CALIFORNIA STATE UNIVERSITY,
BAKERSFIELD
Lee Webb Math Field Day 2010
Team Medley, Junior-Senior Level
Each correct answer is worth ten points. Justification of answers is required
for full credit. Partial credit may be given. An unanswered question is
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. Two different answers to
the same problem will invalidate both.
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.
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 5 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).
2.
Suppose sin   cos  
3.
Find all integer values of x such that x 2  7 x  4 is a perfect square.
4.
Find three odd integers, a, b, c such that a  b  c and
5.
Suppose a, b, c are three distinct real numbers such that
7
. Find sin  cos and sin 3   cos3  .
3
1 1 1 1
   .
a b c 3
a
b
c


. What
bc ca ab
are the possible values for these fractions?
6.
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?
7.
Let S={1,2,3,4,5,6,7,8,9,10} and that you must choose subsets A,B,C,D,E such that
A  B  C  D  E   . How many possible choices are there for A,B,C,D,E (note
some subsets may be empty)? You may leave your answer as a product of smaller
numbers.
8.
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?