Let $ABCD$ be a rectangle where $AB=4$ and $BC=6$. If $AE=CG=3$, $BF=DH=4$, and $S_{AEPH}=5$. Find the area of $PFCG$.
A circle inscribed in $\triangle{ABC}$ (the incircle) is tangent to $BC$ at $X$, to $AC$ at $Y$ , to $AB$ at $Z$. Show that $AX$, $BY$, and $CZ$ are concurrent.
How many rectangles of any size are in the grid shown here?
There are twelve different mixed numbers that can be created by substituting three of the numbers $1$, $2$, $3$ and $5$ for $a$, $b$ and $c$ in the expression $a\frac{b}{c}$ , where $b < c$. What is the mean of these twelve mixed numbers? Express your answer as a mixed number.
If $2016$ consecutive integers are added together, where the $999^{th}$ number in the sequence is $1,244,584$, what is the remainder when this sum is divided by $6$?
In the list of numbers $1, 2, \cdots, 9999$, the digits $0$ through $9$ are replaced with the letters $A$ through $J$, respectively. For example, the number $501$ is replaced by the string $FAB$ and $8243$ is replaced by the string $ICED$. The resulting list of $9999$ strings is sorted alphabetically. How many strings appear before $CHAI$ in this list?
Billy wrote a sequence of five numbers on the board, each an integer between $0$ and $4$, inclusive. Penny then wrote a sequence of five numbers that measured some statistics about Billy’s sequence. In particular, Penny first wrote down the number of $0$s in Billy’s sequence. Then Penny wrote the number of $1$s in Billy’s sequence, and then the number of $2$s, the number of $3$s, and finally the number of $4$s. It turned out that Penny’s sequence was exactly the same as Billy’s! What was this sequence? Express your answer as an ordered $5$-tuple.
Let $ABCDE$ be a pentagon such that $AB=BC=CD=DE=EA$ as shown. If $\angle{ABC}=2\angle{DBE}$, find the measurement of $\angle{ABC}$.
As shown, $E, F, G, H$ are midpoints of the four sides. If $AB\parallel A'B', BC\parallel B'C',$ and $CD\parallel C'D'$, show that $AD\parallel A'D'$.
As shown, two vertices of a square are on the circle and one side is tangent to the circle. If the side length of the square is 8, find the radius of the circle.
$DEB$ is a chord of a circle such that $DE=3$ and $EB=5 .$ Let $O$ be the center of the circle. Join $OE$ and extend $OE$ to cut the circle at $C.$ Given $EC=1,$ find the radius of the circle
Chords $AB$ and $CD$ of a given circle are perpendicular to each other and intersect at a right angle at point $E$. If $BE=16$, $DE=4$, and $AD=5$, find $CE$.
Let $x, y,$ and $z$ be some real numbers such that: $x+2y-z=6$ and $x-y+2z=3$. Find the minimal value of $x^2 + y^2 + z^2$.
Prove the Maximum Area of a Triangle with Fixed Perimeter is Equilateral
Use the Arithmetic Mean-Geometric Mean Inequality to find the maximum volume of a box made from a 25 by 25 square sheet of cardboard by removing a small square from each corner and folding up the sides to form a lidless box.
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.
Simplify $\cos{x}\cos{2x}\cdots\cos{2^{n-1}x}$.
Find the value of $$\binom{n}{0}-\binom{n}{1}+\binom{n}{2}-\binom{n}{3}+\cdots +(-1)^n\binom{n}{n}$$
Show that $$\sum_{k=0}^n\left(2^k\binom{n}{k}\right)=3^n$$
Suppose function $f(x)=\frac{1+x}{1-x}$. Evaluate $$f\Big(\frac{1}{2}\Big)\cdot f\Big(\frac{1}{4}\Big)\cdot f\Big(\frac{1}{6}\Big)\cdots f\Big(\frac{1}{2014}\Big)$$
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.
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.
Let $a_1, a_2,\cdots, a_n > 0, n\ge 2,$ and $a_1+a_2+\cdots+a_n=1$. Prove $$\frac{a_1}{2-a_1} + \frac{a_2}{2-a_2}+\cdots+\frac{a_n}{2-a_n}\ge\frac{n}{2n-1}$$
Suppose all the terms in a geometric sequence $\{a_n\}$ are positive. If $|a_2-a_3|=14$ and $|a_1a_2a_3|=343$, find $a_5$.