How many different triangles have vertices selected from the seven points (-4, 0), (-2, 0), (0,0), (2,0), (4,0), (0,2), and (0,4)?
Three circular cylinders are strapped together as shown. The cross-section of each cylinder is a circle of radius 1. Presuming that the strap used to bind the cylinders together has no thickness and no extra length, how long is the binding strap?
What are the coordinates of the reflection of (6,0) across the graph of $y=3x$?
If the area of a circle's inscribed square is 60, what is the area of its circumscribed square?
What is the area of a trapezoid the lengths of whose bases are 10 and 16, and the lengths of whose legs are 8 and 10?
A diagonal of a square intersects a segment that connects one vertex of the square to the midpoint of an opposite side, as shown. If the length of the shorter section of the diagonal is 2, what is the area of the square?
Let $\triangle{ABC}$ be a Pythagorean triangle. If $\triangle{ABC}$'s circumstance is 30, find its circumcircle's area.
Find the area of shaded area if the side length of the square is 1.
A parabola with equation $y=ax^2+bx+c$ is reflected about the $x$-axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of $y=f(x)$ and $y=g(x)$, respectively. Which of the following describes the graph of $y=(f+g)(x)$?
A grid point is defined as a point whose $x$ and $y$ coordinates are both integers. How many grid points are there on the circle which is centered at (199, 0) with a radius of 199?
In triangle $ABC$ , side $AC$ and the perpendicular bisector of $BC$ meet in point $D$, and $BD$ bisects $\angle ABC$. If $AD=9$ and $DC=7$, what is the area of triangle ABD?
Find the degree measure of an angle whose complement is 25% of its supplement.
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.
A $45^\circ$ arc of circle A is equal in length to a $30^\circ$ arc of circle B. What is the ratio of circle A's area and circle B's area?
Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let $B$ be the total area of the blue triangles, $W$ the total area of the white squares, and $R$ the area of the red square. Which of the following is correct?
Let $C_1$ and $C_2$ be circles defined by $(x-10)^2 + y^2 = 36$ and $(x+15)^2 + y^2 = 81$ respectively. What is the length of the shortest line segment $PQ$ that is tangent to $C_1$ at $P$ and to $C_2$ at $Q$?
In triangle $ABC$, $E$ is a point on $AC$ and $F$ is a point on $AB$. $BE$ and $CF$ intersect at $D$. If the areas of triangles $BDF$, $BCD$ and $CDE$ are 3, 7 and 7 respectively, what is the area of the quadrilateral $AEDF$?
A farmer has four straight fences, with respective lengths 1, 4, 7 and 8 metres. What is the maximum area of the quadrilateral the farmer can enclose?
In the diagram , $PA = QB = PC = QC = PD = QD = 1, CE = CF = EF$ and $EA = BF = 2AB$. Determine $BD$.
In rectangle $ABCD$, we have $AB=8$, $BC=9$, $H$ is on $BC$ with $BH=6$, $E$ is on $AD$ with $DE=4$, line $EC$ intersects line $AH$ at $G$, and $F$ is on line $AD$ with $GF \perp AF$. Find the length of $GF$.
Find all right triangles whose sides' lengths are all integers, and areas equal circumstance numerically.
A triangle's perimeter is 2016, and the ratio of its three altitudes is 3:5:7. Find the area of this triangle.
Solve in positive integers the equation $3^x + 4^y = 5^z$ .
A line that passes through the origin intersects both the line $x=1$ and the line $y=1+\frac{\sqrt{3}}{3}x$. The three lines create an equilateral triangle. What is the perimeter of the triangle?
Consider the lines that meet the graph $y = 2x^4 + 7x^3 + 3x - 5$ in four distinct points $P_i = (x_i, y_i), i = 1, 2, 3, 4$. Prove that
$$\frac{x_1 + x_2 + x_3 + x_44}{4}$$
is independent of the line, and compute its value.