Let A be be a point outside the circle C, and AB and AC be the two tangents from A to this circle C. Let L be an arbitrary tangent to C that cuts AB and AC in P and Q. A line through P parallel to AC cuts BC in R. Prove that while L varies, QR passes through a fixed point. :)

$ABCDE$ is a pentagon whose vertices lie on circle $\omega$ where $\angle DAB=90^{\circ}$. Let $EB$ and $AC$ intersect at $F$, $EC$ meet $BD$ at $G$. $M$ is the midpoint of arc $AB$ on $\omega$, not containing $C$. If $FG\parallel DE\parallel CM$ holds, then what is the value of $\frac{|GE|}{|GD|}$?

A rectangle inscribed in a triangle has its base coinciding with the base $b$ of the triangle. If the altitude of the triangle is $h$, and the altitude $x$ of the rectangle is half the base of the rectangle, then: $\textbf{(A)}\ x=\dfrac{1}{2}h \qquad \textbf{(B)}\ x=\dfrac{bh}{b+h} \qquad \textbf{(C)}\ x=\dfrac{bh}{2h+b} \qquad \textbf{(D)}\ x=\sqrt{\dfrac{hb}{2}} \qquad \textbf{(E)}\ x=\dfrac{1}{2}b$

On sides $BC$, $CA$, $AB$ of triangle $ABC$, points $A_1$, $B_1$, $C_1$ are chosen, respectively, so that the medians $A_1A_2$, $B_1B_2$, $C_1C_2$ of the triangle $A_1B_1C_1$ are respectively parallel to straight lines $AB$, $BC$, $CA$. Determine in what ratio points $A_1$, $B_1$, $C_1$ divide the sides of the triangle $ABC$.

Find a real number $c$ and a positive number $L$ for which \[\lim_{r\to\infty}\frac{r^c\int_0^{\pi/2}x^r\sin x\,dx}{\int_0^{\pi/2}x^r\cos x\,dx}=L.\]

A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. Let the rotational inertia of the rod about its center be $md^2$. Find the ratio $L/d$. $ \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 2\sqrt{3}\qquad\textbf{(E)}\ \text{none of the above} $

Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width? [asy]unitsize(8mm); defaultpen(linewidth(.8pt)); draw(scale(4)*unitsquare); draw((0,3)--(4,3)); draw((1,3)--(1,4)); draw((2,3)--(2,4)); draw((3,3)--(3,4));[/asy]$ \textbf{(A)}\ \frac {5}{4} \qquad \textbf{(B)}\ \frac {4}{3} \qquad \textbf{(C)}\ \frac {3}{2} \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3$

Find all triples $(a,b,c)$ satisfying the following conditions: (i) $a,b,c$ are prime numbers, where $a<b<c<100$. (ii) $a+1,b+1,c+1$ form a geometric sequence.

Given a right triangle $ABC$ with the hypotenuse $AB$. Let $K$ be any interior point of triangle $ABC$ and points $L, M$ are symmetric of point $K$ wrt lines $BC, AC$ respectively. Specify all possible values for $S_{ABLM} / S_{ABC}$, where $S_{XY ... Z}$ indicates the area of the polygon $XY...Z$ .

Triangle $ ABC$ has a right angle at $ B$. Point $ D$ is the foot of the altitude from $ B$, $ AD\equal{}3$, and $ DC\equal{}4$. What is the area of $ \triangle{ABC}$? [asy]unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair B=(0,0), C=(sqrt(28),0), A=(0,sqrt(21)); pair D=foot(B,A,C); pair[] ps={B,C,A,D}; draw(A--B--C--cycle); draw(B--D); draw(rightanglemark(B,D,C)); dot(ps); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,NE); label("$3$",midpoint(A--D),NE); label("$4$",midpoint(D--C),NE);[/asy]$ \textbf{(A)}\ 4\sqrt3 \qquad \textbf{(B)}\ 7\sqrt3 \qquad \textbf{(C)}\ 21 \qquad \textbf{(D)}\ 14\sqrt3 \qquad \textbf{(E)}\ 42$

Circles with centers $ A$ and $ B$ have radii 3 and 8, respectively. A common internal tangent intersects the circles at $ C$ and $ D$, respectively. Lines $ AB$ and $ CD$ intersect at $ E$, and $ AE \equal{} 5$. What is $ CD$? [asy]unitsize(2.5mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=3; pair A=(0,0), Ep=(5,0), B=(5+40/3,0); pair M=midpoint(A--Ep); pair C=intersectionpoints(Circle(M,2.5),Circle(A,3))[1]; pair D=B+8*dir(180+degrees(C)); dot(A); dot(C); dot(B); dot(D); draw(C--D); draw(A--B); draw(Circle(A,3)); draw(Circle(B,8)); label("$A$",A,W); label("$B$",B,E); label("$C$",C,SE); label("$E$",Ep,SSE); label("$D$",D,NW);[/asy]$ \textbf{(A) } 13\qquad \textbf{(B) } \frac {44}{3}\qquad \textbf{(C) } \sqrt {221}\qquad \textbf{(D) } \sqrt {255}\qquad \textbf{(E) } \frac {55}{3}$

Square ABCD has side length 7. Let $A_1$, $B_1$, $C_1$, and $D_1$ be points on rays $\overrightarrow{AB}$, $\overrightarrow{BC}$, $\overrightarrow{CD}$, and $\overrightarrow{DA}$, respectively, where each point is $3$ units from the end of the ray so that $A_1B_1C_1D_1$ forms a second square as shown. SImilarly, let $A_2$, $B_2$, $C_2$, and $D_2$ be points on segments $A_1B_1$, $B_1C_1$, $C_1D_1$, and $D_1A_1$, respectively, forming another square where $A_2$ divides segment $A_1B_1$ into two pieces whose lengths are in the same ratio as $AA_1$ is to $A_1B$. Continue this process to construct square $A_nB_nC_nD_n$ for each positive integer $n$. Find the total of all the perimeters of all the squares. [asy] size(180); pair[] A={(-1,-1),(-1,1),(1,1),(1,-1),(-1,-1)}; string[] X={"A","B","C","D"}; for(int k=0;k<10;++k) { for(int m=0;m<4;++m) { if(k==0) label("$"+X[m]+"$",A[m],A[m]); if(k==1) label("$"+X[m]+"_1$",A[m],A[m]); draw(A[m]--A[m+1]); A[m]+=3/7*(A[m+1]-A[m]); } A[4]=A[0]; }[/asy]

Find an upper bound for the ratio $$\frac{x_1x_2+2x_2x_3+x_3x_4}{x_1^2+x_2^2+x_3^2+x_4^2}$$ over all quadruples of real numbers $(x_1,x_2,x_3,x_4)\neq (0,0,0,0)$. [i]Note.[/i] The smaller the bound, the better the solution.

In a triangle $ABC$, the angle bisector at $A,B,C$ meet the opposite sides at $A_1,B_1,C_1$, respectively. Prove that if the quadrilateral $BA_1B_1C_1$ is cyclic, then $$\frac{AC}{AB+BC}=\frac{AB}{AC+CB}+\frac{BC}{BA+AC}.$$

In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$