Cyclic pentagon $ ABCDE$ has a right angle $ \angle{ABC} \equal{} 90^{\circ}$ and side lengths $ AB \equal{} 15$ and $ BC \equal{} 20$. Supposing that $ AB \equal{} DE \equal{} EA$, find $ CD$.

An [i]annulus[/i] is the region between two concentric circles. The concentric circles in the figure have radii $ b$ and $ c$, with $ b > c$. Let $ \overline{OX}$ be a radius of the larger circle, let $ \overline{XZ}$ be tangent to the smaller circle at $ Z$, and let $ \overline{OY}$ be the radius of the larger circle that contains $ Z$. Let $ a \equal{} XZ$, $ d \equal{} YZ$, and $ e \equal{} XY$. What is the area of the annulus? $ \textbf{(A)}\ \pi a^2 \qquad \textbf{(B)}\ \pi b^2 \qquad \textbf{(C)}\ \pi c^2 \qquad \textbf{(D)}\ \pi d^2 \qquad \textbf{(E)}\ \pi e^2$ [asy]unitsize(1.4cm); defaultpen(linewidth(.8pt)); dotfactor=3; real r1=1.0, r2=1.8; pair O=(0,0), Z=r1*dir(90), Y=r2*dir(90); pair X=intersectionpoints(Z--(Z.x+100,Z.y), Circle(O,r2))[0]; pair[] points={X,O,Y,Z}; filldraw(Circle(O,r2),mediumgray,black); filldraw(Circle(O,r1),white,black); dot(points); draw(X--Y--O--cycle--Z); label("$O$",O,SSW,fontsize(10pt)); label("$Z$",Z,SW,fontsize(10pt)); label("$Y$",Y,N,fontsize(10pt)); label("$X$",X,NE,fontsize(10pt)); defaultpen(fontsize(8pt)); label("$c$",midpoint(O--Z),W); label("$d$",midpoint(Z--Y),W); label("$e$",midpoint(X--Y),NE); label("$a$",midpoint(X--Z),N); label("$b$",midpoint(O--X),SE);[/asy]

Two right angled triangles are given, such that the incircle of the first one is equal to the circumcircle of the second one. Let $S$ (respectively $S'$) be the area of the first triangle (respectively of the second triangle). Prove that $\frac{S}{S'}\geq 3+2\sqrt{2}$.

In the plane we are given the circles $\Gamma$ and $\Delta$ tangent to each other and $\Gamma$ contains $\Delta$. The radius of $\Gamma$ is $R$ and of $\Delta$ is $\frac{R}{2}$. Prove that for each positive integer $n \geq 3$, the equation: \[ (p(1) - p(n))^2 = (n-1)^2 \cdot (2 \cdot (p(1) + p(n)) - (n-1)^2 - 8) \] is the necessary and sufficient condition for $n$ to exist $n$ distinct circles $\Upsilon_1, \Upsilon_2, \ldots, \Upsilon_n$ such that all these circles are tangent to $\Gamma$ and $\Delta$ and $\Upsilon_i$ is tangent to $\Upsilon_{i+1}$, and $\Upsilon_1$ has radius $\frac{R}{p(1)}$ and $\Upsilon_n$ has radius $\frac{R}{p(n)}$.

Square $ EFGH$ is inside the square $ ABCD$ so that each side of $ EFGH$ can be extended to pass through a vertex of $ ABCD$. Square $ ABCD$ has side length $ \sqrt {50}$ and $ BE \equal{} 1$. What is the area of the inner square $ EFGH$? [asy]unitsize(4cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair D=(0,0), C=(1,0), B=(1,1), A=(0,1); pair F=intersectionpoints(Circle(D,2/sqrt(5)),Circle(A,1))[0]; pair G=foot(A,D,F), H=foot(B,A,G), E=foot(C,B,H); draw(A--B--C--D--cycle); draw(D--F); draw(C--E); draw(B--H); draw(A--G); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$E$",E,NNW); label("$F$",F,ENE); label("$G$",G,SSE); label("$H$",H,WSW);[/asy]$ \textbf{(A)}\ 25\qquad \textbf{(B)}\ 32\qquad \textbf{(C)}\ 36\qquad \textbf{(D)}\ 40\qquad \textbf{(E)}\ 42$

A ball was floating in a lake when the lake froze. The ball was removed (without breaking the ice), leaving a hole $24$ cm across as the top and $8$ cm deep. What was the radius of the ball (in centimeters)? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 13 \qquad\textbf{(D)}\ 8\sqrt{3} \qquad\textbf{(E)}\ 6\sqrt{6} $

In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners $ABCD$, he starts at $A$ and walks to $C$. Then, he walks to the midpoint of side $AD$, say $A_1$. Then, he walks to the midpoint of side $CD$ say $C_1$, and then the midpoint of $A_1D$ which is $A_2$. He continues in this fashion, indefinitely. The total length of his path if $AB=5$ and $BC=12$ is of the form $a + b\sqrt{c}$. Find $\displaystyle\frac{abc}{4}$.

A ball is placed on a plane and a point on the ball is marked. Our goal is to roll the ball on a polygon in the plane in a way that it comes back to where it started and the marked point comes to the top of it. Note that We are not allowed to rotate without moving, but only rolling. Prove that it is possible. Time allowed for this problem was 90 minutes.

Let $\triangle ABC$ be a triangle with $AB = 3$, $BC = 4$, and $AC = 5$. Let $I$ be the center of the circle inscribed in $\triangle ABC$. What is the product of $AI$, $BI$, and $CI$?

In $\triangle ABC$, $AB = 30$, $BC = 40$, and $CA = 50$. Squares $A_1A_2BC$, $B_1B_2AC$, and $C_1C_2AB$ are erected outside $\triangle ABC$, and the pairwise intersections of lines $A_1A_2$, $B_1B_2$, and $C_1C_2$ are $P$, $Q$, and $R$. Compute the length of the shortest altitude of $\triangle PQR$. [i]Proposed by Lewis Chen[/i]

Let $ABC$ be a triangle with incenter $I$ and $AB = 1400$, $AC = 1800$, $BC = 2014$. The circle centered at $I$ passing through $A$ intersects line $BC$ at two points $X$ and $Y$. Compute the length $XY$. [i]Proposed by Evan Chen[/i]

The area of trapezoid $ ABCD$ is $ 164 \text{cm}^2$. The altitude is $ 8 \text{cm}$, $ AB$ is $ 10 \text{cm}$, and $ CD$ is $ 17 \text{cm}$. What is $ BC$, in centimeters? [asy]/* AMC8 2003 #21 Problem */ size(4inch,2inch); draw((0,0)--(31,0)--(16,8)--(6,8)--cycle); draw((11,8)--(11,0), linetype("8 4")); draw((11,1)--(12,1)--(12,0)); label("$A$", (0,0), SW); label("$D$", (31,0), SE); label("$B$", (6,8), NW); label("$C$", (16,8), NE); label("10", (3,5), W); label("8", (11,4), E); label("17", (22.5,5), E);[/asy] $ \textbf{(A)}\ 9\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20$

$ \overline{AB}$ is the hypotenuse of a right triangle $ ABC$. Median $ \overline{AD}$ has length $ 7$ and median $ \overline{BE}$ has length $ 4$. The length of $ \overline{AB}$ is: $ \textbf{(A)}\ 10\qquad \textbf{(B)}\ 5\sqrt{3}\qquad \textbf{(C)}\ 5\sqrt{2}\qquad \textbf{(D)}\ 2\sqrt{13}\qquad \textbf{(E)}\ 2\sqrt{15}$

Square $ABCD$ is inscribed in a circle. Square $EFGH$ has vertices $E$ and $F$ on $\overline{CD}$ and vertices $G$ and $H$ on the circle. The ratio of the area of square $EFGH$ to the area of square $ABCD$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers and $m<n$. Find $10n+m$.

In a circle of radius $ 5$ units, $ CD$ and $ AB$ are perpendicular diameters. A chord $ CH$ cutting $ AB$ at $ K$ is $ 8$ units long. The diameter $ AB$ is divided into two segments whose dimensions are: $ \textbf{(A)}\ 1.25, 8.75 \qquad\textbf{(B)}\ 2.75,7.25 \qquad\textbf{(C)}\ 2,8 \qquad\textbf{(D)}\ 4,6$ $ \textbf{(E)}\ \text{none of these}$

Let $ ABCD$ be a unit square (that is, the labels $ A, B, C, D$ appear in that order around the square). Let $ X$ be a point outside of the square such that the distance from $ X$ to $ AC$ is equal to the distance from $ X$ to $ BD$, and also that $ AX \equal{} \frac {\sqrt {2}}{2}$. Determine the value of $ CX^2$.

Alice and Bob stand atop two different towers in the Arctic. Both towers are a positive integer number of meters tall and are a positive (not necessarily integer) distance away from each other. One night, the sea between them has frozen completely into reflective ice. Alice shines her flashlight directly at the top of Bob's tower, and Bob shines his flashlight at the top of Alice's tower by first reflecting it off the ice. The light from Alice's tower travels $16$ meters to get to Bob's tower, while the light from Bob's tower travels $26$ meters to get to Alice's tower. Assuming that the lights are both shown from exactly the top of their respective towers, what are the possibilities for the height of Alice's tower?

In $\triangle RED, RD =1, \angle DRE = 75^\circ$ and $\angle RED = 45^\circ$. Let $M$ be the midpoint of segment $\overline{RD}$. Point $C$ lies on side $\overline{ED}$ such that $\overline{RC} \perp \overline{EM}$. Extend segment $\overline{DE}$ through $E$ to point $A$ such that $CA = AR$. Then $AE = \tfrac{a-\sqrt{b}}{c},$ where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer. Find $a+b+c$.

The sides of a triangle have lengths $11$,$15$, and $k$, where $k$ is an integer. For how many values of $k$ is the triangle obtuse? $\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 13 \qquad \textbf{(E)}\ 14$

Consider all triangles $ ABC$ satisfying the following conditions: $ AB \equal{} AC$, $ D$ is a point on $ \overline{AC}$ for which $ \overline{BD} \perp \overline{AC}$, $ AD$ and $ CD$ are integers, and $ BD^2 \equal{} 57$. Among all such triangles, the smallest possible value of $ AC$ is $ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 10 \qquad \textbf{(C)}\ 11 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 13$ [asy]defaultpen(linewidth(.8pt)); dotfactor=4; pair B = (0,0); pair C = (5,0); pair A = (2.5,7.5); pair D = foot(B,A,C); dot(A);dot(B);dot(C);dot(D); label("$A$", A, N);label("$B$", B, SW);label("$C$", C, SE);label("$D$", D, NE); draw(A--B--C--cycle);draw(B--D);[/asy]

Triangle $ ABC$ has $ AB\equal{}13$ and $ AC\equal{}15$, and the altitude to $ \overline{BC}$ has length $ 12$. What is the sum of the two possible values of $ BC$? $ \textbf{(A)}\ 15\qquad \textbf{(B)}\ 16\qquad \textbf{(C)}\ 17\qquad \textbf{(D)}\ 18\qquad \textbf{(E)}\ 19$

Let $ABC$ be a triangle with $|AB|=|AC|=26$, $|BC|=20$. The altitudes of $\triangle ABC$ from $A$ and $B$ cut the opposite sides at $D$ and $E$, respectively. Calculate the radius of the circle passing through $D$ and tangent to $AC$ at $E$.

A rectangular piece of paper with side lengths 5 by 8 is folded along the dashed lines shown below, so that the folded flaps just touch at the corners as shown by the dotted lines. Find the area of the resulting trapezoid. [asy] size(150); defaultpen(linewidth(0.8)); draw(origin--(8,0)--(8,5)--(0,5)--cycle,linewidth(1)); draw(origin--(8/3,5)^^(16/3,5)--(8,0),linetype("4 4")); draw(origin--(4,3)--(8,0)^^(8/3,5)--(4,3)--(16/3,5),linetype("0 4")); label("$5$",(0,5/2),W); label("$8$",(4,0),S); [/asy]

Let $ABCD$ be a tetrahedron and let $H_{a},H_{b},H_{c},H_{d}$ be the orthocenters of triangles $BCD,CDA,DAB,ABC$, respectively. Prove that lines $AH_{a},BH_{b},CH_{c}, DH_{d}$ are concurrent if and only if $AB^2 + CD^2 = AC^2 + BD^2 = AD^2 + BC^2$

Point $P$ is taken interior to a square with side-length $a$ and such that is it equally distant from two consecutive vertices and from the side opposite these vertices. If $d$ represents the common distance, then $d$ equals: $\textbf{(A)}\ \dfrac{3a}{5} \qquad \textbf{(B)}\ \dfrac{5a}{8} \qquad \textbf{(C)}\ \dfrac{3a}{8} \qquad \textbf{(D)}\ \dfrac{a\sqrt{2}}{2} \qquad \textbf{(E)}\ \dfrac{a}{2}$