39th Annual
Lee Webb Math Field Day
March 13, 2010
Varsity Math Bowl
Before We Begin:
• Please turn off all cell phones while
Math Bowl is in progress.
• The students participating in Rounds 1
& 2 will act as checkers for one another,
as will the students participating in
Rounds 3 & 4.
• There is to be no talking among the
students on stage once the round has
begun.
Answers that are turned in by the checkers
are examined at the scorekeepers’ table.
An answer that is incorrect or in
unacceptable form will be subject to a
penalty. Points will be deducted from the
team score according to how many points
would have been received if the answer
were correct (5 points will be deducted for
an incorrect first place answer, 3 for
second, etc.).
• Correct solutions not placed in the given
answer space are not correct answers!
• Rationalize all denominators.
• Reduce all fractions. Do not leave
fractions as complex fractions.
• FOA stands for “form of answer”. This will
appear at the bottom of some questions.
Your answer should be written in this form.
2010
Math Bowl
Varsity
Round 1
Practice Problem – 10 seconds
What is the area of a
circle of radius  ?
Problem 1.1 – 45 seconds
Which of these points
A(-3,-1), B(-1,3),
C(0,3), D(1,3), E(2,5)
is closest to the line
2y=x+4?
Problem 1.2 – 15 seconds
Simplify:
log5 25  log 25 5
Problem 1.3 – 30 seconds
A hexagonal prism
has how many
edges?
Problem 1.4 – 60 seconds
Patti, Qiu, and Randy each have a
parcel to mail. The ratio of the
weights of Patti’s and Qiu’s parcels is
11:6. The ratio of Randy’s to Qiu’s is
4:3. Randy’s parcel weighs 7.2 lb.
What is the weight, in pounds, of
Patti’s parcel?
.
Problem 1.5 – 30 seconds
If you are driving at 30 mph
(=44 ft/s) and texting at 2
chars/s, how many feet will
you travel while typing a 10
character message?
disclaimer
Please note: the math
field day staff strongly
discourages texting
and driving
Problem 1.6 – 30 seconds
What is the prime factorization
of the geometric mean of the
following numbers:
2 5 7
2 3
3
2 35  7
2
2 3 57
3
2
2
Problem 1.7 – 45 seconds
The x=y, y=z, and z=x planes
cut the sphere
x  y  z  36
2
2
2
into how many parts?
Problem 1.8 – 30 seconds
What is the volume of
each of the parts
described in the
previous problem?
Problem 1.9 – 45 seconds
Multiply out the following
product
( z  i )( z  i )( z  2i )( z  2i )
Problem 1.10 – 60 seconds
A square of side length 1 has
equilateral triangles attached to
the outside of each side. The
total enclosed area can be written
in the form (a  b c ) / d where
a,b,c,d are all relatively prime
natural numbers. Find the sum
a+b+c+d.
Problem 1.11 – 60 seconds
Solve for y:
x 1 y 1

x 1 y 1
Problem 1.12 – 30 seconds
Suppose 2 checker pieces are
placed randomly on a standard
8x8 checker board. What is the
probability the 2 pieces are not in
the same row or column? Answer
as a fraction in lowest terms.
Round 2
Problem 2.1 – 30 seconds
A snowboarder leaves the
half–pipe with a vertical
speed of 48 ft/s. For how
many seconds will she be
above her take-off point?
Problem 2.2 – 30 seconds
A map is drawn with a
1000:1 scale. On the
map a certain lot is 1.44
square inches in area.
How many square feet in
area is the actual lot?
Problem 2.3 – 30 seconds
Let g  x   x
Simplify
2
 3.
g a  b  g a
b
.
Problem 2.4 – 30 seconds
Simplify
ln 4ln9
e
Problem 2.5 – 30 seconds
Find a
simplified
expression
for sin 2
x

2
Problem 2.6 – 15 seconds
The locus of points that
have a constant
difference in distance
from two given points is
a _____________.
Problem 2.7 – 45 seconds
Expand:
( x  1)( x  1)( x  1)( x  1)( x  1)
2
4
8
Problem 2.8 – 45 seconds
3x  2
Suppose f ( x) 
5x  3
1
The domain of f ( x) is all real
numbers except_____.
Problem 2.9 – 15 seconds
An icosahedron has
how many faces?
Problem 2.10 – 30 seconds
Ten players entered a
tournament. Each player played
4 matches (each match was
between 2 players). How many
matches were played?
Problem 2.11 – 30 seconds
If the following is expanded,
how many digits will it have?
10
20
Problem 2.12 – 45 seconds
1 3 3 1 is a row of Pascal’s
triangle. What are the first
three entries of the first row
after this one that has only odd
entries?
Round 3
Practice Problem – 20 seconds
Simplify
1
log 2 16  log 2 4  log 2
32
Problem 3.1 – 30 seconds
In a circle, chord AB has
length 9. Chord CD
intersects AB at E so that
AE=3 and CE=2. What is
the length of DE?
Problem 3.2 – 45 seconds
How many values of x are
there in the interval (0, 2 )
that satisfy the following
equation?
2
(cos x  sin x)  1
Problem 3.3 – 60 seconds
Find all solutions to
||| x | 2 | 2 | 2
Problem 3.4 – 60 seconds
A regular
dodecahedron has
how many edges?
Problem 3.5 – 60 seconds
What is the least common
multiple of
1,2,3,4,5,6,7,8,9,10?
Problem 3.6 – 45 seconds
The main diagonal of a
cube is 18 inches. What is
the area of one face? (in
square inches)
Problem 3.7 – 45 seconds
6 is a perfect number
because it equals the sum
of its proper divisors. What
is the next smallest perfect
number?
Problem 3.8 – 45 seconds
Write
0.1232323232323…
as a simplified fraction.
Problem 3.9 – 45 seconds
Given that x   / 6
simplify
cos x sin x tan x


sec x csc x cot x
Problem 3.10 – 60 seconds
Joey has typed four letters and
four envelopes. But then Mary
put them in the envelopes
randomly. What is probability that
no letter is in the correct
envelope?
Answer in reduced fraction form.
Problem 3.11 – 60 seconds
A round cake is 1 foot in diameter
and 3 inches high. A slice equal
to ¼ of the cake has been cut
away. Find the exposed surface
area of the cake. (i.e. don’t count
the surface that is on the plate).
Answer in sq. in and in terms of
Problem 3.12 – 60 seconds
 is a function such that  (1)=1, (p)=-1
for all primes p, and (ab)= (a)
(b) if a
and b have no common factors greater
than 1 and (n)=0 if n is divisible by any
square greater than 1. What is the
smallest non-prime n such that (n)= -1.
Round 4
Problem 4.1 – 45 seconds
What is the maximum
number of acute angles a
convex decagon can have?
Problem 4.2 – 45 seconds
Seven cubes are the same size.
Six are glued so that they exactly
cover the faces of the last one.
How many faces are exposed on
the resulting arrangement?
Problem 4.3 – 60 seconds
A triangle has
vertices at (2,11),
(4,1), and (6,4).
What is its area?
Problem 4.4 – 45 seconds
Laila and Darnell begin a
chess game. How many
possible legal combinations
are there for their first two
moves (i.e. one move
each)?
Problem 4.5 – 60 seconds
Car A costs $20,000 and gets 30
mpg. Car B costs $21,000 and
gets 40 mpg. If you drive 12,000
mi/yr, and gas costs $2.00 per
gal, after how many years will car
B be a better bargain?
Problem 4.6 – 45 seconds
Suppose
cos(3t ) cos(2t )  sin(3t )sin(2t )
 k cos(t )
Find k in simplest form.
Problem 4.7 – 45 seconds
Simplify:
d
(u sin x)
du
Problem 4.8 – 60 seconds
Simplify:
 / arctan(2)
 1  2i 


 5 
Problem 4.9 – 60 seconds
In the interval (0, 2 ) , how
many solutions are there to
the equation:
cot(2 x)   x / 2  3
Problem 4.10 – 60 seconds

is a function such that  (1)=1,
 (p)= -1 for all primes p, and  (ab)=
 (a)(b) if a and b have no common
factor greater than 1 and (n)=0 if n is
divisible by any square greater than
10
1. Evaluate
  (n)
n 1
Problem 4.11 – 45 seconds
In number theory  ( x ) is the number
of primes that are less than x.
Evaluate
 (50)   (35)
Problem 4.12 – 60 seconds
Two positive numbers have
difference and quotient equal to
5. Find the larger of the two
numbers.