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 $\displaystyle\sum_{i=0}^{50} |a_i|$.
Find the number of sets $\{a,b,c\}$ of three distinct positive integers with the property that the product of $a,b,$ and $c$ is equal to the product of $11,21,31,41,51,61$.
The sequences of positive integers $1,a_2, a_3,...$ and $1,b_2, b_3,...$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$. Find $c_k$.
As shown, in quadrilateral $ABCD$, $AB=AD$, $\angle{BAD} = \angle{DCB} = 90^\circ$. Draw altitude from $A$ towards $BC$ and let the foot be $E$. If $AE=1$, find the area of $ABCD$.
In pentagon $ABCDE$, if $AB=AE$, $BC+DE=CD$, and $\angle{ABC} + \angle{AED} = 180^\circ$, show that $\angle{ADE}=\angle{ADC}$.
Let $\triangle{ABC}$ be an isosceles right triangle where $\angle{C}=90^\circ$. If points $M$ and $N$ are on $AB$ such that $\angle{MCN}=45^\circ$, $AM=4$, and $BN=3$, find the length of $MN$.
In $\triangle{ABC}$, let $AB=c$, $AC=b$, and $\angle{BAC}=\alpha$. If $AD$ bisects $\angle{BAC}$ and intersects $BC$ at $D$, find the length of $AD$.
As shown, prove $$\frac{\sin(\alpha+\beta)}{PC}=\frac{\sin{\alpha}}{PB}+\frac{\sin{\beta}}{PA}$$
(Weitzenbock's Inequality) Let $a, b, c$, and $S$ be a triangle's three sides' lengths and its area, respectively. Show that $$a^2 + b^2 + c^2 \ge 4\sqrt{3}\cdot S$$
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 $PFCG$.
We are given a triangle with the following property: one of its angles is quadrisected (divided into four equal angles) by the height, the angle bisector, and the median from that vertex. This property uniquely determines the triangle (up to scaling). Find the measure of the quadrisected angle.
Let $P$ be a point inside parallelogram $ABCD$. If $\angle{PAB}=\angle{PCB}$, show $\angle{PBA} = \angle{PDA}$.
Two sides of a triangle are 4 and 9; the median drawn to the third side has length 6. Find the length of the third side.
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.
Three squares are drawn on the sides of $\triangle{ABC}$ (i.e. the square on $AB$ has $AB$ as one of its sides and lies outside $\triangle{ABC}$). Show that the lines drawn from the vertices $A, B, C$ to the centers of the opposite squares are concurrent.
Let $P$ be a point inside a unit square $ABCD$. Find the minimal value of $AP+BP+CP$
The diagonals $AC$ and $CD$ of the regular hexagon $ABCDEF$ are divided by inner points $M$ and $N$ such that $AM:AC = CN:CE=r$. Determine $r$ if $B, M,$ and $N$ are collinear.
How many rectangles of any size are in the grid shown here?
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 $6$?
As shown, $ABCD$ is a square with side length equaling 10 cm, $CE\perp BE$, and $CE=8$. Find the area of the shaded triangle.
As shown, a regular hexagon is inscribed in the bigger circle. If the area of the bigger circle is 2016 $cm^2$, find the total area of shaded regions.
Find all positive integer $n$ such that $n^2 + 2^n$ is a perfect square.
Let $ABCDE$ be a pentagon such that $AB=BC=CD=DE=EA$ as shown. If $\angle{ABC}=2\angle{DBE}$, find the measurement of $\angle{ABC}$.
