38 th Annual
Lee Webb Math Field Day

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.

2009
Math Bowl
Varsity

Practice Problem – 10 seconds


Problem 1.1 – 30 seconds
Find the ordered triple that satisfies the system

 x x y y
2
2 z z

4
0
 x y

0
FOA: ( a,b,c )

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
  x
2
 
3 1

Problem 1.4 – 30 seconds
Arc cos(1/ )
.
e

Arc
Answer in radians.

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:

Problem 1.6 – 15 seconds sec
csc x x

Problem 1.7 – 60 seconds
The x-y, y-z, and z-x planes cut the sphere x
2  y
2  z
2 
36 into 8 parts. What is the volume of one of these parts?

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

3 2 i

3

Problem 1.10 – 45 seconds
Consider the sequence of digits
1234567891011121314...
What is the 100 th digit?

Problem 1.11 – 30 seconds
y
log
5 y
 log
5
 y 4

1

Problem 1.12 – 30 seconds
i i

Problem 2.1 – 15 seconds
e

Problem 2.2 – 30 seconds
23 30 '36".

Problem 2.3 – 30 seconds
   
Find .
b

Problem 2.4 – 30 seconds
log
3
9

Problem 2.5 – 15 seconds
sec


2 x

Problem 2.6 – 30 seconds
For the following parabola, how far is the focus from the vertex?
y
  x
2

Problem 2.7 – 60 seconds
k: n k 

5 n

5040

Problem 2.8 – 15 seconds

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
314 n


3


314 m


3 m



Problem 2.12 – 45 seconds
Let ( )
 
[ ]
Put the following in increasing order

FOA: a,b,c,d (e.g)

Practice Problem – 30 seconds
2

2

1
2
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 x
 such that the expression
x
 x
2 is minimal.

Problem 3.3 – 60 seconds
n
10 

2


 n
2 n
2

1



FOA: fraction in lowest terms

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:



3
10
2 x



1


 

 x
10

2
3




Problem 3.7 – 60 seconds
x
2  y
2 
25

Problem 3.8 – 30 seconds
The sum of the infinite series
1
1 1 1

1

4 9 16 25 is equal to for what polynomial ?

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
 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.

Problem 4.1 – 60 seconds
Find the first five digits after the decimal point of the following rational number:
1
7

1
15

1
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

FOA: 1 st , 2 nd , or 3 rd etc.?

Problem 4.7 – 45 seconds

0
 x
2 dx

1

Problem 4.8 – 45 seconds number such that

2

1 0.
Find

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
22 / 7 is the best rational
 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.

Problem 4.12 – 60 seconds
Let be the number of points in the 1 st quadrant with integer coordinates whose distance
Determine lim x
 x
2