The circles $k_1$ and $k_2$ are tangent at the point $P$. A line is drawn through $P$, cutting $k_1$ at $A_1$ and $k_2$ at $A_2$. A second line is drawn through $P$, cutting $k_1$ at $B_1$ and $k_2$ at $B_2$. Prove that the triangles $PA_1B_1$ and $PA_2B_2$ are similar.

A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units? [asy] import three; defaultpen(linewidth(0.8)); real r=0.5; currentprojection=orthographic(1,1/2,1/4); draw(unitcube, white, thick(), nolight); draw(shift(1,0,0)*unitcube, white, thick(), nolight); draw(shift(1,-1,0)*unitcube, white, thick(), nolight); draw(shift(1,0,-1)*unitcube, white, thick(), nolight); draw(shift(2,0,0)*unitcube, white, thick(), nolight); draw(shift(1,1,0)*unitcube, white, thick(), nolight); draw(shift(1,0,1)*unitcube, white, thick(), nolight);[/asy] $\textbf{(A)} \:1 : 6 \qquad\textbf{ (B)}\: 7 : 36 \qquad\textbf{(C)}\: 1 : 5 \qquad\textbf{(D)}\: 7 : 30\qquad\textbf{ (E)}\: 6 : 25$

$ABCDE$ is a regular pentagon. Determine the smallest value of the expression \[\frac{|PA|+|PB|}{|PC|+|PD|+|PE|},\] where $P$ is an arbitrary point lying in the plane of the pentagon $ABCDE$.

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\omega$. Let $F_A, F_B, F_C$ and $F_D$ be the midpoints of arcs $AB, BC, CD$ and $DA$ of $\omega$. Let $I_A, I_B, I_C$ and $I_D$ be the incenters of triangles $DAB, ABC, BCD$ and $CDA$, respectively. Let $\omega_A$ denote the circle that is tangent to $\omega$ at $F_A$ and also tangent to line segment $CD$. Similarly, let $\omega_C$ denote the circle that is tangent to $\omega$ at $F_C$ and tangent to line segment $AB$. Finally, let $T_B$ denote the second intersection of $\omega$ and circle $F_BI_BI_C$ different from $F_B$, and let $T_D$ denote the second intersection of $\omega$ and circle $F_DI_DI_A$. Prove that the radical axis of circles $\omega_A$ and $\omega_C$ passes through points $T_B$ and $T_D$.

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.

In an acute non-isosceles triangle $ABC$, the altitude $AA'$ is drawn and point $H$ is the intersection point of the altitudes and and $O$ is the center of the circumscribed circle. Prove that the point symmetric to the circumcenter of triangle $HOA'$ wrt straight line $HO$, lies on a midline of triangle $ABC$.

In regular triangular pyramid $P-ABC$, $PO$ is its height, $M$ is the midpoint of $PO$. Draw the plane that passes $AM$ and parallel to $BC$. Now the triangular pyramid is divided into two parts. Find the ratio of their volume.

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.

A right pyramid has regular octagon $ABCDEFGH$ with side length $1$ as its base and apex $V.$ Segments $\overline{AV}$ and $\overline{DV}$ are perpendicular. What is the square of the height of the pyramid? $ \textbf{(A) }1 \qquad \textbf{(B) }\frac{1+\sqrt2}{2} \qquad \textbf{(C) }\sqrt2 \qquad \textbf{(D) }\frac32 \qquad \textbf{(E) }\frac{2+\sqrt2}{3} \qquad $

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.

A unit sphere is centered at $(0, 0, 1)$. There is a point light source located at $(1, 0, 4)$ that sends out light uniformly in every direction but is blocked by the sphere. What is the area of the sphere’s shadow on the $x-y$ plane? (A point $(a, b, c)$ denotes the point in three dimensions with $x$-coordinate $a$, $y$-coordinate $b$, and $z$-coordinate $c$)

Let $D$ be a point on the side $BC$ of an equilateral triangle $ABC$ where $D$ is different than the vertices. Let $I$ be the excenter of the triangle $ABD$ opposite to the side $AB$ and $J$ be the excenter of the triangle $ACD$ opposite to the side $AC$. Let $E$ be the second intersection point of the circumcircles of triangles $AIB$ and $AJC$. Prove that $A$ is the incenter of the triangle $IEJ$.

In $ \triangle ABC$ points $ D$ and $ E$ lie on $ \overline{BC}$ and $ \overline{AC}$, respectively. If $ \overline{AD}$ and $ \overline{BE}$ intersect at $ T$ so that $ AT/DT \equal{} 3$ and $ BT/ET \equal{} 4$, what is $ CD/BD$? [asy]unitsize(2cm); defaultpen(linewidth(.8pt)); pair A = (0,0); pair C = (2,0); pair B = dir(57.5)*2; pair E = waypoint(C--A,0.25); pair D = waypoint(C--B,0.25); pair T = intersectionpoint(D--A,E--B); label("$B$",B,NW);label("$A$",A,SW);label("$C$",C,SE);label("$D$",D,NE);label("$E$",E,S);label("$T$",T,2*W+N); draw(A--B--C--cycle); draw(A--D); draw(B--E);[/asy]$ \textbf{(A)}\ \frac {1}{8}\qquad \textbf{(B)}\ \frac {2}{9}\qquad \textbf{(C)}\ \frac {3}{10}\qquad \textbf{(D)}\ \frac {4}{11}\qquad \textbf{(E)}\ \frac {5}{12}$

Let $ABCD$ be a convex quadrilateral such that $BC=AD$. Let $M$ and $N$ be the midpoints of $AB$ and $CD$, respectively. The lines $AD$ and $BC$ meet the line $MN$ at $P$ and $Q$, respectively. Prove that $CQ=DP$.

Consider $\triangle ABC$ an isosceles triangle such that $AB = BC$. Let $P$ be a point satisfying $$\angle ABP = 80^\circ, \angle CBP = 20^\circ, \textrm{and} \hspace{0.17cm} AC = BP$$ Find all possible values of $\angle BCP$.

Given an isosceles trapezoid $ABCD$ with $AB = 6$, $CD = 12$, and area $36$, find $BC$.

In a $\triangle {ABC}$, consider a point $E$ in $BC$ such that $AE \perp BC$. Prove that $AE=\frac{bc}{2r}$, where $r$ is the radio of the circle circumscripte, $b=AC$ and $c=AB$.

$B$ 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 $BB'$.

In an isosceles trapezoid, one of the bases is three times larger than the other. Angle at a greater basis is equal to $45^o$. Show how to cut this trapezium into three parts and make a square with them. Justify your answer.

Vlad draws 100 rays in the Euclidean plane. David then draws a line $\ell$ and pays Vlad one pound for each ray that $\ell$ intersects. Naturally, David wants to pay as little as possible. What is the largest amount of money that Vlad can get from David? [i]Proposed by Vlad Spătaru[/i]

In an acute triangle $ABC$ with $\overline{AB} > \overline{AC}$, let $D, E, F$ be the feet of the altitudes from $A, B, C$, respectively. Let $P$ be an intersection of lines $EF$ and $BC$, and let $Q$ be a point on the segment $BD$ such that $\angle QFD = \angle EPC$. Let $O, H$ denote the circumcenter and the orthocenter of triangle $ABC$, respectively. Suppose that $OH$ is perpendicular to $AQ$. Prove that $P, O, H$ are collinear.

Let the angles of a triangle be $\alpha, \beta$, and $\gamma$, the perimeter $2p$ and the radius of the circumcircle $R$. Prove the inequality $\cot^2 \alpha + \cot^2 \beta + \cot^2 \gamma \ge 3 \left(\frac{9R^2}{p^2}-1\right)$. When is the equality achieved?

Let $ABC$ be an acute triangle. Define $A_1$ the midpoint of the largest arc $BC$ of the circumcircle of $ABC$ . Let $A_2$ and $A_3$ be the feet of the perpendiculars from $A_1$ on the lines $AB$ and $AC$ , respectively. Define $B_1$, $B_2$, $B_3$, $C_1$, $C_2$, and $C_3$ analogously. Show that the lines $A_2A_3$, $B_2B_3$, $C_2C_3$ are concurrent.

Quadrilateral $ABCD$ is both cyclic and circumscribed. Its incircle touches its sides $AB$ and $CD$ at points $X$ and $Y$, respectively. The perpendiculars to $AB$ and $CD$ drawn at $A$ and $D$, respectively, meet at point $U$; those drawn at $X$ and $Y$ meet at point $V$, and finally, those drawn at $B$ and $C$ meet at point $W$. Prove that points $U$, $V$ and $W$ are collinear. [i]Proposed by A. Golovanov[/i]

Triangle $ABC$ has $AB=4$, $BC=6$, and $AC=5$. Let $O$ denote the circumcenter of $ABC$. The circle $\Gamma$ is tangent to and surrounds the circumcircles of triangle $AOB$, $BOC$, and $AOC$. Determine the diameter of $\Gamma$.