The incircle of $ABC$ touches $BC$, $CA$, $AB$ at $K$, $L$, $M$ respectively. The line through $A$ parallel to $LK$ meets $MK$ at $P$, and the line through $A$ parallel to $MK$ meets $LK$ at $Q$. Show that the line $PQ$ bisects $AB$ and bisects $AC$.

A uniform rod is partially in water with one end suspended, as shown in figure. The density of the rod is $5/9$ that of water. At equilibrium, what portion of the rod is above water? $\textbf{(A) } 0.25\\ \textbf{(B) } 0.33\\ \textbf{(C) } 0.5\\ \textbf{(D) } 0.67\\ \textbf{(E) } 0.75$

Let $ABC$ be a triangle and let $X$,$Y$,$Z$ be interior points on the sides $BC$, $CA$, $AB$, respectively. Show that the magnified image of the triangle $XYZ$ under a homothety of factor $4$ from its centroid covers at least one of the vertices $A$, $B$, $C$.

Point $D$ lies inside triangle $ABC$ such that $\angle DAC = \angle DCA = 30^{\circ}$ and $\angle DBA = 60^{\circ}$. Point $E$ is the midpoint of segment $BC$. Point $F$ lies on segment $AC$ with $AF = 2FC$. Prove that $DE \perp EF$.

In right triangle $ABC$ with right angle $C$, $CA=30$ and $CB=16$. Its legs $\overline{CA}$ and $\overline{CB}$ are extended beyond $A$ and $B$. Points $O_{1}$ and $O_{2}$ lie in the exterior of the triangle and are the centers of two circles with equal radii. The circle with center $O_{1}$ is tangent to the hypotenuse and to the extension of leg CA, the circle with center $O_{2}$ is tangent to the hypotenuse and to the extension of leg CB, and the circles are externally tangent to each other. The length of the radius of either circle can be expressed as $p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

In a triangle $ABC$, let $H$ be the point of intersection of altitudes, $I$ the center of incircle, $O$ the center of circumcircle, $K$ the point where the incircle touches $BC$. Given that $IO$ is parallel to $BC$, prove that $AO$ is parallel to $HK$.

Let $P$ be a point in the plane of $\triangle ABC$, and $\gamma$ a line passing through $P$. Let $A', B', C'$ be the points where the reflections of lines $PA, PB, PC$ with respect to $\gamma$ intersect lines $BC, AC, AB$ respectively. Prove that $A', B', C'$ are collinear.

Each of the squares of an $8 \times 8$ board can be colored white or black. Find the number of colorings of the board such that every $2 \times 2$ square contains exactly 2 black squares and 2 white squares.

In triangle $ABC$ the midpoint of $BC$ is called $M$. Let $P$ be a variable interior point of the triangle such that $\angle CPM=\angle PAB$. Let $\Gamma$ be the circumcircle of triangle $ABP$. The line $MP$ intersects $\Gamma$ a second time at $Q$. Define $R$ as the reflection of $P$ in the tangent to $\Gamma$ at $B$. Prove that the length $|QR|$ is independent of the position of $P$ inside the triangle.

Let $ABC$ be a triangle. The incircle of triangle $ABC$ touches the side $BC$ at $A^{\prime}$, and the line $AA^{\prime}$ meets the incircle again at a point $P$. Let the lines $CP$ and $BP$ meet the incircle of triangle $ABC$ again at $N$ and $M$, respectively. Prove that the lines $AA^{\prime}$, $BN$ and $CM$ are concurrent.

$ ABCD$ is a quadrilateral. $ A'BCD'$ is the reflection of $ ABCD$ in $ BC,$ $ A''B'CD'$ is the reflection of $ A'BCD'$ in $ CD'$ and $ A''B''C'D'$ is the reflection of $ A''B'CD'$ in $ D'A''.$ Show that; if the lines $ AA''$ and $ BB''$ are parallel, then ABCD is a cyclic quadrilateral.

In a triangle $ABC$, with $AB\neq AC$ and $A\neq 60^{0},120^{0}$, $D$ is a point on line $AC$ different from $C$. Suppose that the circumcentres and orthocentres of triangles $ABC$ and $ABD$ lie on a circle. Prove that $\angle ABD=\angle ACB$.

Let $ ABP, BCQ, CAR$ be three non-overlapping triangles erected outside of acute triangle $ ABC$. Let $ M$ be the midpoint of segment $ AP$. Given that $ \angle PAB \equal{} \angle CQB \equal{} 45^\circ$, $ \angle ABP \equal{} \angle QBC \equal{} 75^\circ$, $ \angle RAC \equal{} 105^\circ$, and $ RQ^2 \equal{} 6CM^2$, compute $ AC^2/AR^2$. [i]Zuming Feng.[/i]

$m,n,p$ are positive integers which do not simultaneously equal to zero. $3$D Cartesian space is divided into unit cubes by planes each perpendicular to one of $3$ axes and cutting corresponding axis at integer coordinates. Each unit cube is filled with an integer from $1$ to $60$. A filling of integers is called [i]Dien Bien[/i] if, for each rectangular box of size $\{2m+1,2n+1,2p+1\}$, the number in the unit cube which has common centre with the rectangular box is the average of the $8$ numbers of the $8$ unit cubes at the $8$ corners of that rectangular box. How many [i]Dien Bien[/i] fillings are there? Two fillings are the same if one filling can be transformed to the other filling via a translation. [hide]translation from [url=http://artofproblemsolving.com/community/c6h592875p3515526]here[/url][/hide]