38th Annual Lee Webb Math Field Day

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.

2009Math Bowl

Varsity

Round 1

Practice Problem – 10 seconds

What is the area of a circle of radius ?

Problem 1.1 – 30 seconds

Find the ordered triple that satisfies the system

FOA: (a,b,c)

2 4

2 0

0

x y z

x y z

x y

Problem 1.2 – 30 seconds

Several cannon balls are stacked in six layers, so that there is a 6x6 square on the bottom, with a 5x5 layer above that, etc. How many cannon balls are there?

Problem 1.3 – 30 seconds

Let and

.

Evaluate .

214 3f x x

3 1g x x

3f g

Problem 1.4 – 30 seconds

Determine

Answer in radians.

.

cos(1/ ) sin(1/ )Arc e Arc e

Problem 1.5 – 75 seconds

Square ABCD has area 16. E and F are on sides BC and CD such that AE and AF trisect the corner at A. What is the area of quadrilateral AECF?

FOA: a b c

Problem 1.6 – 15 seconds

Write as a

simple trigonometric

function.

sec

csc

x

x

Problem 1.7 – 60 seconds

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

into 8 parts. What is the volume of one of these parts?

2 2 2 36x y z

Problem 1.8 – 45 seconds

A CD player changes the speed of the disc in order to read the encoded bits at the same rate. If the disc spins at 250 rpm for a track that is 60 mm from the center, how many rpm are required for another track that is 20 mm from the center?

Problem 1.9 – 45 seconds

Find the real part of

33 2i

Problem 1.10 – 45 seconds

Consider the sequence of digits

What is the 100th digit?

1234567891011121314...

Problem 1.11 – 30 seconds

Solve for y:

5 5log log 4 1y y

Problem 1.12 – 30 seconds

What is the principal value of

ii

Round 2

Problem 2.1 – 15 seconds

Simplify

3ln 1/ xe

Problem 2.2 – 30 seconds

An angle is reported to be

In decimal notation, this is how many degrees?

23 30 '36".

Problem 2.3 – 30 seconds

Let .

Find .

2 3g x x

g a b g a

b

Problem 2.4 – 30 seconds

Find the exact value

of .3

log 9

Problem 2.5 – 15 seconds

Find an expression

for sec x

2

Problem 2.6 – 30 seconds

For the following parabola, how far is the focus from the vertex?

2y x

Problem 2.7 – 60 seconds

Solve for k:

5

5040k

n

n

Problem 2.8 – 15 seconds

Fill in the blank:

The orthocenter of a triangle is the

intersection of its ___________ .

Problem 2.9 – 60 seconds

Jane and Carlos and their guests had pie for dessert. They used a special pie-cutter that cuts central angles of any integer degree. Everyone got exactly one piece of pie of exactly the same size. How many possibilities are there for the number of guests (do not count the 0 guest case)?

Problem 2.10 – 75 seconds

Joey clothes-pinned a card on the front wheel of his bicycle. The card clicks every time a spoke strikes it. The wheel is 24” in diameter and has 32 spokes. If Joey rides 11 ft per second, how many clicks are there per second?

Round off to the nearest integer.

Problem 2.11 – 30 seconds

Simplify: 314314

3 3

log( ) logn m

n m

Problem 2.12 – 45 seconds

Let . Put the following in increasing order

FOA: a,b,c,d (e.g)

( ) [ ]f x x x

a) ( .2) b) (1)

c) (3 / 2) d) ( )

f f

f f

Round 3

Practice Problem – 30 seconds

Simplify

2 2 2

1log 16 log 4 log

32

Problem 3.1 – 45 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

Find the value of such that the expression

is minimal.

2(cos sin )x x

(0, )x

Problem 3.3 – 60 seconds

Calculate

FOA: fraction in lowest terms

210

22 1n

n

n

Problem 3.4 – 60 seconds

A polyhedron has 24 vertices. Two regular hexagons and one square meet at each vertex. In all there are 8 hexagons. How many squares are there?

Problem 3.5 – 30 seconds

In the polyhedron of the previous problem, there are 24 vertices, 8 hexagonal faces, and 6 square faces. How many edges does the polyhedron have?

Problem 3.6 – 30 seconds

Solve for x:1

3 2 2

10 10 3

x

x

Problem 3.7 – 60 seconds

How many points with integer coordinates satisfy

2 2 25x y

Problem 3.8 – 30 seconds

The sum of the infinite series

is equal to for what polynomial ?

1 1 1 11

4 9 16 25

( )f ( )f x

Problem 3.9 – 60 seconds

Zacky’s Pizzeria offers a choice of 3 different sizes, 2 different kinds of crusts, and 10 different kinds of toppings. How many different pizzas can be ordered (with at least one topping)?

Problem 3.10 – 30 seconds

A rhombus has side length 10 and area 50. What is the measure, in radians, of its smallest angle?

Problem 3.11 – 60 seconds

The light in a lighthouse makes 10 revolutions per minute. How fast does the light flash by on the side of a boat that is 600 feet directly offshore? Answer in feet per second in terms of

Problem 3.12 – 60 seconds

Suppose T1, T2, T3, … is an infinite sequence of similar triangles. The perimeter of each triangle is 80% as much as the previous triangle. If the area of the first triangle is 63, find the sum of the areas of all the triangles.

Round 4

Problem 4.1 – 60 seconds

Find the first five digits after the decimal point of the following rational number: 1

17

115

1

Problem 4.2 – 45 seconds

A gum manufacturer randomly puts a coupon in 1 of every 4 packages. What is the probability of getting at least one coupon if 4 packages are purchased?

Problem 4.3 – 60 seconds

A triangle has vertices at (3,4), (6,9), and (11,2). What is its area?

Problem 4.4 – 45 seconds

A rectangle of length 36 and height 6 is centered at the origin. What is the equation of the circle that

goes through all the vertices of the rectangle?

Problem 4.5 – 30 seconds

If you draw two cards randomly from a standard deck, what is the probability that you get two of a kind (2 kings or 2 sevens, etc)?

Problem 4.6 – 15 seconds

Which letter of the Greek alphabet is

? FOA: 1st , 2nd, or 3rd etc.?

Problem 4.7 – 45 seconds

Evaluate:

20 1

dx

x

Problem 4.8 – 45 seconds

Let be a complex number such that

Find

2 1 0.

Problem 4.9 – 60 seconds

It takes 7 days for 5 chickens to lay 2 dozen eggs. How many days will it take 21 chickens to lay 30 dozen eggs?

Problem 4.10 – 30 seconds

Randy and forty-four other people are situated in a circle. Randy passes a soccer ball to the twelfth person on his right. This is repeated until the ball comes back to Randy. How many people do not touch the ball?

Problem 4.11 – 60 seconds

is the best rational approximation to that has denominator less than 10. It is accurate to 2 places. There is another approximation with denominator 113 that is accurate to 6 places. Find its numerator.

22 / 7

Problem 4.12 – 60 seconds

Let be the number of points in the 1st quadrant with integer coordinates whose distance back to the origin is less than . Determine

( )f x

x

2

( )limx

f x

x