Circles
SEE DIAGRAM. The figure below shows a large white circle with a number of smaller white and shaded circles in its interior. What fraction of the interior of the large white circle is shaded?
11/36
In triangle ABC, D is a point on side AC such that BD = DC. ∠BCD measures 70°. What is the degree measure of ∠ADB?
140º
Camden constructs a square window using 8 equal-size panes of glass. The ratio of the height to width for each pane is 5 : 2, and the borders around and between the panes are 2 inches wide. In inches, what is the side length of the square window? (SEE DIAGRAM)
26
What is the volume of a cube whose surface area is twice that of a cube with volume 1?
2√2
Isaiah cuts open a cardboard cube along some of its edges to form a net that has an area of 18 square centimeters. What is the volume of the cube in cubic centimeters?
3√3
Two congruent circles centered at points A and B each pass through the other circle’s center. The line containing both A and B is extended to intersect the circles at points C and D. The circles intersect at two points, one of which is E. What is the degree measure of ∠CED?
120º
What is the area of the triangle formed by the lines y = 5, y = 1 + x, and y = 1 - x?
16
Point E is the midpoint of side CD in square ABCD. BE meets diagonal AC at point F. The area of quadrilateral AFED is 45. What is the area of ABCD?
108
A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper. The four corners of the wrapping paper will be folded up over the sides and brought together to meet at the center of the top of the box. The box has base length w and height h. What is the area of the sheet of wrapping paper? (SEE DIAGRAM)
2(w+h)^2
Rectangle ABCD has sides CD = 3 and DA = 5. A circle with a radius of 1 is centered at A, a circle with a radius of 2 is centered at B, and a circle with a radius of 3 is centered C. What is the closest integer to the area of the region inside the rectangle but outside all three circles?
4
In a plane, four circles with radii 1, 3, 5 and 7 are tangent to line L at the same point A but they may be on either side of L. Region S consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region S?
65pi
Triangle AMC is isosceles with AM = AC. V is the midpoint of AC, and U is the midpoint of AM. MV and CU are perpendicular to each other, and MV=CU=12. What is the area of triangle AMC?
96
The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle?
(√5-1)/2
A three-quarter sector of a circle of radius 4 inches along with its interior is the 2-D net that forms the lateral surface area of a right circular cone by taping together along the two exposed radii. What is the volume of the cone in cubic inches?
3pi√7
In rectangle ABCD, DC = 2 • CB and points E and F lie on AB so that ED and FD trisect ∠ADC. What is the ratio of the area of triangle DEF to the area of rectangle ABCD?
√3/6
Externally tangent circles with centers at points A and B have radii of lengths 5 and 3, respectively. A line externally tangent to both circles intersects ray AB at point C. What is BC ?
12
In triangle ABC, AB = 13, BC = 14, AC = 15. Let D denote the midpoint of BC. The bisector of angle BAC intersects BC at point E. What is the area of triangle ADE?
3
Take Rhombus ABCD. E and F are two points on CA that, when joined with B and D, form a rhombus BFDE similar to ABCD. The area of rhombus ABCD is 24 and m<BAD = 60˚. What is the area of rhombus BFDE? (SEE DIAGRAM)
8
An ice-cream novelty item consists of a cup in the shape of a 4-inch-tall frustum of a right circular cone, with a 2-inch-diameter base at the bottom and a 4-inch-diameter base at the top, packed solid with ice cream, together with a solid cone of ice cream of height 4 inches, whose base, at the bottom, is the top base of the frustum. What is the total volume of the ice cream, in cubic inches?
44pi/3
Six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the region inside the hexagon but outside all of the semicircles? SEE DIAGRAM
3√3-pi
In triangle ABC, AB = 86, and AC = 97. A circle with center A and radius AB intersects BC at points B and X. Moreover BX and CX have integer lengths. What is BC?
61
Let ABC be a scalene triangle. Point P lies on BC so that AP bisects ∠BAC. The line through B perpendicular to AP intersects the line through A parallel to BC at point D. Suppose BP = 2 and PC = 3. What is AD?
10
Isosceles trapezoid ABCD has parallel sides AD and BC, with BC < AD and AB = CD. There is a point P in the plane such that PA = 1, PB = 2, PC = 3, and PD = 4. What is BC/AD?
1/3
Take the cube ABCDEFGH with opposite vertices C and E. J and I are the midpoints of segments FB and HD respectively. Let R be the ratio of the area of the cross-section EJCI to the area of one of the faces of the cube. What is R^2? (SEE DIAGRAM)
3/2
Quadrilateral ABCD with side lengths AB = 7, BC = 24, CD = 20, DA = 15 is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form (api - b)/c, where a, b, and c are positive integers such that a and c have no common prime factor. What is a + b + c?
1565