The ratio $ \frac{2^{2001}\cdot3^{2003}}{6^{2002}}$ is $ \textbf{(A)}\ \frac{1}{6} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{1}{2} \qquad \textbf{(D)}\ \frac{2}{3} \qquad \textbf{(E)}\ \frac{3}{2}$

Prove that there exists a positive $ c$ such that for every positive integer $ N$ among any $ N$ positive integers not exceeding $ 2N$ there are two numbers whose greatest common divisor is greater than $ cN$.

Let $ ABC$ be a triangle, and $ I$ its incenter. Let the incircle of $ ABC$ touch side $ BC$ at $ D$, and let lines $ BI$ and $ CI$ meet the circle with diameter $ AI$ at points $ P$ and $ Q$, respectively. Given $ BI \equal{} 6, CI \equal{} 5, DI \equal{} 3$, determine the value of $ \left( DP / DQ \right)^2$.

In rectangle $ ABCD$, points $ F$ and $ G$ lie on $ \overline{AB}$ so that $ AF \equal{} FG \equal{} GB$ and $ E$ is the midpoint of $ \overline{DC}$. Also, $ \overline{AC}$ intersects $ \overline{EF}$ at $ H$ and $ \overline{EG}$ at $ J$. The area of the rectangle $ ABCD$ is $ 70$. Find the area of triangle $ EHJ$. [asy] size(180); pair A, B, C, D, E, F, G, H, J; A = origin; real length = 6; real width = 3.5; B = length*dir(0); C = (length, width); D = width*dir(90); F = length/3*dir(0); G = 2*length/3*dir(0); E = (length/2, width); H = extension(A, C, E, F); J = extension(A, C, E, G); draw(A--B--C--D--cycle); draw(G--E--F); draw(A--C); label("$A$", A, dir(180)); label("$D$", D, dir(180)); label("$B$", B, dir(0)); label("$C$", C, dir(0)); label("$F$", F, dir(270)); label("$E$", E, dir(90)); label("$G$", G, dir(270)); label("$H$", H, dir(140)); label("$J$", J, dir(340)); [/asy] $ \displaystyle \textbf{(A)} \ \frac {5}{2} \qquad \textbf{(B)} \ \frac {35}{12} \qquad \textbf{(C)} \ 3 \qquad \textbf{(D)} \ \frac {7}{2} \qquad \textbf{(E)} \ \frac {35}{8}$

In the following, the point of intersection of two lines $ g$ and $ h$ will be abbreviated as $ g\cap h$. Suppose $ ABC$ is a triangle in which $ \angle A \equal{} 90^{\circ}$ and $ \angle B > \angle C$. Let $ O$ be the circumcircle of the triangle $ ABC$. Let $ l_{A}$ and $ l_{B}$ be the tangents to the circle $ O$ at $ A$ and $ B$, respectively. Let $ BC \cap l_{A} \equal{} S$ and $ AC \cap l_{B} \equal{} D$. Furthermore, let $ AB \cap DS \equal{} E$, and let $ CE \cap l_{A} \equal{} T$. Denote by $ P$ the foot of the perpendicular from $ E$ on $ l_{A}$. Denote by $ Q$ the point of intersection of the line $ CP$ with the circle $ O$ (different from $ C$). Denote by $ R$ be the point of intersection of the line $ QT$ with the circle $ O$ (different from $ Q$). Finally, define $ U \equal{} BR \cap l_{A}$. Prove that \[ \frac {SU \cdot SP}{TU \cdot TP} \equal{} \frac {SA^{2}}{TA^{2}}. \]

Given a triangle $ABC$, let $D$ be the midpoint of the side $AC$ and let $M$ be the point that divides the segment $BD$ in the ratio $1/2$; that is, $MB/MD=1/2$. The rays $AM$ and $CM$ meet the sides $BC$ and $AB$ at points $E$ and $F$, respectively. Assume the two rays perpendicular: $AM\perp CM$. Show that the quadrangle $AFED$ is cyclic if and only if the median from $A$ in triangle $ABC$ meets the line $EF$ at a point situated on the circle $ABC$.

Let $ ABC$ be an acute triangle, let $ M,N$ be the midpoints of minor arcs $ \widehat{CA},\widehat{AB}$ of the circumcircle of triangle $ ABC,$ point $ D$ is the midpoint of segment $ MN,$ point $ G$ lies on minor arc $ \widehat{BC}.$ Denote by $ I,I_{1},I_{2}$ the incenters of triangle $ ABC,ABG,ACG$ respectively.Let $ P$ be the second intersection of the circumcircle of triangle $ GI_{1}I_{2}$ with the circumcircle of triangle $ ABC.$ Prove that three points $ D,I,P$ are collinear.

Volume $A$ equals one fourth of the sum of the volumes $B$ and $C$, while volume $B$ equals one sixth of the sum of the volumes $A$ and $C$. There are relatively prime positive integers $m$ and $n$ so that the ratio of volume $C$ to the sum of the other two volumes is $\frac{m}{n}$. Find $m+n$.

Given two triangles and such that the lengths of the sides of the first triangle are the lengths of the medians of the second triangle. Determine the ratio of the areas of these triangles.

Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$. [i]Proposed by David Stoner[/i]

A circle is inscribed in the trapezoid [i]ABCD[/i]. Let [i]K, L, M, N[/i] be the points of tangency of this circle with the diagonals [i]AC[/i] and [i]BD[/i], respectively ([i]K[/i] is between [i]A[/i] and [i]L[/i], and [i]M[/i] is between [i]B[/i] and [i]N[/i]). Given that $AK\cdot LC=16$ and $BM\cdot ND=\frac94$, find the radius of the circle. [color=red][Moderator edit: A solution of this problem can be found on http://www.ajorza.org/math/mathfiles/scans/belarus.pdf , page 20 (the statement of the problem is on page 6). The author of the problem is I. Voronovich.][/color]

Let $ABC$ be a triangle in which no angle is $90^{\circ}$. For any point $P$ in the plane of the triangle, let $A_1, B_1, C_1$ denote the reflections of $P$ in the sides $BC,CA,AB$ respectively. Prove that (i) If $P$ is the incenter or an excentre of $ABC$, then $P$ is the circumenter of $A_1B_1C_1$; (ii) If $P$ is the circumcentre of $ABC$, then $P$ is the orthocentre of $A_1B_1C_1$; (iii) If $P$ is the orthocentre of $ABC$, then $P$ is either the incentre or an excentre of $A_1B_1C_1$.

The sides of a right triangle are $ a$ and $ b$ and the hypotenuse is $ c$. A perpendicular from the vertex divides $ c$ into segments $ r$ and $ s$, adjacent respectively to $ a$ and $ b$. If $ a : b \equal{} 1 : 3$, then the ratio of $ r$ to $ s$ is: $ \textbf{(A)}\ 1 : 3\qquad \textbf{(B)}\ 1 : 9\qquad \textbf{(C)}\ 1 : 10\qquad \textbf{(D)}\ 3 : 10\qquad \textbf{(E)}\ 1 : \sqrt{10}$

Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $1/8,$ and the second term of both series can be written in the form $\frac{\sqrt{m}-n}{p},$ where $m,$ $n,$ and $p$ are positive integers and $m$ is not divisible by the square of any prime. Find $100m+10n+p.$