Mu Alpha Theta National Convention: Hawaii, 2005
State Bowl – Mu Division
Round 1
Round 1 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
State Bowl – Mu Division

3
(2 x  1) 2 dx
1.
Evaluate:
2.
In an arithmetic sequence, the 950th term is 1055 and the 1055th term is 950. What is the
2005th term?
3.
The product of a set of distinct positive integers is 84. What is the smallest possible sum of
these integers?
4.
Evaluate: lim
0
h 0
7(h  1)5  7
h
Round 2 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
State Bowl – Mu Division
Round 2
Round 3 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
State Bowl – Mu Division
5.
 7 3   2 1
What is the trace of the matrix given by 

?
 3 2 5 3
6.
Evaluate:
7.
If the equation of the line tangent to the curve y  x 2  3x at (6, 54) is of the form
y  ax  b where a and b are integers, find the value of a + b.
8.
If each person shakes hands exactly once with every other person in a group of twenty-five
people, how many handshakes will transpire?

6
4
| x  3 | dx
Round 4 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
State Bowl – Mu Division
Round 3
Round 5 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
State Bowl – Mu Division
9.
If the ratio of the sum of the n smallest positive perfect cubes to the sum of the n smallest
natural numbers is 120, find the value of n.
10. Evaluate: lim
x 0
x  sin x
x3
11. What is the area, in square meters, of an isosceles triangle with a side measuring 12 meters
opposite the vertex angle measuring 120?
12. What is the average value of f ( x)  3x( x  1) over 1  x  3 ?
Round 6 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
State Bowl – Mu Division
Round 4
Round 7 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
State Bowl – Mu Division
13. Find the area of the region bounded by the polar curve r  16cos .
14. For 0  A 

, sin A 
2
common fraction.
3
. If f ( x)  sec2 x , evaluate f ( A) . Express your answer as a
5
15. Between 12 P.M. and 12 A.M., a strange watch runs for 60 minutes, then stops for 1 minute,
then runs for another 60 minutes, then stops for 2 minutes, and so on. When this watch is
finally at 12 A.M., how many minutes is it behind the true time?
16. Two fair, six-sided dice are rolled. What is the probability that either the values of the top
faces are equal or one is a multiple of the other?
Round 8 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
State Bowl – Mu Division
Round 5
Round 9 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
State Bowl – Mu Division
17. In a recent survey of 57 vegetarians, 31 liked broccoli, 17 liked cauliflower, and 18 liked
neither. How many liked both broccoli and cauliflower?
18. Find
dy
at the point (0, –1) if xy  cos x  y 2 .
dx
19. When graphed in the complex plane, how many of the seventeenth roots of 2 lie in the
fourth quadrant?
20. In an arithmetic sequence of functions of x, the fourteenth term is 14x + 39 and the twentyseventh term is 27x + 78. Let f ( x) be the sum of the first ten terms of this sequence. Find
the value of f (2005) .
Round 10 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
State Bowl – Mu Division
Round 6
Round 11 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
State Bowl – Mu Division
21. If f is a linear function and

12
8
f ( x) dx  1024 , evaluate

3
1
f (2 x  6)
dx .
4
22. In triangle ABC, A = (–1, –2), B = (5, 2), and the centroid of ABC is at (4, 5). What is the
distance from C to the origin?
(1)n (49 2 ) n

(36)n (2n)!
n 0

23. Evaluate:
24. Two roots of p( z )  3z 4  az 3  bz 2  cz  64 are 4 and 1  i . Find the value of a. (Note:
For this problem, i  1 and a, b, and c are real numbers.)
Round 12 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
State Bowl – Mu Division
Round 7
Round 13 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
State Bowl – Mu Division
25. If the geometric mean of all the positive integral factors of a positive number is 56, what is
the number?
26. Suppose that f (0)  3 and f ( x)  8 for all x. Find the largest possible value of f (2) .
27. What is the area, in square meters, of a triangle with side lengths 9, 10, and 17 meters?
d2 f
28. If f ( x  h)  f ( x)  10h  20 xh  3h , find the value of
at x = 2005.
dx 2
2
Round 14 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
State Bowl – Mu Division
Round 8
Round 15 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
State Bowl – Mu Division
29. Find the maximum value of f ( x) 
1 x
e (1  x) .
2
30. A number N is randomly picked from the set {, 2, 3, 4, 5}, where each number is
equally likely to be chosen. What is the expected value of

N
0
sin x dx ? Express your
answer as a decimal.
31. Function f is periodic with period 5. If two solutions to the equation f ( x)  2005 are 3 and
–4, what is the minimum number of other solutions that exist on the interval [0, 50]?
32. In a finite geometric sequence, the sum of all the terms except for the first is 136 and the
sum of all the terms except for the last is 8. What is the common ratio of this sequence?
Round 16 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
State Bowl – Mu Division
Round 9
Round 17 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
State Bowl – Mu Division


33. If 4 f ( x)  3 f   x   sin x for all x, what is the amplitude of the graph of y  f ( x) ?
2

34. If F ( x)  
cos x
0

d2F
sin u du , find the value of
at x  .
2
2
dx
35. Using two iterations of Newton’s Method, approximate a root of f ( x)  x 2 
initial guess value of x0 
1
. Express your answer as a common fraction.
2
36. If y  1  4log4 x and y  3  log2 x , find the value of xy.
Round 18 of 20
1
using an
2
Mu Alpha Theta National Convention: Hawaii, 2005
State Bowl – Mu Division
Round 10
Round 19 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
State Bowl – Mu Division
37. What is the sum of the squares of the eigenvalues of a 2  2 matrix with determinant 30 and
a trace equal to 11?
38. How many distinguishable ways can each of three congruent strips of cloth making up a
square blanket be colored one of four different colors? Assume that the blanket is a single
thickness of cloth the same color on each side, and that the blanket can be turned over.
39. Find the x-value that satisfies the conclusion of the Mean Value Theorem for derivatives for
the function f ( x)  3x 2  17 x  5 on the interval 1  x  7 .
40. If P( x)  x7 and Q( x)  x3  3x , evaluate

1
1
P(Q( x))  Q( P( x)) dx .
Round 20 of 20