Florida Association of Mu Alpha Theta’s
State Convention 2010 Interschool
General Instructions
This is a chapter-wide team test. Anyone involved with your chapter may collaborate on this
test. Only one answer sheet should be submitted per chapter. Ties will be broken by the sudden
death method. You may use calculators, computers, or any other resource materials.
All submitted answers should be complete and exact. Approximations will not be accepted
unless otherwise specified. Fractional answers or parts thereof should be left as improper
fractions with relatively prime numerator and denominator.
Every question is worth one point unless otherwise noted. If the question is broken into parts
(labeled a, b, c, etc.), then each part will be worth a corresponding fraction of the question’s
value.
This test is due on Friday, 4/16, at 9:30 p.m. at the convention registration desk.
Good luck and enjoy!
The real numbers are totally ordered. That is, for any two real numbers a and b, we have that
either a < b or b < a. The complex numbers, however, have no such ordering. Suppose that I
impose an order on them called the “dictionary order”. The order is defined as follows:
a + bi < c + di if 1) a < c OR 2) b < d when a = c .
(For instance, 1 + 5i < 3 + 2i and 2 + 8i < 2 + 10i.)
Finally, a transformation f is called “order-preserving” if for all x and y in an ordered set
with x < y, then f(x) < f(y).
1. Which of the following transformations are order preserving on the complex numbers?
(If a transformation is order preserving, answer “OP”. If not, give a counterexample. For this
problem only, a correct answer to each part is worth 1 point, as opposed to 1/5 of a point.)
a. f(z) = z
b. f(z) = |z|
c. f(z) = z 2
d. f(z) = Re(z)
e. f(z) = Im(z)
2. Assuming that temperature is a continuous function of position on Earth, there are always
exist two diametrically opposite points on the equator such that the temperatures at those
points are equal. This can be proved using which famous theorem? (Hint: It's one of the big
three “value” theorems learned in calculus.)
3. Prior to 2006, there had been 91 presentations of the World Series, all of which crowned a
winner after a team won four out of seven games. Of the 91 World Series, 17 of them lasted
four games, 20 lasted five games, 21 lasted six games, and 33 lasted all seven games.
Assuming that X is a random variable denoting the length of the World Series, basic
probability tell us that the distribution of X is
x

1


4
x

4 x

4
4


P
(
X

x
)

p
(
1

p
)

p
(
1

p
)
,


3




where p is the probability that the weaker team wins (thus, 0 < p < 0.5). Based on the data,
the average number of games played is roughly 5.77. Compute the value of p to four decimal
places such that E(X ) = 5.77.
4. Determine y (x) , given that
dy
(xy
1
)2. The graph of y (x) passes through the origin.
dx
dy
y y2x
3
5. Determine y (x) , given that 2
. Again, y (x) passes through the origin.
dx
6. Can you KenKen?
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8. Suppose we have two estimators ˆ1 , ˆ2 of a parameter  with the following properties:

  

^ ^
^
^
^^
^





 

E

E

Var

4
Var

8
Let

a

(
1

a
)
1
2
1
2
3
1
2













 

a. What is Var(ˆ3 ) if a  0.5 ? Assume ˆ1 , ˆ2 are independent.
b. What value of a minimizes Var(ˆ ) ?
3
9. Find the smallest positive integer whose first digit is 1 and which has the property that if
this digit is transferred to the end of the number, the resulting number is 3 times as large as
the original. For example, 139 would be transformed to 391 which is not quite 3 times as
large.
10. Chris's score y on a certain test varies as y = ax + bz + c, where x denotes the hours spent
studying and z denotes the hours slept. Chris scores a 3 after studying for 1 hour and sleeping
for 2. He scores 2 after studying for 5 hours and sleeping for 6. Finally, he scores 4 after
studying for 2 hours and sleeping for 1. What score will Chris earn if he studies for 6 hours
and sleeps for 5?
x1 1 5 2 9


11. Given the data x2 2 6 1 3, use least squares regression to find the plane of best

y 3 2 4 1

bx
c. What value of y is predicted when x1  177 and x2  354? (Round
fit: yax
1
2
your answer to two decimal places)
12. Find the radius of the circle below.
13. Find the shaded area below. Equaliteral triangle
ADE and square ABCD both have side length 1.
14. Find the number of positive integral solutions ( x, y ) with x  y such that
1 1 1
 
.
x y 100
15. Three three-digit numbers a < b < c use each of the non-zero digits exactly once and are
in the ratio 1:3:5. What is the value of b?
16. Evaluate the expression
. Give only correct answers.
7
7
7
7

...
17. Alice, Bob, and Cathy play a game with money. When a turn ends, the loser doubles the
amount of money the other two have by giving up their own. Each player loses exactly once
and they all have $48. How much money did each player have to start with if Alice had the
least amount to start and Cathy had the most to start?
18. Three pegs are placed at random into a 4x4 grid as shown below. What is the probability
that at least two markers end up in the same row or column?
19. What is the largest power of 2010 that divides 2010! ?
20. A rousing game of Humans vs. Zombies has broken out at the University of Florida.
Spencer decides to model the number of zombies with a logistics curve of the following form:
A
y
C t ,
1Be
where A, B, and C are positive real numbers and t is the number of days since the game
began. It is known that 1080 people are playing (and thus there should be 1080 zombies by
the end of the game). It is also known that at the beginning of the game there are 2 zombies.
Finally, there are 160 zombies 24 hours since the game's beginning. How many zombies does
the model predict 48 hours after the game's beginning? (Round to the nearest whole zombie.)
21. The lifetime of a machine part has a continuous distribution on the interval (0, 20) with
3
probability density function proportional to (5  x) . What is the probability that the
machine part has a lifetime less than 10? (Round your answer to three decimal places)
22. An insurance company finds that all policyholders are twice as likely to file one claim
than three claims. If the amount of claims follows a Poisson distribution, what is the mean
number of claims filed?
23. An input-output matrix can describe the economic inputs required to produce certain
outputs. Consider the matrix below, where A, L, and C represent agriculture, labor, and
capital respectively:

A

L

C
A L C
.3 .2 .3
.1 .4 .1

.3 .2 .3
The
first
row
tells
us
that
in
order to
produce
1
unit
of
agricultur
e,
we
need
.3
units
of
agricultur
e,
.2
units
of
labor,
and
.3
units
of
capital.
a. How many units of labor are needed to produce 30 units of capital?
b. A company demands 40 units of agriculture, 60 units of labor, and 50 units of capital. How
much of each commodity must be generated in order to meet this demand?
2n 3n
24. Evaluate:  n
5
n1


25. Does the series
ln(n)
converge? Why or why not?
2
n1 n

t2
t,y
t3
4
t intersects itself. Take t to be in the domain of
26. The parametric graph x
real numbers. For what positive value of t is the intersection attained?
27. To whom is the following quote attributed?
“Mathematics consists in proving the most obvious thing in the least obvious way.”