Quadrilateral $APBQ$, shown here, has vertices $A(0, 0)$ and $B(8, 0)$, and vertices $P$ and $Q$ lie on the line given by the equation $4x + 3y = 19$. If $PQ = 3$ units, what is the area of quadrilateral $APBQ$? Express your answer as a common fraction.
Eight blue and five orange tiles are arranged in an ordered line such that the tile on the left must be blue and every tile must be adjacent to at least one tile of the same color. For example, if an arrangement of four tiles was made, the only possibilities would be $BBBB$ or $BBOO$. How many different arrangements are possible if all thirteen tiles must be used?
Spring City is replanting the grass around a circular fountain in the center of the city. The fountain’s diameter is 10 feet, and the grass extends out from the edge of the fountain 20 feet in every direction. Grass seed is sold in bags that will each cover 300 $ft^2$ of grass. How many whole bags of grass seed will the city need to purchase?
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.
As shown, $D$ is the midpoint of $BC$. Point $E$ is on $AD$ such that $BE=AF$. Show that $AF=EF$.
A 12-sided game die has the shape of a hexagonal bipyramid, which consists of two pyramids, each with a regular hexagonal base of side length 1 cm and with height 1 cm, glued together along their hexagons. When this game die is rolled and lands on one of its triangular faces, how high off the ground is the opposite face? Express your answer as a common fraction in simplest radical form.
The function $f (n) = a ⋅ n! + b$, where a and b are positive integers, is defined for all positive integers. If the range of $f$ contains two numbers that differ by 20, what is the least possible value of $f (1)$?
If 738 consecutive integers are added together, where the 178th number in the sequence is 4,256,815, what is the remainder when this sum is divided by 6?
Rectangle $ABCD$ is shown with $AB = 6$ units and $AD = 5$ units. If $AC$ is extended to point $E$ such that $AC$ is congruent to $CE$, what is the length of $DE$?
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 $\angle{ABC}$.
Let $ABCD$ be a trapezoid. Points $M$ and $N$ are the mid points of its diagonal $AC$ and $BD$, respectively. Show that $MN \parallel AB$ and $MN = \frac{1}{2}\mid AB - CD\mid$.
A plane passing through the vertex $A$ and the center of its inscribed sphere of a tetrahedron $ABCD$ intersects its edge $BC$ and $CD$ at point $E$ and $F$, as shown. If $AEF$ divides this tetrahedron into two equal volume parts: $A-BDEF$ and $A-CEF$, what is the relationship between these two parts' surface areas $S_1$ and $S_2$ where $S_1 = S_{A-BDEF}$ and $S_1=S_{A-CEF}$?
$(A) S_1 < S_2\quad(B) S_1 > S_2\quad (C) S_1 = S_2 \quad(D) $ cannot determine
Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of whose angles is twice as large as another.
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.
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.
Given that $a, b,$ and $c$ are positive integers such that $a^b\cdot b^c$ is a multiple of 2016. Compute the least possible value of $a+b+c$.
Triangle $ABC$ is isosceles. An ant begins at $A$, walks exactly halfway along the perimeter of $\triangle{ABC}$, and then returns directly $A$, cutting through the interior of the triangle. The ant's path surround exactly 90% of the area of $\triangle{ABC}$. Compute the maximum value of $\tan{A}$.
Compute the number of permutations $x_1, \cdots, x_6$ of integers $1, \cdots, 6$ such that $x_{i+1}\le 2x_i$ for all $i, 1\le i < 6$.
Compute the least possible non-zero value of $A^2+B^2+C^2$ such that $A, B,$ and $C$ are integers satisfying $A\log16+B\log18+C\log24=0$.
In $\triangle{LEO}$, point $J$ lies on $\overline{LO}$ such that $\overline{JE}\perp\overline{EO}$, and point $S$ lies on $\overline{LE}$ such that $\overline{JS}\perp\overline{LE}$. Given that $JS=9, EO=20,$ and $JO+SE=37$, compute the perimeter of $\triangle{LEO}$.
Compute the least possible area of a non-degenerate right triangle with sides of lengths $\sin{x}$, $\cos{x}$ and $\tan{x}$ where $x$ is a real number.
Let $P(x)$ be the polynomial $x^3 + Ax^2 +Bx+C$ for some constants $A, B,$ and $C$. There exists constant $D$ and $E$ such that for all $x$, $P(x+1)=x^3 + Dx^2 + 54x +37$ and $P(x+2)=x^3 + 26x + Ex+115$. Compute the ordered triple $(A, B, C)$.