A circle with center $O$ passes through points $A$ and $C$ and intersects the sides $AB$ and $BC$ of triangle $ABC$ at points $K$ and $N$, respectively. The circumcircles of triangles $ABC$ and $KBN$ meet at distinct points $B$ and $M$. Prove that $\angle OMB = 90^o$.

In parallelogram $ABCD$, angles $B$ and $D$ are acute while angles $A$ and $C$ are obtuse. The perpendicular from $C$ to $AB$ and the perpendicular from $A$ to $BC$ intersect at a point $P$ inside the parallelogram. If $PB=700$ and $PD=821$, what is $AC$?

Given a regular triangle $ABC$, whose area is $1$, and the point $P$ on its circumscribed circle. Lines $AP, BP, CP$ intersect, respectively, lines $BC, CA, AB$ at points $A', B', C'$. Find the area of the triangle $A'B'C'$.

Let $ABCD$ be a rhombus. $P$ is a point on side $ BC$ and $Q$ is a point on side $CD$ such that $BP = CQ$. Prove that centroid of triangle $APQ$ lies on the segment $BD.$ [i](6 points)[/i]

Square $ ABCD $ has side length $ 10 $. Point $ E $ is on $ \overline{BC} $, and the area of $ \bigtriangleup ABE $ is $ 40 $. What is $ BE $? $\textbf{(A)} \ 4 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \ 7 \qquad \textbf{(E)} \ 8 \qquad $ [asy] pair A,B,C,D,E; A=(0,0); B=(0,50); C=(50,50); D=(50,0); E = (30,50); draw(A--B); draw(B--E); draw(E--C); draw(C--D); draw(D--A); draw(A--E); dot(A); dot(B); dot(C); dot(D); dot(E); label("A",A,SW); label("B",B,NW); label("C",C,NE); label("D",D,SE); label("E",E,N); [/asy]

The polygon $P$, cut out of paper, is bent in a straight line and both halves are glued. Can the perimeter of the polygon $Q$ obtained by gluing be larger than the perimeter of the polygon $P$?

Find whether among all quadrilaterals, whose interiors lie inside a semi-circle of radius $r$, there exist one (or more) with maximum area. If so, determine their shape and area.

Given a cyclic quadrilateral $ABCD$. Let $X$ be a point on segment $BC$ ($X \not= BC$) such that line $AX$ is perpendicular to the angle bisector of $\angle CBD$, and $Y$ be a point on segment $AD$ ($Y \not= D)$ such that $BY$ is perpendicular to the angle bisector of $\angle CAD$. Prove that $XY$ is parallel to $CD$.

There are $ n \geq4 $ points on the plane, the distance between any two of which is an integer. Prove that there are at least $ \frac {1} {6} $ distances, each of which is divisible by $3$.

Assuming $a,b,c$ are the side-lengths of a triangle, show that \begin{align*} \frac{a^2+b^2-c^2}{ab} + \frac{b^2+c^2-a^2}{bc} + \frac{c^2+a^2-b^2}{ca} > 2. \end{align*} Also show that the inequality does not necessarily hold if you replace $2$ (on the right-hand side) by a bigger by a bigger number.

In a cyclic quadrilateral $ABCD$ with $AB=AD$ points $M$,$N$ lie on the sides $BC$ and $CD$ respectively so that $MN=BM+DN$ . Lines $AM$ and $AN$ meet the circumcircle of $ABCD$ again at points $P$ and $Q$ respectively. Prove that the orthocenter of the triangle $APQ$ lies on the segment $MN$ .

A sphere of radius $1$ is covered in ink and rolling around between concentric spheres of radii $3$ and $5$. If this process traces a region of area 1 on the larger sphere, what is the area of the region traced on the smaller sphere?

Let $P_1, P_2, \dots , P_n$ be distinct points of the plane, $n \geq 2$. Prove that \[ \max_{1\leq i<j\leq n} P_iP_j > \frac{\sqrt 3}{2}(\sqrt n -1) \min_{1\leq i<j\leq n} P_iP_j \]

Let $A,B,C,D,E,F$ be the midpoints of consecutive sides of a hexagon with parallel opposite sides. Prove that the points $AB\cap DE$, $BC\cap EF$, $AC\cap DF$ lie on a line.

In an acute-angled triangle $\triangle ABC$, D is the point on BC such that AD bisects ∠BAC, E and F are the feet of the perpendiculars from D onto AB and AC respectively. The segments BF and CE intersect at K. Prove that AK is perpendicular to BC.

Given $P$ a point m inside a triangle acute-angled $ABC$ and $DEF$ intersections of lines with that $AP$, $BP$, $CP$ with$\left[ BC \right],\left[ CA \right],$respective $\left[ AB \right]$ a) Show that the area of the triangle $DEF$ is at most a quarter of the area of the triangle $ABC$ b) Show that the radius of the circle inscribed in the triangle $DEF$ is at most a quarter of the radius of the circle circumscribed on triangle $4ABC.$

A domino is a $2$ by $1$ rectangle . For what integers $m$ and $n$ can we cover an $m*n$ rectangle with non-overlapping dominoes???

Let $A$ and $B$ be points on the circle $k$ with center $O$, so that $AB> AO$. Let $C$ be the intersection of the bisectors of $\angle OAB$ and $k$, different from $A$. Let $D$ be the intersection of the straight line $AB$ with the circumcircle of the triangle $OBC$, different from $B$. Show that $AD = AO$ .

The figure below shows a semicircle inscribed in a right triangle. [asy] draw((0, 0) -- (15, 0) -- (0, 8) -- cycle); real r = 120 / 23; real theta = -aTan(8/15); draw(arc((r, r), r, theta + 180, theta + 360)); [/asy] The triangle has legs of length 8 and 15. The semicircle is tangent to the two legs, and its diameter is on the hypotenuse. What is the radius of the semicircle?

Given a closed broken line $A_1A_2A_3...A_n$ in space and a plane intersecting all its segments, $A_1A_2$ at $B_1, A_2A_3$ at $B_2$ ,$... $, $A_nA_1$ at $B_n$, prove that $$\frac{A_1B_1}{B_1A_2}\cdot \frac{A_2B_2}{B_2A_3}\cdot \frac{A_3B_3}{B_3A_4}\cdot ...\cdot \frac{A_nB_n}{B_nA_1}= 1$$.

Let $I$ be the incenter of an acute triangle $ABC$. Assume that $K_1$ is the point such that $AK_1 \perp BC$ and the circle with center $K_1$ of radius $K_1A$ is internally tangent to the incircle of triangle $ABC$ at $A_1$. The points $B_1, C_1$ are defined similarly. a) Prove that $AA_1, BB_1, CC_1$ are concurrent at a point $P$. b) Let $\omega_1,\omega_2,\omega_3$ be the excircles of triangle $ABC$ with respect to $A, B, C$, respectively. The circles $\gamma_1,\gamma_2\gamma_3$ are the reflections of $\omega_1,\omega_2,\omega_3$ with respect to the midpoints of $BC, CA, AB$, respectively. Prove that P is the radical center of $\gamma_1,\gamma_2,\gamma_3$.

The edges of a regular tetrahedron with vertices $A ,~ B,~ C$, and $D$ each have length one. Find the least possible distance between a pair of points $P$ and $Q$, where $P$ is on edge $AB$ and $Q$ is on edge $CD$. $\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{3}{4}\qquad\textbf{(C) }\frac{\sqrt{2}}{2}\qquad\textbf{(D) }\frac{\sqrt{3}}{2}\qquad\textbf{(E) }\frac{\sqrt{3}}{3}$ [asy] size(150); import patterns; pair D=(0,0),C=(1,-1),B=(2.5,-0.2),A=(1,2),AA,BB,CC,DD,P,Q,aux; add("hatch",hatch()); //AA=new A and etc. draw(rotate(100,D)*(A--B--C--D--cycle)); AA=rotate(100,D)*A; BB=rotate(100,D)*D; CC=rotate(100,D)*C; DD=rotate(100,D)*B; aux=midpoint(AA--BB); draw(BB--DD); P=midpoint(AA--aux); aux=midpoint(CC--DD); Q=midpoint(CC--aux); draw(AA--CC,dashed); dot(P); dot(Q); fill(DD--BB--CC--cycle,pattern("hatch")); label("$A$",AA,W); label("$B$",BB,S); label("$C$",CC,E); label("$D$",DD,N); label("$P$",P,S); label("$Q$",Q,E); //Credit to TheMaskedMagician for the diagram [/asy]

In acute-angled triangle $ABC$ ($AC > AB$), point $H$ is the orthocenter and point $M$ is the midpoint of the segment $BC$. The median $AM$ intersects the circumcircle of triangle $ABC$ at $X$. The line $CH$ intersects the perpendicular bisector of $BC$ at $E$ and the circumcircle of the triangle $ABC$ again at $F$. Point $J$ lies on circle $\omega$, passing through $X, E,$ and $F$, such that $BCHJ$ is a trapezoid ($CB \parallel HJ$). Prove that $JB$ and $EM$ meet on $\omega$. [i]Proposed by Alireza Dadgarnia[/i]

Consider a polyhedron having $100$ edges. (a) Find the maximal possible number of its edges which can be intersected by a plane (not containing any vertices of the polyhedron) if the polyhedron is convex. (b) Prove that for a non-convex polyhedron this number i. can be as great as $96$, ii. cannot be as great as $100$. (A Andjans, Riga

Consider $\triangle OAB$ with acute angle $AOB$. Thorugh a point $M \neq O$ perpendiculars are drawn to $OA$ and $OB$, the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\triangle OPQ$ is $H$. What is the locus of $H$ if $M$ is permitted to range over a) the side $AB$; b) the interior of $\triangle OAB$.