Line $PT$ is tangent to circle $O$ at point $T$. $PA$ intersects circle $O$ and its diameter $CT$ at $B$, $D$, and $A$ in that order. If $CD=2, AD=3,$ and $BD=6$, find the length of $PB$.
In tetrahedron $ABCD$, $\angle{ADB} = \angle{BDC} = \angle{CDA} = 60^\circ$, $AD=BD=3$, and $CD=2$. Find the radius of $ABCD$'s circumsphere.
In tetrahedron $P-ABC$, $AB=BC=CA$ and $PA=PB=PC$. If $AB=1$ and the altitude from $P$ to $ABC$ is $\sqrt{2}$, find the radius of $P-ABC$'s inscribed sphere.
Let $AB=2$ is a diameter of circle $O$. If $AC=AO$, $AC\perp AB$, $BD=\frac{3}{2}\cdot AB$, $BD\perp AB$ and $P$ is a point on arc $AB$. Find the largest possible area of the enclosed polygon $ABDPC$.
Two circles, $O_1$ and $O_2$ are tangent. Let $AB$ be their common tangent line which touches $O_1$ at point $A$ and touches $O_2$ at point $B$. Extend $AO_1$ and intersects $O_1$ at another point $C$. Line $CD$ is tangent to circle $O_2$ at point $D$. Show that $AC=CD$.
Let $P$ be a point inside square $ABCD$ such that $AP=1, BP = 3,$ nd $DP=\sqrt{7}$. Find the area of $ABCD$.
Try to find at least two solutions.
Let $M$ be the midpoint of $AB$ which is the hypotenuse of a non-isosceles right triangle ${ABC}$. If $DM\perp AB$ and $DC$ bisects $\angle{ACB}$, show $CM=DM$.
Let $ABCD$ be a square. Point $E$ is outside $ABCD$ such that $DE=CD$ and $\angle{AED}=15^\circ$. Show $\triangle{DCE}$ is equilateral.
Compute the value of $$\sum_{n=2019}^\infty\frac{1}{\binom{n}{2019}}$$
Show that $\sin{x}+2\sin{2x}+\cdots + n\sin{nx}=\frac{(n+1)\sin{nx} - n\sin{(n+1)x}}{2(1-\cos{x})}$
Simplify $\cos{x}\cos{2x}\cdots\cos{2^{n-1}x}$.
Evaluate $\cos\frac{2\pi}{2n+1}+\cos\frac{4\pi}{2n+1}+\cdots+\cos\frac{2n\pi}{2n+1}$.
Show that $C_n^0-C_n^2+C_n^4-C_n^6+\cdots=2^{\frac{n}{2}}\cos\frac{n\pi}{4}$.
Show that
\begin{align*}
C_n^0-C_n^2+C_n^4-C_n^6+\cdots &=2^{\frac{n}{2}}\cos\frac{n\pi}{4}\\
C_n^1-C_n^3+C_n^5-C_n^7+\cdots &=2^{\frac{n}{2}}\sin\frac{n\pi}{4}
\end{align*}
Show that $$\binom{n}{1}-\frac{1}{2}\binom{n}{2}+\frac{1}{3}\binom{n}{3}-\cdots+(-1)^{n+1}\binom{n}{n}= 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$$
Find the value of $$\binom{n}{0}-\binom{n}{1}+\binom{n}{2}-\binom{n}{3}+\cdots +(-1)^n\binom{n}{n}$$
John walks from point $A$ to $C$ while Mary goes from point $B$ to $D$. Both of them will move along the grid, either right or up, so they take shortest routes. How many different possibilities are there such that their routes do not intersect?
Show that $$\sum_{k=0}^n\left(2^k\binom{n}{k}\right)=3^n$$
Let the sum of first $n$ terms of arithmetic sequence $\{a_n\}$ be $S_n$, and the sum of first $n$ terms of arithmetic sequence $\{b_n\}$ be $T_n$. If $\frac{S_n}{T_n}=\frac{2n}{3n+7}$, compute the value of $\frac{a_8}{b_6}$.
Suppose every term in the sequence
$$1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, \cdots$$
is either $1$ or $2$. If there are exactly $(2k-1)$ twos between the $k^{th}$ one and the $(k+1)^{th}$ one, find the sum of its first $2014$ terms.
Let $c_1, c_2, c_3, \cdots$ be a series of concentric circles whose radii form a geometric sequence with common ratio as $r$. Suppose the areas of rings which are formed by two adjacent circles are $S_1, S_2, S_3, \cdots$. Which statement below is correct regarding the sequence $\{S_n\}$?
A) It is not a geometric sequence
B) It is a geometric sequence and its common ratio is $r$
C) It is a geometric sequence and its common ratio is $r^2$
D) It is a geometric sequence and its common ratio is $r^2-1$
Given the sequence $\{a_n\}$ satisfies $a_n+a_m=a_{n+m}$ for any positive integers $n$ and $m$. Suppose $a_1=\frac{1}{2013}$. Find the sum of its first $2013$ terms.
What is the Ptolemy Theorem?
Let sequence $\{a_n\}$ satisfy $a_1=2$ and $a_{n+1}=\frac{2(n+2)}{n+1}a_n$ where $n\in \mathbb{Z}^+$. Compute the value of $$\frac{a_{2014}}{a_1+a_2+\cdots+a_{2013}}$$
Let $n$ be a positive integer. Show that $$\Big(1+\frac{1}{3}\Big)\Big(1+\frac{1}{3^2}\Big)\cdots\Big(1+\frac{1}{3^n}\Big) < 2$$