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.

2010Math 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:

5 25log 25 log 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:

3 2 22 3 5 7

2 5 7 22 3 5 7

32 3

Problem 1.7 – 45 seconds

The x=y, y=z, and z=x planes cut the sphere

into how many parts?

2 2 2 36x y z

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

( )( )( 2 )( 2 )z i z i z i z i

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 where a,b,c,d are all relatively prime natural numbers. Find the sum a+b+c+d.

( ) /a b c d

Problem 1.11 – 60 seconds

Solve for y:1 1

1 1

x y

x y

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 .

Simplify .

2 3g x x

g a b g a

b

Problem 2.4 – 30 seconds

Simplify

ln 4 ln9e

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:

2 4 8( 1)( 1)( 1)( 1)( 1)x x x x x

Problem 2.8 – 45 seconds

Suppose

The domain of is all real numbers except_____.

.

3 2( )

5 3

xf x

x

1( )f x

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?

1020

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

2 2 2

1log 16 log 4 log

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 that satisfy the following equation?

2(cos sin ) 1x x

(0,2 )

Problem 3.3 – 60 seconds

Find all solutions to

||| | 2 | 2 | 2x

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 simplify

/ 6x

cos sin tan

sec csc cot

x x x

x x 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

Find k in simplest form.

cos(3 )cos(2 ) sin(3 )sin(2 )

cos( )

t t t t

k t

Problem 4.7 – 45 seconds

Simplify:

( sin )du x

du

Problem 4.8 – 60 seconds

Simplify:

/ arctan(2)1 2

5

i

Problem 4.9 – 60 seconds

In the interval , how many solutions are there to the equation:

(0,2 )

cot(2 ) / 2 3x x

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 1. Evaluate

10

1

( )n

n

Problem 4.11 – 45 seconds

In number theory is the number of primes that are less than x. Evaluate

( )x

(50) (35)

Problem 4.12 – 60 seconds

Two positive numbers have difference and quotient equal to 5. Find the larger of the two numbers.