Junior/Senior Math Bowl (2005)
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Transcript Junior/Senior Math Bowl (2005)
Rules of Engagement
• 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 team
members once the round has begun. Any
pairs caught talking, even between questions,
will be ejected from the competition.
• Checkers are more than welcome to take
a chance that the answer their teammate
gave is also correct, though it doesn’t
appear as a possible answer. However,
keep in mind that if the answer is in an
unacceptable form or otherwise incorrect,
points will be deducted from the team
score according to how many points would
have been received if the answer was
correct. (5 points will be deducted for an
incorrect first place answer.)
• Checkers, please remember that
multiplication and addition are
commutative.
• Correct solutions not placed in the given
answer space are not correct answers!
• Rationalize all denominators.
• Reduce all fractions, unless the question
says otherwise. Do not leave fractions as
complex fractions.
• Use only log base 10 or natural log.
• It is only necessary to write an equation
when asked for an equation or a function.
• Answers of the form a b are acceptable,
unless both answers are rational.
• Use interval notation for domains and/or
ranges.
• When units are given in the problem, units
are required in the answer.
• Good luck, and most
importantly, have fun!
2005
Math Bowl
Varsity
Round 1
Practice Problem – 15 seconds
Let
x/ 2
f x 4
.
Find
f 1 .
Problem 1.1 – 25 seconds
Find the ordered
triple that satisfies
the system
x y 2z 4
x y 2z 0
x y
0
Problem 1.2 – 25 seconds
Several logs are stored in a
pile with 20 logs on the
bottom layer, 19 on the
second layer, 18 on the third,
and so on. If the top layer
has one log, how many logs
are in the pile?
Problem 1.3 – 30 seconds
Let f x 5 3x and
g x 3x 1 .
Find the
polynomial f g x .
2
Problem 1.4 – 20 seconds
For the sets A 1,3,5,6,8 ,
B 2,3,6,7 and C 6,8,9,
find B C A .
.
Problem 1.5 – 20 seconds
1, 2
If the point
is
on the graph of
2
f x ax 4, find a.
Problem 1.6 – 15 seconds
Write
sec x
csc x
as a
simple trigonometric
function.
Problem 1.7 – 25 seconds
Determine the
domain of the
function
1
1
1
f x
x x 1 x 2
Problem 1.8 – 15 seconds
Find the
length
of x.
25
x
30
Problem 1.9 – 30 seconds
Find the area of the
parallelogram in the plane
with vertices
A 1, 0 , B 0,1 ,
C 1, 0 , and D 0, 1 .
Problem 1.10 – 25 seconds
Solve for y:
log5 y log5 y 4 1
Problem 1.11 – 30 seconds
Find the arc length
corresponding to a
3
central angle of 14 on
a circle with radius 7 cm.
Problem 1.12 – 20 seconds
Calculate
sin 30 cos 60 sin 60 cos 30
Round 2
Practice Problem – 25 seconds
Simplify
1
log 2 16 log 2 4 log 2
32
Problem 2.1 – 15 seconds
Simplify
3ln 2 x 1
e
Problem 2.2 – 20 seconds
Simplify
m 2 2 m 24
ln e
2
m 36
completely.
Problem 2.3 – 25 seconds
Let
Find
g x 2x 3
.
g a b g a
b
.
Problem 2.4 – 15 seconds
Find the exact value
of
log 3 9
.
Problem 2.5 – 15 seconds
What are the next
two terms in the
sequence
A, c, E, g, …
Problem 2.6 – 35 seconds
Find the center of
the ellipse
4 x y 16 x 6 y 21 0
2
2
Problem 2.7 – 20 seconds
Find the roots of
x 4 x 9 x 36 0
3
2
Problem 2.8 – 25 seconds
If log a 4 .6021, log a 7 .8451,
and log a 9 .9542, find
63
.
log a
4
Problem 2.9 – 20 seconds
Find the next term
of the sequence
20, 17, 13, 8, …
Problem 2.10 – 15 seconds
According to the rational root
theorem, what are the
possible rational roots of
x 4 x 3x x 4 x 3 0?
6
5
4
2
Problem 2.11 – 25 seconds
If z 4 3i ,
find z .
Problem 2.12 – 35 seconds
For what
interval(s) of x
x
y
does 16 9 1 produce
real y values?
2
2
Round 3
Problem 3.1 – 30 seconds
The area of an equilateral
triangle varies directly
with the square of the
length of a side. Find the
constant of
proportionality.
Problem 3.2 – 30 seconds
Solve
tan x tan x 2 0
2
in the interval
, .
2 2
Problem 3.3 – 20 seconds
Calculate
2 3i 6 2i
Problem 3.4 – 25 seconds
Find the
length of
CD in
terms of x.
A
C
30
D
x
45
45
B
Problem 3.5 – 20 seconds
Evaluate
100
6
1
Problem 3.6 – 30 seconds
Find the inverse of
1 1
1 0
Problem 3.7 – 20 seconds
Find the polar
equation for the
Cartesian equation
x y 7
2
2
Problem 3.8 – 30 seconds
Evaluate tan 3 on
the interval
,
2 2 .
1
Problem 3.9 – 40 seconds
Let
1 2
A
1 1
5 1
B
3 0
and
.
Find
det BA
.
Problem 3.10 – 30 seconds
Find the coefficient of
3
x y
4
in the expansion of
7
x y .
Problem 3.11 – 25 seconds
How many times can
the face 5 be
expected to occur in a
sequence of 2016
throws of a fair die?
Problem 3.12 – 25 seconds
If u 3, 2 ,
and v 1, 3 ,
find u v .
Round 4
Problem 4.1 – 20 seconds
Find
x2
lim 2
x2 x 4
.
Problem 4.2 – 35 seconds
Expand
2x 5
2
x 5x 6
into partial fractions.
Problem 4.3 – 20 seconds
12
4
1
3
4
Let r .
Find r ' , with only
positive exponents
in the answer.
Problem 4.4 – 25 seconds
Find the sum of the
first five multiples
of 4.
Problem 4.5 – 20 seconds
A couple is planning their
wedding. They can select
from 2 different chapels, 4
soloists, 3 organists, and 2
ministers. How many
different wedding
arrangements are possible?
Problem 4.6 – 25 seconds
Find the distance
between the
points P 2, 4,3 and
Q 4,7, 3 .
Problem 4.7 – 15 seconds
If P A .3 and
P B A .6 ,
find P A B .
Problem 4.8 – 35 seconds
Find
lim 1 cos csc x
x
6
Problem 4.9 – 35 seconds
Find c in the interval
1
2 , 2 such that
1
f 2 f
2
f 'c
1
2
2
1
f x x
x
if
.
Problem 4.10 – 30 seconds
Evaluate
2
x
5
dx
2
0
Problem 4.11 – 20 seconds
Find the slope of the
tangent line to the
2
graph of f x x 2 ,
at the point 1,3 .
Problem 4.12 – 45 seconds
A gum manufacturer
randomly puts a coupon in
1 of every 5 packages.
What is the probability of
getting at least one coupon
if 4 packages are
purchased?