A circle $\omega$ and a point $T$ outside the circle is given. Let a tangent from $T$ to $\omega$ touch $\omega$ at $A$, and take points $B,C$ lying on $\omega$ such that $T,B,C$ are colinear. The bisector of $\angle ATC$ intersects $AB$ and $AC$ at $P$ and $Q$,respectively. Prove that $PA=\sqrt{PB\cdot QC}$

The polyhedron $PABCDQ$ has the form shown in the figure. It is known that $ABCD$ is parallelogram, the planes of the triangles of the $PAC$ and $PBD$ mutually perpendicular, and also mutually perpendicular are the planes of triangles $QAC$ and $QBC$. Each face of this polyhedron is painted black or white so that the faces that have a common edge are painted in different colors. Prove that the sum of the squares of the areas of the black faces is equal to the sum of the squares of the areas of the white faces. [img]https://1.bp.blogspot.com/-UM5PKEGGWqc/X1V2cXAFmwI/AAAAAAAAMdw/V-Qr94tZmqkj3_q-5mkSICGF1tMu-b_VwCLcBGAsYHQ/s0/2007.5%2Bchampions%2Btourn.png[/img]

Let the inscribed circle $ \omega $ of triangle $ ABC $ touch the side $ BC $ at the point $ A '$. Let $ AA '$ intersect $ \omega $ at $ P \neq A $. Let $ CP $ and $ BP $ intersect $ \omega $, respectively, at points $ N $ and $ M $ other than $ P $. Prove that $ AA ', BN $ and $ CM $ intersect at one point.

Given triangle $ ABC$ and its circumcircle $ \Gamma$, let $ M$ and $ N$ be the midpoints of arcs $ BC$ (that does not contain $ A$) and $ CA$ (that does not contain $ B$), repsectively. Let $ X$ be a variable point on arc $ AB$ that does not contain $ C$. Let $ O_1$ and $ O_2$ be the incenter of triangle $ XAC$ and $ XBC$, respectively. Let the circumcircle of triangle $ XO_1O_2$ meets $ \Gamma$ at $ Q$. (a) Prove that $ QNO_1$ and $ QMO_2$ are similar. (b) Find the locus of $ Q$ as $ X$ varies.

Convex $n$-gon $P$, where $n> 3$, is cut into equal triangles by diagonals that do not intersect inside it. What are the possible values of $n$ if the $n$-gon is cyclic?

Let $ P$ be a point in triangle $ ABC$, and define $ \alpha,\beta,\gamma$ as follows: \[ \alpha\equal{}\angle BPC\minus{}\angle BAC, \quad \beta\equal{}\angle CPA\minus{}\angle \angle CBA, \quad \gamma\equal{}\angle APB\minus{}\angle ACB.\] Prove that \[ PA\dfrac{\sin \angle BAC}{\sin \alpha}\equal{}PB\dfrac{\sin \angle CBA}{\sin \beta}\equal{}PC\dfrac{\sin \angle ACB}{\sin \gamma}.\]

Let $ABC$ be a triangle and $D$ a point on the side $BC$. The tangent line to the circumcircle of the triangle $ABD$ at the point $D$ intersects the side $AC$ at $E$. The tangent line to the circumcircle of the triangle $ACD$ at the the point $D$ intersects the side $AB$ at $F$. Prove that the point $A$ and the circumcenters of the triangles $ABC$ and $DEF$ are collinear. (Malik Talbi)

The given points are $ A $ and $ B $ and the circle $ k $. Draw a circle passing through the points $ A $ and $ B $ and defining, at the intersection with the circle $ k $, a common chord of a given length $ d $.

Let $C$ denote the set of points $(x, y) \in R^2$ such that $x^2 + y^2 \le1$. A sequence $A_i = (x_i, y_i), |i \ge¸ 0$ of points in $R^2$ is ‘centric’ if it satisfies the following properties: $\bullet$ $A_0 = (x_0, y_0) = (0, 0)$, $A_1 = (x_1, y_1) = (1, 0)$. $\bullet$ For all $n\ge 0$, the circumcenter of triangle $A_nA_{n+1}A_{n+2}$ lies in $C$. Let $K$ be the maximum value of $x^2_{2012} + y^2_{2012}$ over all centric sequences. Find all points $(x, y)$ such that $x^2 + y^2 = K$ and there exists a centric sequence such that $A_{2012} = (x, y)$.

Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.

Let $ABC$ be a non-obtuse triangle satisfying $AB>AC$ and $\angle B=45^{\circ}$. The circumcentre $O$ and incentre $I$ of triangle $ABC$ are such that $\sqrt{2}\ OI=AB-AC$. Find the value of $\sin A$.

An ant is at vertex of a cube. Every $10$ minutes it moves to an adjacent vertex along an edge. If $N$ is the number of one hour journeys that end at the starting vertex, find the sum of the squares of the digits of $N$.

A convex polygon of $n$ sides is considered. All its diagonals are drawn and we suppose that any three of them can only intersect on a vertex and that there is no pair of parallel diagonals. Under these conditions, we wish to compute a) The total number of intersection points of these diagonals, excluding the vertices. b) How many points, of these intersections, lie inside the polygon and how many lie outside.

Let the line $L$ be perpendicular to the plane $P$. Three spheres touch each other in pairs so that each sphere touches the plane $P$ and the line $L$. The radius of the larger sphere is $1$. Find the minimum radius of the smallest sphere.

Consider an acute scalene triangle $\triangle{ABC}$. The interior bisector of $A$ intersects $BC$ at $E$ and the minor arc of $\overarc {BC}$ in circumcircle of $\triangle{ABC}$ at $M$. Suppose that $D$ is a point on the minor arc of $\overarc {BC}$ such that $ED=EM$. $P$ is a point on the line segment of $AD$ such that $\angle ABP=\angle ACP \not= 0$. $O$ is the circumcenter of $\triangle{ABC}$. Prove that $OP \perp AM$.

Let $ABC$ be an isosceles triangle with vertex $A$ and $AB> BC$. Let $k$ be the circle with center $A$ passsing through $B$ and $C$. Let $H$ be the second intersection of $k$ with the altitude of the triangle $ABC$ through $B$. Further let $G$ be the second intersection of $k$ with the median through $B$ in triangle $ABC$. Let $X$ be the intersection of the lines $AC$ and $GH$. Show that $C$ is the midpoint of $AX$.

Let $BM$ be the median of right-angled triangle $ABC (\angle B = 90^{\circ})$. The incircle of triangle $ABM$ touches sides $AB, AM$ in points $A_{1},A_{2}$; points $C_{1}, C_{2}$ are defined similarly. Prove that lines $A_{1}A_{2}$ and $C_{1}C_{2}$ meet on the bisector of angle $ABC$.

[u]Round 1[/u] [b]p1.[/b] Compute $(1 - 2(3 - 4(5 - 6)))(7 - (8 - 9))$. [b]p2.[/b] How many numbers are in the set $\{20, 21, 22, ..., 88, 89\}$? [b]p3.[/b] Three times the complement of the supplement of an angle is equal to $60$ degrees less than the angle itself. Find the measure of the angle in degrees. [u]Round 2[/u] [b]p4.[/b] A positive number is decreased by $10\%$, then decreased by $20\%$, and finally increased by $30\%$. By what percent has this number changed from the original? Give a positive answer for a percent increase and a negative answer for a percent decrease. [b]p5.[/b] What is the area of the triangle with vertices at $(2, 3)$, $(8, 11)$, and $(13, 3)$? [b]p6.[/b] There are three bins, each containing red, green, and/or blue pens. The first bin has $0$ red, $0$ green, and $3$ blue pens, the second bin has $0$ red, $2$ green, and $4$ blue pens, and the final bin has $1$ red, $5$ green, and $6$ blue pens. What is the probability that if one pen is drawn from each bin at random, one of each color pen will be drawn? [u]Round 3[/u] [b]p7.[/b] If a and b are positive integers and $a^2 - b^2 = 23$, what is the value of $a$? [b]p8.[/b] Find the prime factorization of the greatest common divisor of $2^3\cdot 3^2\cdot 5^5\cdot 7^4$ and $2^4\cdot 3^1\cdot 5^2\cdot 7^6$. [b]p9.[/b] Given that $$a + 2b + 3c = 5$$ $$2a + 3b + c = -2$$ $$3a + b + 2c = 3,$$ find $3a + 3b + 3c$. [u]Round 4[/u] [b]p10.[/b] How many positive integer divisors does $11^{20}$ have? [b]p11.[/b] Let $\alpha$ be the answer to problem $10$. Find the real value of $x$ such that $2^{x-5} = 64^{x/\alpha}$. [b]p12.[/b] Let $\beta$ be the answer to problem $11$. Triangle $LMT$ has a right angle at $M$, $LM = \beta$, and $LT = 4\beta - 3$. If $Z$ is the midpoint of $LT$, what is the length$ MZ$? PS. You should use hide for answers. Rounds 5-8 are [url=https://artofproblemsolving.com/community/c3h3133709p28395558]here[/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3134133p28400917]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Lengths of five edges of a tetrahedron are $1$, while the last one is $x$. Its volume is $F(x)$. On its domain of definition, we have $\text{(A)}$ $F(x)$ is an increasing function, it has no maximum value. $\text{(B)}$ $F(x)$ is an increasing function, it has maximum value. $\text{(C)}$ $F(x)$ is not an increasing function, it has no maximum value. $\text{(D)}$ $F(x)$ is an increasing function, it has maximum value.

Let $h_a$, $h_b$ and $h_c$ be the altitudes of a triangle $\triangle ABC$ ejected from the vertices $A$,$B$ and $C$, respectively. Similarly, let $h_x$, $h_y$ and $h_z$ be the altitudes of an another triangle $\triangle XYZ$. Show that if $$h_a : h_b : h_c = h_x : h_y : h_z, $$ then the triangles $\triangle ABC$ and $\triangle XYZ$ are similar.

In parallelogram $ ABCD$, point $ M$ is on $ \overline{AB}$ so that $ \frac{AM}{AB} \equal{} \frac{17}{1000}$ and point $ N$ is on $ \overline{AD}$ so that $ \frac{AN}{AD} \equal{} \frac{17}{2009}$. Let $ P$ be the point of intersection of $ \overline{AC}$ and $ \overline{MN}$. Find $ \frac{AC}{AP}$.

In triangle $ABC$, $A'$, $B'$, and $C'$ are on the sides $BC$, $AC$, and $AB$, respectively. Given that $AA'$, $BB'$, and $CC'$ are concurrent at the point $O$, and that \[\frac{AO}{OA'}+\frac{BO}{OB'}+\frac{CO}{OC'}=92,\] find \[\frac{AO}{OA'}\cdot \frac{BO}{OB'}\cdot \frac{CO}{OC'}.\]

A concrete sewer pipe fitting is shaped like a cylinder with diameter $48$ with a cone on top. A cylindrical hole of diameter $30$ is bored all the way through the center of the fitting as shown. The cylindrical portion has height $60$ while the conical top portion has height $20$. Find $N$ such that the volume of the concrete is $N \pi$. [asy] import three; size(250); defaultpen(linewidth(0.7)+fontsize(10)); pen dashes = linewidth(0.7) + linetype("2 2"); currentprojection = orthographic(0,-15,5); draw(circle((0,0,0), 15),dashes); draw(circle((0,0,80), 15)); draw(scale3(24)*((-1,0,0)..(0,-1,0)..(1,0,0))); draw(shift((0,0,60))*scale3(24)*((-1,0,0)..(0,-1,0)..(1,0,0))); draw((-24,0,0)--(-24,0,60)--(-15,0,80)); draw((24,0,0)--(24,0,60)--(15,0,80)); draw((-15,0,0)--(-15,0,80),dashes); draw((15,0,0)--(15,0,80),dashes); draw("48", (-24,0,-20)--(24,0,-20)); draw((-15,0,-20)--(-15,0,-17)); draw((15,0,-20)--(15,0,-17)); label("30", (0,0,-15)); draw("60", (50,0,0)--(50,0,60)); draw("20", (50,0,60)--(50,0,80)); draw((50,0,60)--(47,0,60));[/asy]

Let $ABC$ be an acute triangle with $AB<AC$ and $D,E,F$ be the contact points of the incircle $(I)$ with $BC,AC,AB$. Let $M,N$ be on $EF$ such that $MB \perp BC$ and $NC \perp BC$. $MD$ and $ND$ intersect the $(I)$ in $D$ and $Q$. Prove that $DP=DQ$.

An equilateral triangle of side$ 1$ is rotated around its center, yielding another equilareral triangle. Find the area of the intersection of these two triangles.