In $\triangle{ABC}$, $AB = 33, AC=21,$ and $BC=m$ where $m$ is an integer. There exist points $D$ and $E$ on $AB$ and $AC$, respectively, such that $AD=DE=EC=n$ where $n$ is also an integer. Find all the possible values of $m$.
Let quadrilateral $ABCD$ inscribe a circle. If $BE=ED$, prove $$AB^2+BC^2 +CD^2 + DA^2 = 2AC^2$$
In $\triangle{ABC}$, $AE$ and $AF$ trisects $\angle{A}$, $BF$ and $BD$ trisects $\angle{B}$, $CD$ and $CE$ trisects $\angle{C}$. Show that $\triangle{DEF}$ is equilateral.
As shown, in $\triangle{ABC}$, $AB=AC$, $\angle{A} = 20^\circ$, $\angle{ABE} = 30^\circ$, and $\angle{ACD}=20^\circ$. Find the measurement of $\angle{CDE}$.
Let $O_1$ and $O_2$ be two intersecting circles. Let a common tangent to these two circles touch $O_1$ at $A$ and $O_2$ at $B$. Show that the common chord of these two circles, when extended, bisects segment $AB$.
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.
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 $CM=DM$.
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?