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 $\angle{C}$.
A regular pentagon is inscribed in a unit circle. Find the perimeter of this pentagon.
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 $R=2$ be the circumradius of $\triangle{ABC}$. Compute the value of $$\frac{a+b+c}{\sin{A}+\sin{B}+\sin{C}}$$
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.
Initially Alex, Betty, and Charlie had a total of $444$ peanuts. Charlie had the most peanuts, and Alex had the least. The three numbers of peanuts that each person had formed a geometric progression. Alex eats $5$ of his peanuts, Betty eats $9$ of her peanuts, and Charlie eats $25$ of his peanuts. Now the three numbers of peanuts each person has forms an arithmetic progression. Find the number of peanuts Alex had initially.
There is a $40\%$ chance of rain on Saturday and a $30\%$ chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$.
Let $x,y,$ and $z$ be real numbers satisfying the system $\log_2(xyz-3+\log_5 x)=5$ $\log_3(xyz-3+\log_5 y)=4$ $\log_4(xyz-3+\log_5 z)=4$ Find the value of $|\log_5 x|+|\log_5 y|+|\log_5 z|$.
An $a \times b \times c$ rectangular box is built from $a \cdot b \cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1 \times b \times c$ parallel to the $(b \times c)$ faces of the box contains exactly $9$ red cubes, exactly $12$ green cubes, and some yellow cubes. Each of the $b$ layers of size $a \times 1 \times c$ parallel to the $(a \times c)$ faces of the box contains exactly $20$ green cubes, exactly $25$ yellow cubes, and some red cubes. Find the smallest possible volume of the box.
Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pariwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\overline{AB}$, and for $n \geq 2$, let $C_n$ be the foot of the altitude to $\overline{C_{n-2}B}$ in $\triangle C_{n-2}C_{n-1}B$. The sum $\sum_{i=1}^\infty C_{n-2}C_{n-1} = 6p$. Find $p$.
For polynomial $P(x)=1-\dfrac{1}{3}x+\dfrac{1}{6}x^{2}$, define $Q(x)=P(x)P(x^{3})P(x^{5})P(x^{7})P(x^{9})=\displaystyle\sum_{i=0}^{50} a_ix^{i}$. Find the value of $\displaystyle\sum_{i=0}^{50} |a_i|$.
Squares $ABCD$ and $EFGH$ have a common center at $\overline{AB} || \overline{EF}$. The area of $ABCD$ is 2016, and the area of $EFGH$ is a smaller positive integer. Square $IJKL$ is constructed so that each of its vertices lies on a side of $ABCD$ and each vertex of $EFGH$ lies on a side of $IJKL$. Find the difference between the largest and smallest positive integer values for the area of $IJKL$.