In $\triangle ABC$, the incircle $\omega$ has center $I$ and is tangent to $\overline{CA}$ and $\overline{AB}$ at $E$ and $F$ respectively. The circumcircle of $\triangle{BIC}$ meets $\omega$ at $P$ and $Q$. Lines $AI$ and $BC$ meet at $D$, and the circumcircle of $\triangle PDQ$ meets $\overline{BC}$ again at $X$. Suppose that $EF = PQ = 16$ and $PX + QX = 17$. Then $BC^2$ can be expressed as $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $100m + n$. [i]Proposed by Ankan Bhattacharya and Michael Ren

Two semicircles touch the side of the rectangle, each other and the segment drawn in it as in the figure. What part of the whole rectangle is filled? [img]https://cdn.artofproblemsolving.com/attachments/3/e/70ca8b80240a282553294a58cb3ed807d016be.png[/img]

In a rectangle $ABCD$ two not intersecting circles $\omega_1$ and $\omega_2$ are drawn such that $\omega_1$ is tangent to $AB$ and $AD$ at points $P$ and $S$ respectively, and $\omega_2$ is tangent to $CB$ and $CD$ at $T$ and $Q$ respectively. It is known that $PQ=11, ST=10, BD=14$. Find the distance between centers of circles $\omega_1$ and $\omega_2$. [i]I. Voronovich[/i]

A ladder $10\text{ m}$ long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of $1\text{ m/s}$, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is $6\text{ m}$ from the wall?

Let $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8$ be a regular octagon. Let $M_1$, $M_3$, $M_5$, and $M_7$ be the midpoints of sides $\overline{A_1 A_2}$, $\overline{A_3 A_4}$, $\overline{A_5 A_6}$, and $\overline{A_7 A_8}$, respectively. For $i = 1, 3, 5, 7$, ray $R_i$ is constructed from $M_i$ towards the interior of the octagon such that $R_1 \perp R_3$, $R_3 \perp R_5$, $R_5 \perp R_7$, and $R_7 \perp R_1$. Pairs of rays $R_1$ and $R_3$, $R_3$ and $R_5$, $R_5$ and $R_7$, and $R_7$ and $R_1$ meet at $B_1$, $B_3$, $B_5$, $B_7$ respectively. If $B_1 B_3 = A_1 A_2$, then $\cos 2 \angle A_3 M_3 B_1$ can be written in the form $m - \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.

Segment $\overline{PQ}$ is drawn and squares $ABPQ$ and $CDQP$ are constructed in the plane such that they lie on opposite sides of segment $\overline{PQ}.$ If $PQ=1,$ find $BD.$

A square has sides of length $2$. Set $S$ is the set of all line segments that have length $2$ and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$.

Let $ABCD$ be a convex quadrilateral such that the triangle $ABD$ is equilateral and the triangle $BCD$ is isosceles, with $\angle C = 90^o$. If $E$ is the midpoint of the side $AD$, determine the measure of the angle $\angle CED$.

In $ \triangle ABC $, $ AB = 3 $, $ BC = 4 $, and $ CA = 5 $. Circle $\omega$ intersects $\overline{AB}$ at $E$ and $B$, $\overline{BC}$ at $B$ and $D$, and $\overline{AC}$ at $F$ and $G$. Given that $EF=DF$ and $\tfrac{DG}{EG} = \tfrac{3}{4}$, length $DE=\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. Find $a+b+c$.

A circle of radius 2 is centered at $ O$. Square $ OABC$ has side length 1. Sides $ \overline{AB}$ and $ \overline{CB}$ are extended past $ b$ to meet the circle at $ D$ and $ E$, respectively. What is the area of the shaded region in the figure, which is bounded by $ \overline{BD}$, $ \overline{BE}$, and the minor arc connecting $ D$ and $ E$? [asy] defaultpen(linewidth(0.8)); pair O=origin, A=(1,0), C=(0,1), B=(1,1), D=(1, sqrt(3)), E=(sqrt(3), 1), point=B; fill(Arc(O, 2, 0, 90)--O--cycle, mediumgray); clip(B--Arc(O, 2, 30, 60)--cycle); draw(Circle(origin, 2)); draw((-2,0)--(2,0)^^(0,-2)--(0,2)); draw(A--D^^C--E); label("$A$", A, dir(point--A)); label("$C$", C, dir(point--C)); label("$O$", O, dir(point--O)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$B$", B, SW);[/asy] $ \textbf{(A) } \frac {\pi}3 \plus{} 1 \minus{} \sqrt {3} \qquad \textbf{(B) } \frac {\pi}2\left( 2 \minus{} \sqrt {3}\right) \qquad \textbf{(C) } \pi\left(2 \minus{} \sqrt {3}\right) \qquad \textbf{(D) } \frac {\pi}{6} \plus{} \frac {\sqrt {3} \minus{} 1}{2} \\ \qquad \indent \textbf{(E) } \frac {\pi}{3} \minus{} 1 \plus{} \sqrt {3}$

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: $\textbf{(A)}\ \dfrac{3a}{5} \qquad \textbf{(B)}\ \dfrac{5a}{8} \qquad \textbf{(C)}\ \dfrac{3a}{8} \qquad \textbf{(D)}\ \dfrac{a\sqrt{2}}{2} \qquad \textbf{(E)}\ \dfrac{a}{2}$

Two circles with radii 1 and 2 touch each other and a line as in the figure. In the region between the circles and the line, there is a circle with radius $ r$ which touches the two circles and the line. What is $ r$? [img]http://i250.photobucket.com/albums/gg265/geometry101/GeometryImage2.jpg[/img] A. 1/3 B. $ \frac {1}{\sqrt {5}}$ C. $ \sqrt {3} \minus{} \sqrt {2}$ D. $ 6 \minus{} 4 \sqrt {2}$ E. None of these

Below is a net consisting of $3$ squares, $4$ equilateral triangles, and $1$ regular hexagon. Each polygon has side length $1$. When we fold this net to form a polyhedron, what is the volume of the polyhedron? (This figure is called a "triangular cupola".) Net: [asy] pair A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P; A = origin; B = (0,3); C = 3*dir(150); D = (0,1); E = (0,2); F = C+2*dir(30); G = C+dir(30); H = 2*dir(150); I = dir(150); J = (1,1); K = J+dir(30); L = (1,2); M = F+dir(120); N = G+dir(120); O = H+dir(240); P = I+dir(240); draw(A--B--C--cycle); draw(D--E--F--G--H--I--cycle); draw(D--E--L--J--cycle); draw(F--G--N--M--cycle); draw(H--I--P--O--cycle); draw(J--K--L--cycle); [/asy] Resulting polyhedron: [img]https://upload.wikimedia.org/wikipedia/commons/9/93/Triangular_cupola.png[/img]

Let $ABCD$ be a cyclic quadrilateral satisfying $AD^2 + BC^2 = AB^2$. The diagonals of $ABCD$ intersect at $E$. Let $P$ be a point on side $\overline{AB}$ satisfying $\angle APD = \angle BPC$. Show that line $PE$ bisects $\overline{CD}$. [i]Proposed by Ankan Bhattacharya[/i]

Three circles touch each other externally and all these cirlces also touch a fixed straight line. Let $A,B,C$ be the mutual points of contact of these circles. If $\omega$ denotes the Brocard angle of the triangle $ABC$, prove that $\cot{\omega}$ = 2.