A wheel of radius 10 inches rolls inside a wheel of radius 54 inches. Point P^{1} on the little rolling wheel starts at point P on the fixed wheel. After how many revolutions of the little wheel do points P and P^{1} coincide once more?
Algebra II / Trigonometry
Trapezoid, Similar Triangles
Let ABCD be a trapezoid with the measure of base AB twice that of base DC, and let E be the point of intersection of the diagonals. If the measure of diagonal AC is 11, then what is the measure of segment EC?
 11/3
 15/4
 4
 7/2
 3
Spinning in Circles
We humans are traveling around in circles without even knowing it. We travel a circle around the axis of the earth once a day, and an almostcircle around the sun once a year. Considering each trip independently, how fast do we travel in each circle? (Assume we’re on the equator.) Here are some helpful data:
Give each answer in kilometers per hour.
Find Diameter of Circle
AB is a diameter of a circle. Tangents AD and BC are drawn so that segments AC and BD intersect at a point E on the circle. If AD = a and BC = b, with a ≠ b, then the diameter AB is:
a.  a – b  
b. 


c.  √ab  
d. 


e. 

Stack of Water Pipes
Three of the new water pipes for Main Street are stacked as shown. The external diameter of each pipe is 12 inches. How tall is the stack (h in the picture)?
Gertrude Goldfish
Two young men in tuxedos are walking down the street at about 10 pm. One of them is carrying a round goldfish bowl filled with water. There is a goldfish, named Gertrude, in the bowl. As they pass a round pool in the park, the goldfish gets very excited and jumps out of the bowl into the pool, right at the edge. She starts swimming due north. She hits the wall after she has swum exactly 30 feet. She heads east, and, after going 40 feet, she hits the wall again. Once she regains consciousness after this second collision, she calculates the diameter of the pool. What is it? And what is the story about the two guys in their tuxes?
Clairbuoyant
A fishing boat, the Clairbuoyant, sails 40 miles east, then 80 miles south, and finally 20 miles east again. How far is it from its starting point?
Tetrahedron in Cube
Four of the 8 vertices of a cube are vertices of a regular tetrahedron. Find the ratio of the surface area of the cube to the surface area of the tetrahedron.
 2
 3
 √2
 √3
 2
Line Splits Triangle Into Equal Areas
A vertical line divides the triangle with vertices (0,0), (1,1) and (9,1) in the xyplane into two regions of equal area. What is the equation of the line?
 x = 2.5
 x = 3.0
 x = 3.5
 x = 4.0
 x = 4.5
Radio Antenna
A piece of wire for a radio antenna is to be strung from the roof of Andrea’s house to the roof of the garage. The garage is 40 feet from the house. The roof of the house is 45 feet high at the edge where the wire is fastened, and the roof of the garage is 15 feet high. The wire must be at least how long?