CALIFORNIA STATE UNIVERSITY, BAKERSFIELD
Lee Webb Math Field Day 2010
Team Medley, Freshman-Sophomore Level
Each correct answer is worth ten points. Answers require justification. Partial
credit may be given. Unanswered questions are given zero points.
You have 50 minutes to complete the Exam. When the exam is over, give only
one set of answers per team to the proctor. Multiple solutions to the same problem
will invalidate each other.
Elegance of solutions may affect score and may be used to break ties.
All calculators, cell phones, music players, and other electronic devices should be
put away in backpacks, purses, pockets, etc. Leaving early or otherwise disrupting
other contestants may be cause for disqualification.
Freshman-Sophomore Team Medley
Lee Webb Math Field Day 2010
1. Suppose ABCD is a square. Rays AB, BC , CD, DA divide the plane into five
regions (the square itself and four unbounded regions outside the square). With
4 colors, how many ways can the regions be painted so that no adjacent regions
have the same color (note: not all colors must be used in each of the colorings).
1 1 1 1
   .
a b c 3
2. Find three odd integers, a, b, c such that 0  a  b  c and
3. Two circles are situated so that the center of each circle is on the other circle.
What is the ratio of the area common to both circles to the total area covered by
both circles?
4. Adam, Brian, Cathy, Daniel, and Ella are playing a game for which they need
to break into two groups. How many ways can they do this? Assume a group
must have at least one person and not all of them have to be in a group (i.e. 0 or
1 or 2 or 3 of them may be spectators).
5. Suppose ABC is an equilateral triangle and that P is a point in the same plane
such that PAB, PBC, and PCA are all isosceles triangles. How many such
points P are there?
6. A pool table is has length 12 feet and width 8 feet. A ball is on the side of one
of the long the edges, 3 feet away from a corner. The ball is struck so that it
hits each of the other edges exactly once and then comes to rest at its original
position. At each edge its angle of incidence equals its angle of reflection.
How far did the ball travel?
7. One of the following numbers is a perfect square. Find it.
a. 8344572651 b. 7955896032 c. 1695032253 d . 4906358264
f . 5729581636 g. 3213046377 h. 2032918848 i. 2973562479
e. 1782570645
j. 3567659100
8. Suppose the three faces of a rectangular box that meet at one corner of the box
have areas of 48, 72, and 96 square units. What is the volume of the box?
Freshman-Sophomore Team Medley
Lee Webb Math Field Day 2010