Let $ABCD$ be a cyclic quadrilaterla and let $E$ be the point of intersection of its diagonals $AC$ and $BD$. Suppose $AD$ and $BC$ meet in $F$. Let the midpoints of $AB$ and $CD$ be $G$ and $H$ respectively. If $\Gamma $ is the circumcircle of triangle $EGH$, prove that $FE$ is tangent to $\Gamma $.

The diagonals of a cyclic quadrilateral $ABCD$ meet at $O$. Let $S_1, S_2$ be the circumcircles of triangles $ABO$ and $CDO$ respectively, and $O,K$ their intersection points. The lines through $O$ parallel to $AB$ and $CD$ meet $S_1$ and $S_2$ again at $L$ and $M$, respectively. Points $P$ and $Q$ on segments $OL$ and $OM$ respectively are taken such that $OP : PL = MQ : QO$. Prove that $O,K, P,Q$ lie on a circle.

Let $ABC$ be an isosceles triangle with $AB=AC$ . Let also $\omega$ be a circle of center $K$ tangent to the line $AC$ at $C$ which intersects the segment $BC$ again at $H$ . Prove that $HK \bot AB $.

Let $ABC$ be an acute triangle, with $AB<AC$. Let $K$ be the midpoint of the arch $BC$ that does not contain $A$ and let $P$ be the midpoint of $BC$. Let $I_B,I_C$ be the $B$-excenter and $C$-excenter of $ABC$, respectively. Let $Q$ be the reflection of $K$ with respect to $A$. Prove that the points $P,Q,I_B,I_C$ are concyclic.

A uniform pool ball of radius $r$ and mass $m$ begins at rest on a pool table. The ball is given a horizontal impulse $J$ of fixed magnitude at a distance $\beta r$ above its center, where $-1 \le \beta \le 1$. The coefficient of kinetic friction between the ball and the pool table is $\mu$. You may assume the ball and the table are perfectly rigid. Ignore effects due to deformation. (The moment of inertia about the center of mass of a solid sphere of mass $m$ and radius $r$ is $I_{cm} = \frac{2}{5}mr^2$.) [asy] size(250); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); filldraw(circle((0,0),1),gray(.8)); draw((-3,-1)--(3,-1)); draw((-2.4,0.1)--(-2.4,0.6),EndArrow); draw((-2.5,0)--(2.5,0),dashed); draw((-2.75,0.7)--(-0.8,0.7),EndArrow); label("$J$",(-2.8,0.7),W); label("$\beta r$",(-2.3,0.35),E); draw((0,-1.5)--(0,1.5),dashed); draw((1.7,-0.1)--(1.7,-0.9),BeginArrow,EndArrow); label("$r$",(1.75,-0.5),E); [/asy] (a) Find an expression for the final speed of the ball as a function of $J$, $m$, and $\beta$. (b) For what value of $\beta$ does the ball immediately begin to roll without slipping, regardless of the value of $\mu$?

Consider the sequence $ a_n\equal{}\sqrt[3]{n^3\plus{}3n^2\plus{}2n\plus{}1}\plus{}a\sqrt[5]{n^5\plus{}5n^4\plus{}1}\plus{}\frac{ln(e^{n^2}\plus{}n\plus{}2)}{n\plus{}2}\plus{}b$. Find $ a,b \in \mathbb{R}$ such that $ \displaystyle\lim_{n\to\infty}a_n\equal{}5$.

We have an acute-angled triangle $ABC$, and $AA',BB'$ are its altitudes. A point $D$ is chosen on the arc $ACB$ of the circumcircle of $ABC$. If $P=AA'\cap BD,Q=BB'\cap AD$, show that the midpoint of $PQ$ lies on $A'B'$.

A convex quadrilateral $ABCD$ is inscribed in a circle with center $O$. The diagonals $AC$, $BD$ of $ABCD$ meet at $P$. Circumcircles of $\triangle ABP$ and $\triangle CDP$ meet at $P$ and $Q$ ($O,P,Q$ are pairwise distinct). Show that $\angle OQP=90^{\circ}$.

Acute triangle $ABC$ is inscribed in a circle with center $O$. The reflections of $O$ across the three altitudes of the triangle are called $U$, $V$, $W$: $U$ over the altitude from $A$, $V$ over the altitude from $B$, and $W$ over the altitude from $C$. Let $\ell_A$ be a line through $A$ parallel to $VW$, and define $\ell_B$, $\ell_C$ similarly. Prove that the three lines $\ell_A$, $\ell_B$, $\ell_C$ are concurrent.

In triangle $ ABC$ the medians $ \overline{AD}$ and $ \overline{CE}$ have lengths 18 and 27, respectively, and $ AB \equal{} 24$. Extend $ \overline{CE}$ to intersect the circumcircle of $ ABC$ at $ F$. The area of triangle $ AFB$ is $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.

The triangle $ ABC$ with the circumcircle $ k(U,r)$ is given. On the extension of the radii $ UA$ a point $ P$ is chosen. The reflection of the line $ PB$ on the line $ BA$ is called $ g$. Likewise the reflection of the line $ PC$ on the line $ CA$ is called $ h$. The intersection of $ g$ and $ h$ is called $ Q$. Find the geometric location of all possible intersections $ Q$, while $ P$ passes through the extension of the radii $ UA$.

Two circles with radii 1 meet in points $X, Y$, and the distance between these points also is equal to $1$. Point $C$ lies on the first circle, and lines $CA, CB$ are tangents to the second one. These tangents meet the first circle for the second time in points $B', A'$. Lines $AA'$ and $BB'$ meet in point $Z$. Find angle $XZY$.

In trapezoid $ ABCD$ with $ \overline{BC}\parallel\overline{AD}$, let $ BC\equal{}1000$ and $ AD\equal{}2008$. Let $ \angle A\equal{}37^\circ$, $ \angle D\equal{}53^\circ$, and $ m$ and $ n$ be the midpoints of $ \overline{BC}$ and $ \overline{AD}$, respectively. Find the length $ MN$.

Let a non-isosceles acute triangle ABC with tha attitude AD, BE, CF and the orthocenter H. DE, DF intersect (AD) at M, N respectively. $P\in AB,Q\in AC$ satisfy $NP\perp AB,MQ\perp AC$ a) Prove that EF is the tangent line of (APQ) b) Let T be the tangency point of (APQ) with EF,.DT $\cap$ MN={K}. L is the reflection of A in MN. Prove that MN, EF ,(DLK) pass through a piont

$ABC$ be a triangle. Its incircle touches the sides $CB, AC, AB$ respectively at $N_{A},N_{B},N_{C}$. The orthic triangle of $ABC$ is $H_{A}H_{B}H_{C}$ with $H_{A}, H_{B}, H_{C}$ are respectively on $BC, AC, AB$. The incenter of $AH_{C}H_{B}$ is $I_{A}$; $I_{B}$ and $I_{C}$ were defined similarly. Prove that the hexagon $I_{A}N_{B}I_{C}N_{A}I_{B}N_{C}$ has all sides equal.

Let $(O)$ be a fixed circle with the radius $R$. Let $A$ and $B$ be fixed points in $(O)$ such that $A,B,O$ are not collinear. Consider a variable point $C$ lying on $(O)$ ($C\neq A,B$). Construct two circles $(O_1),(O_2)$ passing through $A,B$ and tangent to $BC,AC$ at $C$, respectively. The circle $(O_1)$ intersects the circle $(O_2)$ in $D$ ($D\neq C$). Prove that: a) \[ CD\leq R \] b) The line $CD$ passes through a point independent of $C$ (i.e. there exists a fixed point on the line $CD$ when $C$ lies on $(O)$).

Let $ ABC$ be an acute-angled triangle. We consider the equilateral triangle $ A'UV$, where $ A' \in (BC)$, $ U\in (AC)$ and $ V\in(AB)$ such that $ UV \parallel BC$. We define the points $ B',C'$ in the same way. Prove that $ AA'$, $ BB'$ and $ CC'$ are concurrent.

Let $P$ be a regular polygon with $ 4k + 1 $ sides (where $ k $ is a natural) whose vertices are $ A_1, A_2, ..., A_ {4k + 1} $ (in that order ). Each vertex $ A_j $ of $P$ is assigned a natural of the set $ \{1,2, ..., 4k + 1 \} $ such that no two vertices are assigned the same number. On $P$ the following operation is performed: Let $ B_j $ be the midpoint of the side $ A_jA_ {j + 1} $ for $ j = 1,2, ..., 4k + 1 $ (where is consider $ A_ {4k + 2} = A_1 $). If $ a $, $ b $ are the numbers assigned to $ A_ {j} $ and $ A_ {j + 1} $, respectively, the midpoint $ B_j $ is written the number $ 7a-3b $. By doing this with each of the $ 4k + 1 $ sides, the $ 4k + 1 $ vertices initially arranged are erased. We will say that a natural $ m $ is [i]fatal [/i] if for all natural $ k $ , no matter how the vertices of $P$ are initially arranged, it is impossible to obtain $ 4k + 1 $ equal numbers through a finite amount of operations from $ m $. a) Determine if the $ 2010 $ is fatal or not. Justify. b) Prove that there are infinite fatal numbers. [color=#f00]PS. A help in translation of the 2nd paragraph is welcome[/color]. [hide=Original wording]Diremos que un natural $m$ es fatal si no importa cómo se disponen inicialmente los vértices de ${P}$, es imposible obtener mediante una cantidad finita de operaciones $4k+1$ números iguales a $m$.[/hide]

When real numbers $x,\ y$ moves in the constraint with $x^2+xy+y^2=6.$ Find the range of $x^2y+xy^2-x^2-2xy-y^2+x+y.$ 30 points

Let $ ABC$ be a triangle; $ \Gamma_A,\Gamma_B,\Gamma_C$ be three equal, disjoint circles inside $ ABC$ such that $ \Gamma_A$ touches $ AB$ and $ AC$; $ \Gamma_B$ touches $ AB$ and $ BC$; and $ \Gamma_C$ touches $ BC$ and $ CA$. Let $ \Gamma$ be a circle touching circles $ \Gamma_A, \Gamma_B, \Gamma_C$ externally. Prove that the line joining the circum-centre $ O$ and the in-centre $ I$ of triangle $ ABC$ passes through the centre of $ \Gamma$.

Given any set of $9$ points in the plane such that there is no $3$ of them collinear, show that for each point $P$ of the set, the number of triangles with its vertices on the other $8$ points and that contain $P$ on its interior is even.

In a plane the circles $\mathcal K_1$ and $\mathcal K_2$ with centers $I_1$ and $I_2$, respectively, intersect in two points $A$ and $B$. Assume that $\angle I_1AI_2$ is obtuse. The tangent to $\mathcal K_1$ in $A$ intersects $\mathcal K_2$ again in $C$ and the tangent to $\mathcal K_2$ in $A$ intersects $\mathcal K_1$ again in $D$. Let $\mathcal K_3$ be the circumcircle of the triangle $BCD$. Let $E$ be the midpoint of that arc $CD$ of $\mathcal K_3$ that contains $B$. The lines $AC$ and $AD$ intersect $\mathcal K_3$ again in $K$ and $L$, respectively. Prove that the line $AE$ is perpendicular to $KL$.

In triangle $ABC$ with incenter $I$ and circumcenter $O$, let $A',B',C'$ be the points of tangency of its circumcircle with its $A,B,C$-mixtilinear circles, respectively. Let $\omega_A$ be the circle through $A'$ that is tangent to $AI$ at $I$, and define $\omega_B, \omega_C$ similarly. Prove that $\omega_A,\omega_B,\omega_C$ have a common point $X$ other than $I$, and that $\angle AXO = \angle OXA'$. [i]Proposed by Sammy Luo[/i]

Two of the squares of a $ 7\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible?

Let $ABCD$ be a quadrilateral, and let $E$ and $F$ be points on sides $AD$ and $BC$, respectively, such that $\frac{AE}{ED} = \frac{BF}{FC}$. Ray $FE$ meets rays $BA$ and $CD$ at $S$ and $T$, respectively. Prove that the circumcircles of triangles $SAE$, $SBF$, $TCF$, and $TDE$ pass through a common point.