Let $ABCD$ be a unit square. A circle with radius $\frac{32}{49}$ passes through point $D$ and is tangent to side $AB$ at point $E$. Then $DE = \frac{m}{n}$ , where $m$, $n$ are positive integers and gcd $(m, n) = 1$. Find $100m + n$.

a) In a triangle $ MNP$, the lenghts of the sides are less than $ 2$. Prove that the lenght of the altitude corresponding to the side $ MN$ is less than $ \sqrt {4 \minus{} \frac {MN^2}{4}}$. b) In a tetrahedron $ ABCD$, at least $ 5$ edges have their lenghts less than $ 2$.Prove that the volume of the tetrahedron is less than $ 1$.

Circles $\omega_1$ and $\omega_2$ intersect at two points $P$ and $Q$, and their common tangent line closer to $P$ intersects $\omega_1$ and $\omega_2$ at points $A$ and $B$, respectively. The line parallel to line $AB$ that passes through $P$ intersects $\omega_1$ and $\omega_2$ for the second time at points $X$ and $Y$, respectively. Suppose $PX = 10, PY = 14,$ and $PQ = 5$. Then the area of trapezoid $XABY$ is $m\sqrt{n}$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$.

For positive real numbers $s$, let $\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$. The set of all $s$ for which $\tau(s)$ is nonempty, but all triangles in $\tau(s)$ are congruent, is an interval $[a,b)$. Find $a^2+b^2$.

The digits $1$, $2$, $3$, $4$, and $5$ are each used once to write a five-digit number $PQRST$. The three-digit number $PQR$ is divisible by $4$, the three-digit number $QRS$ is divisible by $5$, and the three-digit number $RST$ is divisible by $3$. What is $P$? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }5$

Let $ABC$ be a triangle with $AB=5$, $BC=7$, $CA=8$, and circumcircle $\omega$. Let $P$ be a point inside $ABC$ such that $PA:PB:PC=2:3:6$. Let rays $\overrightarrow{AP}$, $\overrightarrow{BP}$, and $\overrightarrow{CP}$ intersect $\omega$ again at $X$, $Y$, and $Z$, respectively. The area of $XYZ$ can be expressed in the form $\dfrac{p\sqrt q}{r}$ where $p$ and $r$ are relatively prime positive integers and $q$ is a positive integer not divisible by the square of any prime. What is $p+q+r$? [i]Proposed by James Lin[/i]

Let right $\triangle ABC$ have $AC = 3$, $BC = 4$, and right angle at $C$. Let $D$ be the projection from $C$ to $\overline{AB}$. Let $\omega$ be a circle with center $D$ and radius $\overline{CD}$, and let $E$ be a variable point on the circumference of $\omega$. Let $F$ be the reflection of $E$ over point $D$, and let $O$ be the center of the circumcircle of $\triangle ABE$. Let $H$ be the intersection of the altitudes of $\triangle EFO$. As $E$ varies, the path of $H$ traces a region $\mathcal R$. The area of $\mathcal R$ can be written as $\tfrac{m\pi}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $\sqrt{m}+\sqrt{n}$. [i]Proposed by ApraTrip[/i]

Circles with centers $P, Q$ and $R$, having radii $1, 2$ and $3$, respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$, respectively, with $Q'$ between $P'$ and $R'$. The circle with center $Q$ is externally tangent to each of the other two circles. What is the area of triangle $PQR$? $\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}$

Triangle $ABC$ has sidelengths $AB = 14, BC = 15,$ and $CA = 13$. We draw a circle with diameter $AB$ such that it passes $BC$ again at $D$ and passes $CA$ again at $E$. If the circumradius of $\triangle CDE$ can be expressed as $\tfrac{m}{n}$ where $m, n$ are coprime positive integers, determine $100m+n$. [i]Proposed by Lewis Chen[/i]

Let $ABC$ be a triangle with $\angle C=90^\circ$, and $D$ be a point outside $ABC$, such that $\angle ADC=\angle BAC$. The segments $CD$ and $AB$ meet at point $E$. It is known that the distance from $E$ to $AC$ is equal to the circumradius of triangle $ADE$. Find the angles of triangle $ABC$.

A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$.

Let $ABC$ be a triangle with $AB=5$, $BC=7$, and $CA=8$. Let $D$ be a point on $BC$, and define points $B'$ and $C'$ on line $AD$ (or its extension) such that $BB'\perp AD$ and $CC'\perp AD$. If $B'A=B'C'$, then the ratio $BD:DC$ can be expressed in the form $m:n$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$. [i]Proposed by Michael Ren[/i]

A disk with radius $1$ is externally tangent to a disk with radius $5$. Let $A$ be the point where the disks are tangent, $C$ be the center of the smaller disk, and $E$ be the center of the larger disk. While the larger disk remains fixed, the smaller disk is allowed to roll along the outside of the larger disk until the smaller disk has turned through an angle of $360^\circ$. That is, if the center of the smaller disk has moved to the point $D$, and the point on the smaller disk that began at $A$ has now moved to point $B$, then $\overline{AC}$ is parallel to $\overline{BD}$. Then $\sin^2(\angle BEA)=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

In triangle $ABC$, $AB=7$, $AC=9$, and $BC=8$. The angle bisector of $\angle{BAC}$ intersects side $BC$ at $D$, and the angle bisector of $\angle{ABC}$ intersects $AD$ at $E$. Compute $AE^2$. [i]2016 CCA Math Bonanza Lightning #2.2[/i]