Let $ABC$ be a triangle and $M$ be the midpoint of $BC$, and let $AM$ meet the circumcircle of $ABC$ again at $R$. A line passing through $R$ and parallel to $BC$ meet the circumcircle of $ABC$ again at $S$. Let $U$ be the foot from $R$ to $BC$, and $T$ be the reflection of $U$ in $R$. $D$ lies in $BC$ such that $AD$ is an altitude. $N$ is the midpoint of $AD$. Finally let $AS$ and $MN$ meets at $K$. Prove that $AT$ bisector $MK$.

A right that passes through the incircle $ I$ of the triangle $ \Delta ABC$ intersects the side $ AB$ and $ CA$ in $ P$, respective $ Q$. We denote $ BC\equal{}a\ , \ AC\equal{}b\ ,\ AB\equal{}c$ and $ \frac{PB}{PA}\equal{}p\ ,\ \frac{QC}{QA}\equal{}q$. i) Prove that: \[ a(1\plus{}p)\cdot \overrightarrow{IP}\equal{}(a\minus{}pb)\overrightarrow{IB}\minus{}pc\overrightarrow{IC}\] ii) Show that $ a\equal{}bp\plus{}cq$. iii) If $ a^2\equal{}4bcpq$, then the rights $ AI\ ,\ BQ$ and $ CP$ are concurrents.

Let $ n$ be a positive integer. Consider a rectangle $ (90n\plus{}1)\times(90n\plus{}5)$ consisting of unit squares. Let $ S$ be the set of the vertices of these squares. Prove that the number of distinct lines passing through at least two points of $ S$ is divisible by $ 4$.

How many ways are there to remove an $11\times11$ square from a $2011\times2011$ square so that the remaining part can be tiled with dominoes ($1\times 2$ rectangles)?

Given an $\triangle ABC$ isoceles with base $BC$ we note with $M$ the midpoint of said base. Let $X$ be any point on the shortest arc $AM$ of the circumcircle of $\triangle ABM$ and let $T$ be a point on the inside $\angle BMA$ such that $\angle TMX = 90^o$ and $TX = BX$. Show that $\angle MTB - \angle CTM$ does not depend on $X$.

On the plane is a set of at least four points. If any one point from this set is removed, the resulting set has an axis of symmetry. Is it necessarily true that the whole set has an axis of symmetry?

A cuboid of volume $V$ contains a convex polyhedron $M$. The orthogonal projection of $M$ onto each face of the cuboid covers the entire face. What is the smallest possible volume of polyhedron $M$?

Let $ABC$ be an acute triangle with $AB\neq AC$, $M$ be the median of $BC$, and $H$ be the orthocenter of $\triangle ABC$. The circumcircle of $B$, $H$, and $C$ intersects the median $AM$ at $N$. Show that $\angle ANH=90^\circ$.

Constant points $B$ and $C$ lie on the circle $\omega$. The point middle of $BC$ is named $M$ by us. Assume that $A$ is a variable point on the $\omega$ and $H$ is the orthocenter of the triangle $ABC$. From the point $H$ we drop a perpendicular line to $MH$ to intersect the lines $AB$ and $AC$ at $X$ and $Y$ respectively. Prove that with the movement of $A$ on the $\omega$, the orthocenter of the triangle $AXY$ also moves on a circle.

Prove that in all triangles $\vartriangle ABC$ with $\angle A = 2 \angle B$ it holds that, if $D$ is the foot of the perpendicular from $C$ to the perpendicular bisector of $AB$, $\frac{AC}{DC}$ is constant for any value of $\angle B$.

Suppose the altitudes of a triangle are $10, 12$ and $15$. What is its semi-perimeter?

Let $S_1$ and $S_2$ be two spheres with distinct radii that touch externally. The spheres lie inside a cone $C$, and each sphere touches the cone in a full circle. Inside the cone there are $n$ additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone $C$, both of the spheres $S_1$ and $S_2$ externally, as well as the two neighboring solid spheres. What are the possible values of $n$? [i]Proposed by Iceland.[/i]

The diagram below shows four regular hexagons each with side length $1$ meter attached to the sides of a square. This figure is drawn onto a thin sheet of metal and cut out. The hexagons are then bent upward along the sides of the square so that $A_1$ meets $A_2$, $B_1$ meets $B_2$, $C_1$ meets $C_2$, and $D_1$ meets $D_2$. If the resulting dish is filled with water, the water will rise to the height of the corner where the $A_1$ and $A_2$ meet. there are relatively prime positive integers $m$ and $n$ so that the number of cubic meters of water the dish will hold is $\sqrt{\frac{m}{n}}$. Find $m+n$. [asy] /* File unicodetex not found. */ /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(7cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.3, xmax = 14.52, ymin = -8.3, ymax = 6.3; /* image dimensions */ draw((0,1)--(0,0)--(1,0)--(1,1)--cycle); draw((1,1)--(1,0)--(1.87,-0.5)--(2.73,0)--(2.73,1)--(1.87,1.5)--cycle); draw((0,1)--(1,1)--(1.5,1.87)--(1,2.73)--(0,2.73)--(-0.5,1.87)--cycle); draw((0,0)--(1,0)--(1.5,-0.87)--(1,-1.73)--(0,-1.73)--(-0.5,-0.87)--cycle); draw((0,1)--(0,0)--(-0.87,-0.5)--(-1.73,0)--(-1.73,1)--(-0.87,1.5)--cycle); /* draw figures */ draw((0,1)--(0,0)); draw((0,0)--(1,0)); draw((1,0)--(1,1)); draw((1,1)--(0,1)); draw((1,1)--(1,0)); draw((1,0)--(1.87,-0.5)); draw((1.87,-0.5)--(2.73,0)); draw((2.73,0)--(2.73,1)); draw((2.73,1)--(1.87,1.5)); draw((1.87,1.5)--(1,1)); draw((0,1)--(1,1)); draw((1,1)--(1.5,1.87)); draw((1.5,1.87)--(1,2.73)); draw((1,2.73)--(0,2.73)); draw((0,2.73)--(-0.5,1.87)); draw((-0.5,1.87)--(0,1)); /* dots and labels */ dot((1.87,-0.5),dotstyle); label("$C_1$", (1.72,-0.1), NE * labelscalefactor); dot((1.87,1.5),dotstyle); label("$B_2$", (1.76,1.04), NE * labelscalefactor); dot((1.5,1.87),dotstyle); label("$B_1$", (0.96,1.8), NE * labelscalefactor); dot((-0.5,1.87),dotstyle); label("$A_2$", (-0.26,1.78), NE * labelscalefactor); dot((-0.87,1.5),dotstyle); label("$A_1$", (-0.96,1.08), NE * labelscalefactor); dot((-0.87,-0.5),dotstyle); label("$D_2$", (-1.02,-0.18), NE * labelscalefactor); dot((-0.5,-0.87),dotstyle); label("$D_1$", (-0.22,-0.96), NE * labelscalefactor); dot((1.5,-0.87),dotstyle); label("$C_2$", (0.9,-0.94), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]

Let $G$ be the centroid of a triangle $ABC$ and let $d$ be a line that intersects $AB$ and $AC$ at $B_{1}$ and $C_{1}$, respectively, such that the points $A$ and $G$ are not separated by $d$. Prove that: $[BB_{1}GC_{1}]+[CC_{1}GB_{1}] \geq \frac{4}{9}[ABC]$.

$E,F$ are points on $AB,AC$ that satisfies $(B,E,F,C)$ cyclic. $D$ is the intersection of $BC$ and the perpendicular bisecter of $EF$, and $B',C'$ are the reflections of $B,C$ on $AD$. $X$ is a point on the circumcircle of $\triangle{BEC'}$ that $AB$ is perpendicular to $BX$,and $Y$ is a point on the circumcircle of $\triangle{CFB'}$ that $AC$ is perpendicular to $CY$. Show that $DX=DY$.

In the triangle $ABC$, points $D, E, F$ are on the sides $BC, CA$ and $AB$ respectively such that $FE$ is parallel to $BC$ and $DF$ is parallel to $CA$, Let P be the intersection of $BE$ and $DF$, and $Q$ the intersection of $FE$ and $AD$. Prove that $PQ$ is parallel to $AB$.

Let $X$ be an interior point of a rectangle $ABCD$. Let the bisectors of $\angle DAX$ and $\angle CBX$ intersect in $P$. A point $Q$ satisfies $\angle QAP=\angle QBP=90^\circ$. Show that $PX=QX$.

Let $ABCD$ be a square, and $C$ the circle whose diameter is $AB.$ Let $Q$ be an arbitrary point on the segment $CD.$ We know that $QA$ meets $C$ on $E$ and $QB$ meets it on $F.$ Also $CF$ and $DE$ intersect in $M.$ show that $M$ belongs to $C.$

Prove that a regular polygon with an odd number of edges cannot be partitioned into four pieces with equal areas by two lines that pass through the center of polygon.

$ABCDEF$ is an equiangular hexagon of perimeter $21$. Given that $AB = 3, CD = 4$, and $EF = 5$, compute the area of hexagon $ABCDEF$.

Let $X$ be an arbitrary point inside the circumcircle of a triangle $ABC$. The lines $BX$ and $CX$ meet the circumcircle in points $K$ and $L$ respectively. The line $LK$ intersects $BA$ and $AC$ at points $E$ and $F$ respectively. Find the locus of points $X$ such that the circumcircles of triangles $AFK$ and $AEL$ touch.

Let $P$ be a point inside the equilateral triangle $ABC$ such that $6\angle PBC = 3\angle PAC = 2\angle PCA$. Find the measure of the angle $\angle PBC$ .

On a Cartesian plane, $n$ parabolas are drawn, all of which are graphs of quadratic trinomials. No two of them are tangent. They divide the plane into many areas, one of which is above all the parabolas. Prove that the border of this area has no more than $2(n-1)$ corners (i.e. the intersections of a pair of parabolas).

Let $ABC$ be triangle with incenter $I$ . Let $AI$ intersect $BC$ at $D$. Point $P,Q$ lies inside triangle $ABC$ such that $\angle BPA + \angle CQA = 180^\circ$ and $B,Q,I,P,C$ concyclic in order . $BP$ intersect $CQ$ at $X$. Prove that the intersection of $(ABC)$ and $(APQ)$ lies on line $XD$.

Let $ABCD$ be a non-cyclic convex quadrilateral. The feet of perpendiculars from $A$ to $BC,BD,CD$ are $P,Q,R$ respectively, where $P,Q$ lie on segments $BC,BD$ and $R$ lies on $CD$ extended. The feet of perpendiculars from $D$ to $AC,BC,AB$ are $X,Y,Z$ respectively, where $X,Y$ lie on segments $AC,BC$ and $Z$ lies on $BA$ extended. Let the orthocenter of $\triangle ABD$ be $H$. Prove that the common chord of circumcircles of $\triangle PQR$ and $\triangle XYZ$ bisects $BH$.