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 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!

a b

2005Math Bowl

Varsity

Round 1

Practice Problem – 15 seconds

Let .

Find .

/ 24 xf x

1f

Problem 1.1 – 25 seconds

Find the ordered triple that satisfies

the system2 42 0

0

x y zx y zx y

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 and .

Find the polynomial .

25 3f x x

3 1g x x

f g x

Problem 1.4 – 20 seconds

For the sets , and ,

find ..

1,3,5,6,8A

2,3,6,7B 6,8,9C

B C A

Problem 1.5 – 20 seconds

If the point is on the graph of

, find a.

1,2

2 4f x ax

Problem 1.6 – 15 seconds

Write as a

simple trigonometric function.

seccscxx

Problem 1.7 – 25 seconds

Determine the domain of the

function 1 1 1

1 2f x

x x x

Problem 1.8 – 15 seconds

Find the length of x.

x25

30

Problem 1.9 – 30 seconds

Find the area of the parallelogram in the plane

with vertices

1,0 , 0,1 ,

1,0 , and 0, 1 .

A B

C D

Problem 1.10 – 25 seconds

Solve for y:

5 5log log 4 1y y

Problem 1.11 – 30 seconds

Find the arc length corresponding to a

central angle of on a circle with radius 7 cm.

314

Problem 1.12 – 20 seconds

Calculate

sin 30 cos 60 sin 60 cos30

Round 2

Practice Problem – 25 seconds

Simplify

2 2 21log 16 log 4 log32

Problem 2.1 – 15 seconds

Simplify

3ln 2 1xe

Problem 2.2 – 20 seconds

Simplify

completely.

2 2 24

2

ln36

m mem

Problem 2.3 – 25 seconds

Let .

Find .

2 3g x x

g a b g ab

Problem 2.4 – 15 seconds

Find the exact value

of .3

log 9

Problem 2.5 – 15 seconds

What are the next two terms in the

sequenceA, c, E, g, …

Problem 2.6 – 35 seconds

Find the center of the ellipse

2 24 16 6 21 0x y x y

Problem 2.7 – 20 seconds

Find the roots of

3 24 9 36 0x x x

Problem 2.8 – 25 seconds

If , , and , find

.

log 4 .6021a log 7 .8451a log 9 .9542a

63log4a

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

6 5 4 24 3 4 3 0?x x x x x

Problem 2.11 – 25 seconds

If ,find .

4 3z i

z

Problem 2.12 – 35 seconds

For whatinterval(s) of x

does producereal y values?

2 2

116 9x y

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

in the interval .

2tan tan 2 0x x

,2 2

Problem 3.3 – 20 seconds

Calculate

2 3 6 2i i

Problem 3.4 – 25 seconds

Find thelength of

CD in terms of x.

A

C

B

x D30

4545

Problem 3.5 – 20 seconds

Evaluate100

1

6

Problem 3.6 – 30 seconds

Find the inverse of1 1

1 0

Problem 3.7 – 20 seconds

Find the polar equation for the

Cartesian equation 2 2 7x y

Problem 3.8 – 30 seconds

Evaluate on the interval .

1tan 3

,2 2

Problem 3.9 – 40 seconds

Let and .

Find .

1 21 1

A

5 13 0

B

det BA

Problem 3.10 – 30 seconds

Find the coefficient of

in the expansion of .

3 4x y

7x 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 , and ,find .

3,2u

1, 3v

u v

Round 4

Problem 4.1 – 20 seconds

Find .22

2lim4x

xx

Problem 4.2 – 35 seconds

Expand

into partial fractions.

2

2 55 6x

x x

Problem 4.3 – 20 seconds

Let .Find , with only positive exponents

in the answer.

3 4

12 4 1r

'r

Problem 4.4 – 25 seconds

Find the sum of the first five multiples

of 4.

Problem 4.5 – 20 secondsA 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 and .

2, 4,3P

4,7, 3Q

Problem 4.7 – 15 seconds

If and ,

find .

.3P A

.6P B A

P A B

Problem 4.8 – 35 seconds

Find

6

lim 1 cos cscx

x

Problem 4.9 – 35 seconds

Find c in the interval such that

if .

1 ,22

12

2' 122

f ff c

1f x xx

Problem 4.10 – 30 seconds

Evaluate

0

22 5x dx

Problem 4.11 – 20 seconds

Find the slope of the tangent line to the

graph of ,at the point .

2 2f x x

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?