Points $A', B', C'$ are the reflections of vertices $A, B, C$ about the opposite sidelines of triangle $ABC$. Prove that the circles $AB'C', A'BC',$ and $A'B'C$ have a common point.

6. Consider a triangle $ABC$ with $1 < \frac{AB}{AC} < \frac{3}{2}$. Let $M$ and $N$, respectively, be variable points of the sides $AB$ and $AC$, different from $A$, such that $\frac{MB}{AC} - \frac{NC}{AB} = 1$. Show that circumcircle of triangle $AMN$ pass through a fixed point different from $A$.

a) Year 1872 Texas 3 gold miners found a peice of gold. They have a coin that with possibility of $\frac 12$ it will come each side, and they want to give the piece of gold to one of themselves depending on how the coin will come. Design a fair method (It means that each of the 3 miners will win the piece of gold with possibility of $\frac 13$) for the miners. b) Year 2005, faculty of Mathematics, Sharif university of Technolgy Suppose $0<\alpha<1$ and we want to find a way for people name $A$ and $B$ that the possibity of winning of $A$ is $\alpha$. Is it possible to find this way? c) Year 2005 Ahvaz, Takhti Stadium Two soccer teams have a contest. And we want to choose each player's side with the coin, But we don't know that our coin is fair or not. Find a way to find that coin is fair or not? d) Year 2005,summer In the National mathematical Oympiad in Iran. Each student has a coin and must find a way that the possibility of coin being TAIL is $\alpha$ or no. Find a way for the student.

Determine all $n \in \mathbb{Z}^+$ such that a regular hexagon (i.e. all sides equal length, all interior angles same size) can be partitioned in finitely many $n-$gons such that they can be composed into $n$ congruent regular hexagons in a non-overlapping way upon certain rotations and translations.

$B$ is a point on the line which is tangent to a circle at the point $A$. The line segment $AB$ is rotated about the centre of the circle through some angle to the line segment $A'B'$. Prove that the line $AA'$ passes through the midpoint of $BB'$.

We match sets $ M$ of points in the coordinate plane to sets $ M*$ according to the rule that $ (x*,y*) \in M*$ if and only if $ x \cdot x* \plus{} y \cdot y* \leq 1$ whenever $ (x,y) \in M.$ Find all triangles $ Q$ such that $ Q*$ is the reflection of $ Q$ in the origin.

Two circles lie completely outside each other.Let $A$ be the point of intersection of internal common tangents of the circles and let $K$ be the projection of this point onto one of their external common tangents.The tangents,different from the common tangent,to the circles through point $K$ meet the circles at $M_1$ and $M_2$.Prove that the line $AK$ bisects angle $M_1 KM_2$.

Given triangle $ ABC$. Point $ O$ is the center of the excircle touching the side $ BC$. Point $ O_1$ is the reflection of $ O$ in $ BC$. Determine angle $ A$ if $ O_1$ lies on the circumcircle of $ ABC$.

Given an acute angled triangle $ ABC$ , $ O$ is the circumcenter and $ H$ is the orthocenter.Let $ A_1$,$ B_1$,$ C_1$ be the midpoints of the sides $ BC$,$ AC$ and $ AB$ respectively. Rays $ [HA_1$,$ [HB_1$,$ [HC_1$ cut the circumcircle of $ ABC$ at $ A_0$,$ B_0$ and $ C_0$ respectively.Prove that $ O$,$ H$ and $ H_0$ are collinear if $ H_0$ is the orthocenter of $ A_0B_0C_0$

Let $ ABC$ be triangle, $ I$ its in-center; $ A_1,B_1,C_1$ be the reflections of $ I$ in $ BC, CA, AB$ respectively. Suppose the circum-circle of triangle $ A_1B_1C_1$ passes through $ A$. Prove that $ B_1,C_1,I,I_1$ are concylic, where $ I_1$ is the in-center of triangle $ A_1,B_1,C_1$.

Let $ABC$ be a triangle and its bisector at $A$ cuts its circumcircle at $D.$ Let $I$ be the incenter of triangle $ABC,$ $M$ be the midpoint of $BC,$ $P$ is the symmetric to $I$ with respect to $M$ (Assuming $P$ is in the circumcircle). Extend $DP$ until it cuts the circumcircle again at $N.$ Prove that among segments $AN, BN, CN$, there is a segment that is the sum of the other two.

Let $ A_1,A_2,...,A_n$ be the vertices of a closed convex $ n$-gon $ K$ numbered consecutively. Show that at least $ n\minus{}3$ vertices $ A_i$ have the property that the reflection of $ A_i$ with respect to the midpoint of $ A_{i\minus{}1}A_{i\plus{}1}$ is contained in $ K$. (Indices are meant $ \textrm{mod} \;n\ .$)

The circle, which is tangent to the circumcircle of isosceles triangle $ABC$ ($AB=AC$), is tangent $AB$ and $AC$ at $P$ and $Q$, respectively. Prove that the midpoint $I$ of the segment $PQ$ is the center of the excircle (which is tangent to $BC$) of the triangle .

Assume $ABC$ is an isosceles triangle that $AB=AC$ Suppose $P$ is a point on extension of side $BC$. $X$ and $Y$ are points on $AB$ and $AC$ that: \[PX || AC \ , \ PY ||AB \] Also $T$ is midpoint of arc $BC$. Prove that $PT \perp XY$

Let $\triangle ABC$ be an acute triangle with $AB>AC$, let $I$ be the center of the incircle. Let $M,N$ be the midpoint of $AC$ and $AB$ respectively. $D,E$ are on $AC$ and $AB$ respectively such that $BD\parallel IM$ and $CE\parallel IN$. A line through $I$ parallel to $DE$ intersects $BC$ in $P$. Let $Q$ be the projection of $P$ on line $AI$. Prove that $Q$ is on the circumcircle of $\triangle ABC$.

A hexagonal table is given, as the one on the drawing, which has $~$ $2012$ $~$ columns. There are $~$ $2012$ $~$ hexagons in each of the odd columns, and there are $~$ $2013$ $~$ hexagons in each of the even columns. The number $~$ $i$ $~$ is written in each hexagon from the $~$ $i$-th column. Changing the numbers in the table is allowed in the following way: We arbitrarily select three adjacent hexagons, we rotate the numbers, and if the rotation is clockwise then the three numbers decrease by one, and if we rotate them counterclockwise the three numbers increase by one (see the drawing below). What's the maximum number of zeros that can be obtained in the table by using the above-defined steps.

The circle $\omega_1$ with diameter $[AB]$ and the circle $\omega_2$ with center $A$ intersects at points $C$ and $D$. Let $E$ be a point on the circle $\omega_2$, which is outside $\omega_1$ and at the same side as $C$ with respect to the line $AB$. Let the second point of intersection of the line $BE$ with $\omega_2$ be $F$. For a point $K$ on the circle $\omega_1$ which is on the same side as $A$ with respect to the diameter of $\omega_1$ passing through $C$ we have $2\cdot CK \cdot AC = CE \cdot AB$. Let the second point of intersection of the line $KF$ with $\omega_1$ be $L$. Show that the symmetric of the point $D$ with respect to the line $BE$ is on the circumcircle of the triangle $LFC$.

Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.

A finite set of points $X$ and an equilateral triangle $T$ are given on a plane. Suppose that every subset $X'$ of $X$ with no more than $9$ elements can be covered by two images of $T$ under translations. Prove that the whole set $X$ can be covered by two images of $T$ under translations.

Let $P$ be an arbitrary variable point in the plane of a triangle $ABC. A_1$ is the projection of $P$ onto $BC, A_2$ is the midpoint of line segment $PA_1, A_2P$ meets $BC$ at $A_3, A_4$ is the reflection of $P$ about $A_3$. Prove that $PA_4$ has a fixed point. Trần Quang Hùng

How many ways are there to color the vertices of a triangle red, green, blue, or yellow such that no two vertices have the same color? Rotations and reflections are considered distinct.

In a triangle $ABC$, $\angle C=45$. $AD$ is the altitude of the triangle. $X$ is on $AD$ such that $\angle XBC=90-\angle B$ ($X$ is in the triangle). $AD$ and $CX$ cut the circumcircle of $ABC$ in $M$ and $N$ respectively. if tangent to circumcircle of $ABC$ at $M$ cuts $AN$ at $P$, prove that $P$,$B$ and $O$ are collinear.(25 points) the exam time was 4 hours and 30 minutes.

Given a convex quadrilateral $ABCD$. The points $P$ and $Q$ are the midpoints of the diagonals $AC$ and $BD$ respectively. The line $PQ$ intersects the lines $AB$ and $CD$ at $N$ and $M$ respectively. Prove that the circumcircles of triangles $NAP$, $NBQ$, $MQD$, and $MPC$ have a common point.

Three circles touch each other externally and all these cirlces also touch a fixed straight line. Let $A,B,C$ be the mutual points of contact of these circles. If $\omega$ denotes the Brocard angle of the triangle $ABC$, prove that $\cot{\omega}$ = 2.

In $ \triangle{ABC}$ with $ AB = 12$, $ BC = 13$, and $ AC = 15$, let $ M$ be a point on $ \overline{AC}$ such that the incircles of $ \triangle{ABM}$ and $ \triangle{BCM}$ have equal radii. Let $ p$ and $ q$ be positive relatively prime integers such that $ \tfrac{AM}{CM} = \tfrac{p}{q}$. Find $ p + q$.