The Vanderbilt High School
Mathematics Competition
Saturday, April 1, 2006
Vanderbilt University
Nashville, Tennessee
Instructions
1.
2.
3.
4.
Do NOT open this test until you are told to do so.
You will have exactly 60 minutes to complete the test.
Use No. 2 pencil ONLY.
Calculators, books, notes, and any other forms of aid are STRICTLY
PROHIBITED. Any attempt to use said objects will result in immediate
disqualification and removal from the testing room.
5. Hats, sunglasses, and earphones may not be worn in the testing room.
6. Please turn off or silence all cell phones, beepers, and watches.
7. If you choose to leave the testing room, you must turn in your test and you will
not be allowed to return until after the testing period has expired.
8. The test will be scored as follows: 4 points for each correct answer, 0 points for
each question left blank, and -1 point for each question answered incorrectly.
9. Write and label your answers to the 3 tiebreaker questions in the blank space at
the top on the back of your Scantron form. The tiebreaker questions are worth
0.1 points each and incorrect tiebreaker answers will not deduct points from
your score.
10. NOTA denotes “None of the above”. Choose this answer if you believe that
none of the given choices are correct or if you believe there is an error in the
question.
1
1. What is the value of 1 
?
1
2
1
3
1
1
2
6 1
A.
B.
6 1
1
3  ...
4  37
7
C.
D.
4  37
7
E. NOTA
2. A is a point on a circle with radius x. B is a point on a diameter of the circle, 5 units
from the center. What is the distance from A to B, if the line AB is perpendicular to
the diameter?
A. x 2  25
B.
3. Find the value of
A. 4 6
4.
x 2  5x
C.
x 2  25 D. not enough information
E. NOTA
52 6  52 6
B. 2 2
C.
D.
2
3 3
2
E. NOTA
The eigenvalues of a square matrix A are defined as all numbers λ such that

det( A  I )  0 . An eigenvector x is associated with each eigenvalue λ and satisfies


the equation Ax  x . Which of the following is NOT an eigenvector for the matrix
7 3

 ?
3 7
 4
A.  
 4
  10 

B. 
  10 
  4
D.  
 4 
1
C.  
 1
E. NOTA
5. What is the area of the pentagon having vertices (6,2), (-8,1), (-3,8), (1,0), (4,10)?
A. 177
B. 88.5
C. 176
D. 88
E. NOTA
6. Math professors Jack Pessaci and Lilly Romley have come across a startling new
mathematical operation that they call “Tanlin”, denoted by ۩(n). Using the Tanlin
operation, one can easily compute the number of ways to write n as the sum of 10 nonnegative integers, counting order. For example, ۩(1) = 10. Unfortunately they did not release
their secret to anyone except their friend, Owen Irns, who forgot. Help them rediscover the
magic by computing ۩(50).
 60 
A.  
9
 59 
B.  
 10 
 60 
C.  
 10 
 59 
D.  
9
E. NOTA
7. Which of the following is greatest?
A. 624
B. 918
C. 119
D. e35
E. cos2(1078)
 
 2 
 3 
8. Compute the following: cos   cos
  cos 
7
 7 
 7 
A.
3
2
B.
1
2
C. 0
D. 
1
2
E. NOTA
9. How many ordered pairs of integers (X,Y) in [0,4] exist such that XY5 and XY9 are
both prime when converted to base 10?
A. 3
B. 4
C. 5
D. 6
E. 7
10. An infinite collection of lockers, numbered 1,2,3,4,… , all have their doors closed.
For successive numbers n = 1,2,3,4,5,… open all closed lockers and close all open
lockers that are numbered by multiples of n. Which of these lockers will eventually
remain open?
A. 9999
B. 10000
C. 10001
D. 10005
E. 10010
11. How many solutions does the equation cos(10 x)  x 2 have?
A. 0
B. 2
C. 4
D. 5
E. 6

12. Let  1 be 4x + 3y = 3,  2 be 8x = y.  1 is the line formed by reflecting  1 across the

line y = x.  2 is the line formed by reflecting  2 across the x-axis. Find the measure


of the acute angle between  1 and  2 .
 65 
 4 
 29 
 28 
 E. NOTA
 B. cos 1 
 C. cos 1 
 D. cos 1 
A. cos 1 

9
 65 
 5 65 
 5 65 


13. In the figure below (not drawn to scale), AC = 12, BD = 8, AB = 6, BC = 7, AD =10
and  A and  C are supplementary. Find DC.
B
A
D
C
29
26
13
C.
D.
E. NOTA
3
3
3
14. Which of the following is true of the relation h whose domain and range are both
over the set of all integers such that h(n) = 4n – 1?
A. 8
B.
I. It is a Function
II. It is One-to-One
III. It is Onto
IV. It decreases as the value of n increases
A. I only
B. all are true
C. I and II
D. I and III
E. None
15. How many rows of the truth table for p  (q ~ r ) are true?
A. 7
B. 3
C. 6
D. 5
E. NOTA
16. A unicorn is buying rutabaga to feed his young unicorn children, of which he has 6.
Child A eats 17 pounds of food every 13 days. Child B eats 3 pounds of food every
2 days. Child C eats a pound of food per day. Child D eats no food, ever, and
instead survives on dewdrops. Child E eats 10 pounds of food every 8 days. Child
F eats 10 pounds of food every 6.5 days. If he begins feeding his children on day 1,
on what day will the 687th pound of food be eaten?
A. 105
B. 169
C. 104
D. 338
E. NOTA
17. The six digit number 3730A5, where A is the tens digit, is divisible by 21.
What is the sum of all of the integral divisors of A?
A. 4
B. 12
C. 15
D. 7
E. NOTA
18. The latus rectum of a conic section is the width of the function through the focus.
Find the positive difference between the lengths of the latus rectums of
3 y  x 2  4 x  9 and x 2  4 y 2  6 x  16 y  24 .
1
3
5
B. 2
C.
D.
E. NOTA
2
2
2
19. For a positive integer n, define s(n) as the product of the base 4 digits of n. For
example, 31 = 1334 so s(31) = 1 3  3 = 9. What is s(1) + s(2) + … + s(255)?
A.
A. 1496
B. 1554
C. 1572
D. 1596
E. 1624
20. Other than 1, how many positive integers n are there such that 2n + 3n + 4n is a
perfect square?
A. 0
B. 1
C. 2
D. 3
E. more than 3
21. Compute the following determinant:
2 1 ... 1 1
1 2 ... 1 1
. . . . .
1 1 ... 2 1
1 1 ... 1 2 20062006
A. 2006  2007
B. 20062
D. 22006
C. 2007
E. NOTA
22. Find the 3rd digit to the right of the decimal point in the decimal expression of
A. 2
B. 4
C. 6
D. 8
79 .
E. NOTA
23. Set A contains n distinct elements, where n is of the form (2(k+1)2+1)2 and k is a
non-negative integer. If x = the number of subsets of A with an odd # of elements,
and y = the number of subsets of A with an even # of elements, compute x – y.
A. 0
B. k+1
C. n+1
D. 1
E. NOTA
24. Suppose toothpicks of length 2" are aligned point-to-point on a desk in the shape of a
closed rectangle. What is the least area (in inches2) that can be enclosed by this
rectangle if it has perimeter 40" ?
A. 9
B. 400
C. 100
D. 36
E. NOTA
25. Consider the set Ж of all imaginary roots of an equivalence relation of least residue.
If every element of Ж is also the magnitude of an eigenvector of hyperbolic
sinusoidal form, find the number of elements in Ж C. April fools. The answer is A.
You’re welcome.
Tie Breaker 1: What is the sum of the cubes of the integers in the set [-15,17]?
Tie Breaker 2: What is the 47th pentagonal number?
Tie Breaker 3: How many diagonals are there in a regular 27-gon?