \cdot 900^2$ and $(200^2 -1)(201^2 -1)\cdot ... \cdot (900^2 -1) .$

There is a knight in the left down corner of $2009 \times 2009$ chessboard. The row and the column containing this corner are painted. The knight cannot move into painted cell and after its move new row and column that contains a square with knight become painted. Is it possible to paint all rows and columns of the chessboard?

In triangle $ABC$ points $M, N$ are midpoints of $BC, CA$ respectively. Point $P$ is inside $ABC$ such that $\angle BAP = \angle PCA = \angle MAC .$ Prove that $\angle PNA = \angle AMB .$

Find all functions $f : \mathbb R \to \mathbb R$ such that \[f\left(x+xy+f(y)\right)= \left( f(x)+\frac 12 \right) \left( f(y)+\frac 12 \right) \qquad \forall x,y \in \mathbb R.\]

Pairwise distinct real numbers $a, b, c$ satisfies the equality \[a +\frac 1b =b + \frac 1c =c+\frac 1a.\] Find all possible values of $abc .$

Find all prime numbers $p$ and positive integers $m$ such that $2p^2 + p + 9 = m^2.$

Given a $n \times n$ square board. Two players by turn remove some side of unit square if this side is not a bound of $n \times n$ square board. The player lose if after his move $n \times n$ square board became broken into two parts. Who has a winning strategy?

Let $ABCD$ be a parallelogram with $\angle BAC = 45^\circ,$ and $AC > BD .$ Let $w_1$ and $w_2$ be two circles with diameters $AC$ and $DC,$ respectively. The circle $w_1$ intersects $AB$ at $E$ and the circle $w_2$ intersects $AC$ at $O$ and $C$, and $AD$ at $F.$ Find the ratio of areas of triangles $AOE$ and $COF$ if $AO = a,$ and $FO = b .$

Find all possible real values of $a$ for which the system of equations \[\{\begin{array}{cc}x +y +z=0\\\text{ } \\ xy+yz+azx=0\end{array}\] has exactly one solution.

Let $M = \{1, 2, 3, 4, 6, 8,12,16, 24, 48\} .$ Find out which of four-element subsets of $M$ are more: those with product of all elements greater than $2009$ or those with product of all elements less than $2009.$

In triangle $ABC$ let $M$ and $N$ be midpoints of $BC$ and $AC,$ respectively. Point $P$ is inside $ABC$ such that $\angle BAP = \angle PBC = \angle PCA .$ Prove that if $\angle PNA = \angle AMB,$ then $ABC$ is isosceles triangle.

Find all polynomials $P(x)$ with real coefficients such that for all pairwise distinct positive integers $x, y, z, t$ with $x^2 + y^2 + z^2 = 2t^2$ and $\gcd(x, y, z, t ) = 1,$ the following equality holds \[2P^2(t ) + 2P(xy + yz + zx) = P^2(x + y + z) .\] Note. $P^2(k)=\left( P(k) \right)^2.$

Solve the system of equations \[\{\begin{array}{cc}x^3=2y^3+y-2\\ \text{ } \\ y^3=2z^3+z-2 \\ \text{ } \\ z^3 = 2x^3 +x -2\end{array}\]

Find all functions $f : \mathbb Z \to \mathbb Z$ such that \[f (n |m|) + f (n(|m| +2)) = 2f (n(|m| +1)) \qquad \forall m,n \in \mathbb Z.\] Note. $|x|$ denotes the absolute value of the integer $x.$

Point $O$ is inside triangle $ABC$ such that $\angle AOB = \angle BOC = \angle COA = 120^\circ .$ Prove that \[\frac{AO^2}{BC}+\frac{BO^2}{CA}+\frac{CO^2}{AB} \geq \frac{AO+BO+CO}{\sqrt 3}.\]

Let $G$ be a connected graph, with degree of all vertices not less then $m \geq 3$, such that there is no path through all vertices of $G$ being in every vertex exactly once. Find the least possible number of vertices of $G.$