Let $G$ be the centroid of $\triangle{ABC}$. Points $M$ and $N$ are on side $AB$ and $AC$, respectively such that $\overline{AM} = m\cdot\overline{AB}$ and $\overline{AN} = n\cdot\overline{AC}$ where $m$ and $n$ are two positive real numbers. Find the minimal value of $mn$.
An isosceles triangle with equal sides of 5 and bases of 6 is inscribed in a circle. Find the radius of that circle.
A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain $A$ measures $60^{\circ}$. Find $h^2$.
In $\triangle ABC$ let $I$ be the center of the inscribed circle, and let the bisector of $\angle ACB$ intersect $\overline{AB}$ at $L$. The line through $C$ and $L$ intersects the circumscribed circle of $\triangle ABC$ at the two points $C$ and $D$. If $LI=2$ and $LD=3$, then $IC= \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
Centered at each lattice point in the coordinate plane are a circle radius $\frac{1}{10}$ and a square with sides of length $\frac{1}{5}$ whose sides are parallel to the coordinate axes. The line segment from $(0,0)$ to $(1001, 429)$ intersects $m$ of the squares and $n$ of the circles. Find $m + n$.
Circles $\omega_1$ and $\omega_2$ intersect at points $X$ and $Y$. Line $\ell$ is tangent to $\omega_1$ and $\omega_2$ at $A$ and $B$, respectively, with line $AB$ closer to point $X$ than to $Y$. Circle $\omega$ passes through $A$ and $B$ intersecting $\omega_1$ again at $D \neq A$ and intersecting $\omega_2$ again at $C \neq B$. The three points $C$, $Y$, $D$ are collinear, $XC = 67$, $XY = 47$, and $XD = 37$. Find $AB^2$.
Let $ABC$ be a right triangle where $AB=4, BC=3$ and $AC=5$. Draw square $ACEF$ and $BCDG$ as shown. Find the area of $\triangle{CDE}$.
Let $ABCD$ be a convex quadrilateral and the two diagonals intersect at point $O$. If $AC = 12, BD=5$ and $\angle{AOB}=2\angle{BOC}$, find the area of $S_{ABCD}$
In $\triangle{ABC}$, if $(b+c):(c+a):(a+b)=5:6:7$, compute $\sin{A}:\sin{B}:\sin{C}$.
Let quadrilateral $ABCD$ inscribe in a circle. If $\angle{CAD}=30^\circ$, $\angle{ACB}=45^\circ$, and $CD=2$. Find the length of $AB$.
In $\triangle{ABC}$, if $\sin{A}:\sin{B}:\sin{C}=4:5:7$, compute $\cos{C}$ and $\sin{C}$.
In $\triangle{ABC}$, if $(a+b+c)(a+b-c)=3ab$, compute $\angle{C}$.
Two people located 500 yards apart have spotted a hot air balloon. The angle of elevation from one person to the balloon is $67^\circ$. From the second person to the balloon the angle of elevation is $46^\circ$. How high is the balloon when it is spotted? (You can use a calculator. Please keep the result to the 2 decimal places.)
In $\triangle{ABC}$, if $(a^2 +b^2)\sin(A-B)=(a^2-b^2)\sin(A+B)$, determine the shape of $\triangle{ABC}$.
Let $\triangle{ABC}$ be an acute triangle and $a, b, c$ be the three sides opposite to $\angle{A}, \angle{B}, \angle{C}$ respectively. If vectors $m=(a+c,b)$ and $n=(a-c, b-a)$ satisfy $m\cdot n = 0$,
(1) Compute the measurement of $\angle{C}$.
(2) Find the range of $\sin{A} + \sin{B}$.
In $\triangle{ABC}$, if $a\cos{C} + \frac{c}{2} = b$, (1) compute $\angle{A}$. (2) if $a=1$, find the range of the perimeter o f $\triangle{ABC}$.
Let $BD$ be a median in $\triangle{ABC}$. If $AB=\frac{4\sqrt{6}}{3}$, $\cos{B}=\dfrac{\sqrt{6}}{6}$, and $BD=\sqrt{5}$, find the length of $BC$ and the value of $\sin{A}$.
Let $ABCD$ be inscribed in a circle. If $AB=a, BC=b, CD=c,$ and $DA=d$, show that $$\cos{B} = \frac{a^2 + b^2 -c^2 - d^2}{2(ab+cd)}$$
There are four points on a plane as shown. Points $A$ and $B$ are fixed points satisfying $AB=\sqrt{3}$. Points $P$ and $Q$ can move, as long as $AP=PQ=QB=1$. Let $S$ and $T$ be the area of $\triangle{APB}$ and $\triangle{PQB}$, respectively. Find the maximum value of $S^2+T^2$.
Let $R=\frac{7\sqrt{3}}{3}$ be the circumradius of $\triangle{ABC}$. If $\angle{B} = 60^\circ$ and its area $S_{\triangle{ABC}}=10\sqrt{3}$, find the lengths of $a$, $b$, and $c$.
In isosceles right triangle $\triangle{ABC}$, $\angle{C}$ is the right angle. Points $D$ and $E$ are on side $AB$ such that $\angle{DCE}=45^\circ$, $AD=25$, and $EB=16$. Let the length of $DE$ be $x$. Find $x^2$.
Let $ABCD$ be a square. Points $E$ and $F$ are on its sides $BC$ and $CD$ such that $\angle{EAF}=45^\circ$. If point $G$ is on $EF$ such that $AG\perp EF$, show $AG=AD$.
Let $D$ be a point inside an isosceles triangle $\triangle{ABC}$ where $AB=AC$. If $\angle{ADB} > \angle{ADC}$, prove $DC > DB$.
Let $ABCD$ be a square. Points $E$ and $F$ are on side $CD$ and $BC$, respectively such that $AF$ bisects $\angle{EAB}$. Show that $AE=BE+DF$.
Let $ABCD$ be a square with side length of 1. Find the total area of the shaded parts in the diagram.
