In triangle $ABC,$ point $D$ is on $\overline{BC}$ with $CD=2$ and $DB=5,$ point $E$ is on $\overline{AC}$ with $CE=1$ and $EA=3,$ $AB=8,$ and $\overline{AD}$ and $\overline{BE}$ intersect at $P.$ Points $Q$ and $R$ lie on $\overline{AB}$ so that $\overline{PQ}$ is parallel to $\overline{CA}$ and $\overline{PR}$ is parallel to $\overline{CB}.$ It is given that the ratio of the area of triangle $PQR$ to the area of triangle $ABC$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Let $ABCD$ be a convex quadrilateral with $AD\not\parallel BC$. Define the points $E=AD \cap BC$ and $I = AC\cap BD$. Prove that the triangles $EDC$ and $IAB$ have the same centroid if and only if $AB \parallel CD$ and $IC^{2}= IA \cdot AC$. [i]Virgil Nicula[/i]

Let $ w $ be the incircle of triangle $ ABC $. Segments $ BC, CA $ meet with $ w $ at points $ D, E$. A line passing through $ B $ and parallel to $ DE $ meets $ w $ at $ F $ and $ G $. ($ F $ is nearer to $ B $ than $ G $.) Line $ CG $ meets $ w $ at $ H ( \ne G ) $. A line passing through $ G $ and parallel to $ EH $ meets with line $ AC $ at $ I $. Line $ IF $ meets with circle $ w $ at $ J (\ne F ) $. Lines $ CJ $ and $ EG $ meets at $ K $. Let $ l $ be the line passing through $ K $ and parallel to $ JD $. Prove that $ l, IF, ED $ meet at one point.

A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length 4. A plane passes through the midpoints of $\overline{AE}$, $\overline{BC}$, and $\overline{CD}$. The plane's intersection with the pyramid has an area that can be expressed as $\sqrt{p}$. Find $p$.

Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicated in the figure. Find the product $abc$ if $a + b + c = 43$ and $d = 3$. [asy] size(200); defaultpen(fontsize(10)); pair A=origin, B=(14,0), C=(9,12), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), P=centroid(A,B,C); draw(D--A--B--C--A^^B--E^^C--F); dot(A^^B^^C^^P); label("$a$", P--A, dir(-90)*dir(P--A)); label("$b$", P--B, dir(90)*dir(P--B)); label("$c$", P--C, dir(90)*dir(P--C)); label("$d$", P--D, dir(90)*dir(P--D)); label("$d$", P--E, dir(-90)*dir(P--E)); label("$d$", P--F, dir(-90)*dir(P--F)); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, N); label("$P$", P, 1.8*dir(285));[/asy]

A circle passes through the vertices of a triangle with side-lengths of $7\tfrac{1}{2},10,12\tfrac{1}{2}$. The radius of the circle is: $\textbf{(A)}\ \dfrac{15}{4} \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ \dfrac{25}{4} \qquad \textbf{(D)}\ \dfrac{35}{4} \qquad \textbf{(E)}\ \dfrac{15\sqrt2}{2} $

A [i]kite[/i] is a quadrilateral whose diagonals are perpendicular. Let kite $ABCD$ be such that $\angle B = \angle D = 90^\circ$. Let $M$ and $N$ be the points of tangency of the incircle of $ABCD$ to $AB$ and $BC$ respectively. Let $\omega$ be the circle centered at $C$ and tangent to $AB$ and $AD$. Construct another kite $AB^\prime C^\prime D^\prime$ that is similar to $ABCD$ and whose incircle is $\omega$. Let $N^\prime$ be the point of tangency of $B^\prime C^\prime$ to $\omega$. If $MN^\prime \parallel AC$, then what is the ratio of $AB:BC$?

[asy] size(100); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((0,1)--(1,0)); draw((0,0)--(.5,sqrt(3)/2)--(1,0)); label("$A$",(0,0),SW); label("$B$",(1,0),SE); label("$C$",(1,1),NE); label("$D$",(0,1),NW); label("$E$",(.5,sqrt(3)/2),E); label("$F$",intersectionpoint((0,0)--(.5,sqrt(3)/2),(0,1)--(1,0)),2W); //Credit to chezbgone2 for the diagram[/asy] Vertex $E$ of equilateral triangle $ABE$ is in the interior of square $ABCD$, and $F$ is the point of intersection of diagonal $BD$ and line segment $AE$. If length $AB$ is $\sqrt{1+\sqrt{3}}$ then the area of $\triangle ABF$ is $\textbf{(A) }1\qquad\textbf{(B) }\frac{\sqrt{2}}{2}\qquad\textbf{(C) }\frac{\sqrt{3}}{2}$ $\qquad\textbf{(D) }4-2\sqrt{3}\qquad \textbf{(E) }\frac{1}{2}+\frac{\sqrt{3}}{4}$

Given the parallelogram $ABCD$. The circle $S_1$ passes through the vertex $C$ and touches the sides $BA$ and $AD$ at points $P_1$ and $Q_1$, respectively. The circle $S_2$ passes through the vertex $B$ and touches the side $DC$ at points $P_2$ and $Q_2$, respectively. Let $d_1$ and $d_2$ be the distances from $C$ and $B$ to the lines $P_1Q_1$ and $P_2Q_2$, respectively. Find all possible values of the ratio $d_1:d_2$. [i](I. Voronovich)[/i]

Consider a right triangle $ABC$, where $A$ is the right angle, and $AC > AB$. Points $E$ on $AC$ and $D$ on $BC$ are chosen so that$ AB = AE = BD$. Prove that the triangle $ADE$ is right if and only if the ratio $AB : AC : BC$ of sides of the triangle $ABC$ is $3 : 4 : 5$. (A. Parovan)

Let $A,B,C$ be nodes of the lattice $Z\times Z$ such that inside the triangle $ABC$ lies a unique node $P$ of the lattice. Denote $E = AP \cap BC$. Determine max $\frac{AP}{PE}$ , over all such configurations.

Let $ABC$ be an equilateral triangle whose angle bisectors of $B$ and $C$ intersect at $D$. Perpendicular bisectors of $BD$ and $CD$ intersect $BC$ at points $E$ and $Z$ respectively. a) Prove that $BE=EZ=ZC$. b) Find the ratio of the areas of the triangles $BDE$ to $ABC$

For two different regular icosahedrons it is known that some six of their vertices are vertices of a regular octahedron. Find the ratio of the edges of these icosahedrons.

In a tennis tournament, $n$ women and $2n$ men play, and each player plays exactly one match with every other player. If there are no ties and the ratio of the number of matches won by women to the number of matches won by men is $7/5$, then $n$ equals $\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad \textbf{(E) }\text{none of these}$

Consider triangle $ABC$ with $\angle A=2\angle B$. The angle bisectors from $A$ and $C$ intersect at $D$, and the angle bisector from $C$ intersects $\overline{AB}$ at $E$. If $\dfrac{DE}{DC}=\dfrac13$, compute $\dfrac{AB}{AC}$.

In trapezoid $ABCD$, diagonal $AC$ is the bisector of angle $A$. Point $K$ is the midpoint of diagonal $AC$. It is known that $DC = DK$. Find the ratio of the bases $AD: BC$.

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}$

For a particular peculiar pair of dice, the probabilities of rolling 1, 2, 3, 4, 5 and 6 on each die are in the ratio $ 1: 2: 3: 4: 5: 6$. What is the probability of rolling a total of 7 on the two dice? $ \textbf{(A) } \frac 4{63} \qquad \textbf{(B) } \frac 18 \qquad \textbf{(C) } \frac 8{63} \qquad \textbf{(D) } \frac 16 \qquad \textbf{(E) } \frac 27$

Let $ABC$ be a triangle. Let $D,E$ be points on the segment $BC$ such that $BD=DE=EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AD$ in $P$ and $AE$ in $Q$ respectively. Determine the ratio of the area of the triangle $APQ$ to that of the quadrilateral $PDEQ$.

In the Mathematical Competition of HMS (Hellenic Mathematical Society) take part boys and girls who are divided into two groups : [i]Juniors[/i] and [i]seniors.[/i]The number of the boys taking part of this year competition is 55% of the number of all participants. The ratio of the number of juniors boys to the number of senior boys is equal to the ratio of the number of juniors to the number of seniors. Find the ratio of the number of junior boys to the number of junior girls.

The side lengths of a trapezoid are $\sqrt[4]{3}, \sqrt[4]{3}, \sqrt[4]{3}$, and $2 \cdot \sqrt[4]{3}$. Its area is the ratio of two relatively prime positive integers, $m$ and $n$. Find $m + n$.

Given a triangle $ABC$ ($AB \ne AC$). Let $ P$ be a point in the plane containing triangle $ABC$ satisfying the following property: If the projections of $ P$ onto $AB$,$AC$ are $C_1$,$B_1$ respectively, then $\frac{PB}{PC}=\frac{PC_1}{PB_1}=\frac{AB}{AC}$ or $\frac{PB}{PC}=\frac{PB_1}{PC_1}=\frac{AB}{AC}$. Prove that $\angle PBC + \angle PCB = \angle BAC$.

Let $ABC$ be a triangle with integral side lengths. The angle bisector drawn from $B$ and the altitude drawn from $C$ meet at point $P$ inside the triangle. Prove that the ratio of areas of triangles $APB$ and $APC$ is a rational number.

Let $ ABC$ be a triangle such that \[ \frac{BC}{AB \minus{} BC}\equal{}\frac{AB \plus{} BC}{AC}\] Determine the ratio $ \angle A : \angle C$.

Let $ABC$ be a triangle and $I$ its incentre. Let $\varrho_1$ and $\varrho_2$ be the inradii of triangles $IAB$ and $IAC$ respectively. (a) Show that there exists a function $f: ( 0, \pi ) \mapsto \mathbb{R}$ such that \[ \frac{ \varrho_1}{ \varrho_2} = \frac{f(C)}{f(B)} \] where $B = \angle ABC$ and $C = \angle BCA$ (b) Prove that \[ 2 ( \sqrt{2} -1 ) < \frac{ \varrho_1} { \varrho_2} < \frac{ 1 + \sqrt{2}}{2} \]