Point $B$ lies on line segment $\overline{AC}$ with $AB=16$ and $BC=4$. Points $D$ and $E$ lie on the same side of line $AC$ forming equilateral triangles $\triangle ABD$ and $\triangle BCE$. Let $M$ be the midpoint of $\overline{AE}$, and $N$ be the midpoint of $\overline{CD}$. The area of $\triangle BMN$ is $x$. Find $x^2$.
A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$.
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)
The graphs $y=3(x-h)^2+j$ and $y=2(x-h)^2+k$ have y-intercepts of $2013$ and $2014$, respectively, and each graph has two positive integer x-intercepts. Find $h$.
In $\triangle ABC$, $AB = 3$, $BC = 4$, and $CA = 5$. Circle $\omega$ intersects $\overline{AB}$ at $E$ and $B$, $\overline{BC}$ at $B$ and $D$, and $overline{AC}$ at $F$ and $G$. Given that $EF=DF$ and $\frac{DG}{EG} = \frac{3}{4}$, length $DE=\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. Find $a+b+c$.
In the Cartesian plane let $A = (1,0)$ and $B = \left( 2, 2\sqrt{3} \right)$. Equilateral triangle $ABC$ is constructed so that $C$ lies in the first quadrant. Let $P=(x,y)$ be the center of $\triangle ABC$. Then $x \cdot y$ can be written as $\tfrac{p\sqrt{q}}{r}$, where $p$ and $r$ are relatively prime positive integers and $q$ is an integer that is not divisible by the square of any prime. Find $p+q+r$.
Suppose that a parabola has vertex $\left(\frac{1}{4},-\frac{9}{8}\right)$ and equation $y = ax^2 + bx + c$, where $a > 0$ and $a + b + c$ is an integer. The minimum possible value of $a$ can be written in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
For a real number $a$, let $\lfloor a \rfloor$ denominate the greatest integer less than or equal to $a$. Let $\mathcal{R}$ denote the region in the coordinate plane consisting of points $(x,y)$ such that $\Big\lfloor x \Big\rfloor ^2 + \Big\lfloor y \Big\rfloor ^2 = 25$. The region $\mathcal{R}$ is completely contained in a disk of radius $r$ (a disk is the union of a circle and its interior). The minimum value of $r$ can be written as $\frac {\sqrt {m}}{n}$, where $m$ and $n$ are integers and $m$ is not divisible by the square of any prime. Find $m + n$.
Let $\mathcal{R}$ be the region consisting of the set of points in the coordinate plane that satisfy both $|8 - x| + y \le 10$ and $3y - x \ge 15$. When $\mathcal{R}$ is revolved around the line whose equation is $3y - x = 15$, the volume of the resulting solid is $\frac {m\pi}{n\sqrt {p}}$, where $m$, $n$, and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m + n + p$.
The parabolas $y=ax^2 - 2$ and $y=4 - bx^2$ intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area $12$. What is $a+b$?
A circle of radius r passes through both foci of, and exactly four points on, the ellipse with equation $x^2+16y^2=16.$ The set of all possible values of $r$ is an interval $[a,b).$ What is $a+b?$
A bee starts flying from point $P_0$. She flies $1$ inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015}$ she is exactly $a \sqrt{b} + c \sqrt{d}$ inches away from $P_0$, where $a$, $b$, $c$ and $d$ are positive integers and $b$ and $d$ are not divisible by the square of any prime. What is $a+b+c+d$ ?
The line perpendicular to $2x -2y = 2$, and with the same $y$-intercept, is graphed on the coordinate plane. What is the sum of its $x$- and $y$-intercepts?
The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coordinates is it true that $|4x+3y|\leq 1000$?
Let $P$ be the parabola with equation $y=x^2$ and let $Q = (20, 14)$. There are real numbers $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $P$ if and only if $r$ < $m$ < $s$. What is $r + s$?
Let points $A = (0,0) , \ B = (1,2), \ C = (3,3),$ and $D = (4,0)$. Quadrilateral $ABCD$ is cut into equal area pieces by a line passing through $A$. This line intersects $\overline{CD}$ at point $\left (\frac{p}{q}, \frac{r}{s} \right )$, where these fractions are in lowest terms. What is $p + q + r + s$?
Line $l_1$ has equation $3x - 2y = 1$ and goes through $A = (-1, -2)$. Line $l_2$ has equation $y = 1$ and meets line $l_1$ at point $B$. Line $l_3$ has positive slope, goes through point $A$, and meets $l_2$ at point $C$. The area of $\triangle ABC$ is $3$. What is the slope of $l_3$?
Two bees start at the same spot and fly at the same rate in the following directions. Bee $A$ travels $1$ foot north, then $1$ foot east, then $1$ foot upwards, and then continues to repeat this pattern. Bee $B$ travels $1$ foot south, then $1$ foot west, and then continues to repeat this pattern. In what directions are the bees traveling when they are exactly $10$ feet away from each other?
Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point (0,0) and the directrix lines have the form $y=ax+b$ with a and b integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?
A square region $ABCD$ is externally tangent to the circle with equation $x^2+y^2=1$ at the point $(0,1)$ on the side $CD$. Vertices $A$ and $B$ are on the circle with equation $x^2+y^2=4$. What is the side length of this square?
What is the area of the polygon whose vertices are the points of intersection of the curves $x^2 + y^2 =25$ and $(x-4)^2 + 9y^2 = 81 ?$
Two parabolas have equations $y= x^2 + ax +b$ and $y= x^2 + cx +d$, where $a, b, c,$ and $d$ are integers, each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas will have a least one point in common?
Square $PQRS$ lies in the first quadrant. Points $(3,0), (5,0), (7,0),$ and $(13,0)$ lie on lines $SP, RQ, PQ$, and $SR$, respectively. What is the sum of the coordinates of the center of the square $PQRS$?
Suppose $a$ and $b$ are single-digit positive integers chosen independently and at random. What is the probability that the point $(a,b)$ lies above the parabola $y=ax^2-bx$?
A segment through the focus $F$ of a parabola with vertex $V$ is perpendicular to $\overline{FV}$ and intersects the parabola in points $A$ and $B$. What is $\cos\left(\angle AVB\right)$?
For how many integer values of $k$ do the graphs of $x^2+y^2=k^2$ and $xy = k$ not intersect?