Practice (64)
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Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0)$. For $n=1,2,3,\dots$, right after arriving at the point $p_n$, if Aaron can turn $90^\circ$ left and walk one unit to an unvisited point $p_{n+1}$, he does that. Otherwise, he walks one unit straight ahead to reach $p_{n+1}$. Thus the sequence of points continues $p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0)$, and so on in a counterclockwise spiral pattern. What is $p_{2015}$?
The $y$-intercepts, $P$ and $Q$, of two perpendicular lines intersecting at the point $A(6,8)$ have a sum of zero. What is the area of $\triangle APQ$?
In rectangle $ABCD$, $AB=1$, $BC=2$, and points $E$, $F$, and $G$ are midpoints of $\overline{BC}$, $\overline{CD}$, and $\overline{AD}$, respectively. Point $H$ is the midpoint of $\overline{GE}$. What is the area of the shaded region?
A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square?
Positive integers $a$ and $b$ are such that the graphs of $y=ax+5$ and $y=3x+b$ intersect the $x$-axis at the same point. What is the sum of all possible $x$-coordinates of these points of intersection?
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 $(\frac{p}{q}, \frac{r}{s})$, where these fractions are in lowest terms. What is $p+q+r+s$?
Define $a\clubsuit b=a^2b-ab^2$. Which of the following describes the set of points $(x, y)$ for which $x\clubsuit y=y\clubsuit x$?
Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of $\triangle ABC$?
The point in the $xy$-plane with coordinates (1000, 2012) is reflected across the line $y=2000$. What are the coordinates of the reflected point?
A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?
A rectangular region is bounded by the graphs of the equations $y=a, y=-b, x=-c,$ and $x=d$, where $a,b,c,$ and $d$ are all positive numbers. Which of the following represents the area of this region?
A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx +2$ passes through no lattice point with $0 < x \le 100$ for all $m$ such that $\frac{1}{2} < m < a$. What is the maximum possible value of $a$?
Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $(a,0)$ to $(3,3)$, divides the entire region into two regions of equal area. What is $a$?
Triangle $OAB$ has $O=(0,0)$, $B=(5,0)$, and $A$ in the first quadrant. In addition, $\angle ABO=90^\circ$ and $\angle AOB=30^\circ$. Suppose that $OA$ is rotated $90^\circ$ counterclockwise about $O$. What are the coordinates of the image of $A$?
Two points $B$ and $C$ are in a plane. Let $S$ be the set of all points $A$ in the plane for which $\triangle ABC$ has area $1.$ Which of the following describes $S?$
A parabola with equation $y=x^2+bx+c$ passes through the points $(2,3)$ and $(4,3)$. What is $c$?
Which of the following describes the graph of the equation $(x+y)^2=x^2+y^2$?
What is the greatest possible area of a triangle with vertices on or above the $x$-axis and on or below the parabola $y = -(x\u200a - \frac{1}{2})^2+ 3$? Express your answer in simplest radical form.
If the point $(x, x)$ is equidistant from (-2, 5) and (3, -2), what is the value of $x$?
If $f(x) = 3x^2$, what is the x-coordinate of the point of intersection of the graphs of $y = f(x)$ and $y = f(x \u2212 4)$?
A line segment with endpoints A(3, 1) and B(2, 4) is rotated about a point in the plane so that its endpoints are moved to A' (4, 2) and B' (7, 3), respectively. What are the coordinates of the center of rotation? Express your answer as an ordered pair.
What is the length of the shortest segment that can be drawn from the point (4, 1) to 2x - y + 4 = 0? Express your answer as a decimal to the nearest hundredth.
A line passes through the points (-2, 8) and (5, -13). When the equation of the line is written in the form $y = mx + b$, what is the product of $m$ and $b$?
$\triangle{ABC}$ has vertices at A(-3, 4), B(5, 0) and C(1, -4). What is the $x$-coordinate of the point where the median from C intersects $\overline{AB}$?
One line has a slope of \u22121/3 and contains the point (3, 6). Another line has a slope of 5/3 and contains the point (3, 0). We are asked to find the product of the coordinates of the point at which the two lines intersect.