2. The graphs of
and 
intersect in N points. Then
(a) N = 0 (b) N=1 (c) N = 2 (d) N = 3 (e) N > 3
3. The diagonals of a rhombus measure 24 inches and 32 inches. The perimeter of the rhombus (a) cannot be determined uniquely from the given information. (b) must be 80 inches. (c) must be
inches.
(d) must be 160 inches. (e) must be 
inches.
4. For n = 1,2, . . . , let T n= 1 + 2 + . . . + n. Which of the following statements is correct? (a) There is no value of n for which T n is a positive power of 2. (A positive power of two is an integer of the form 2kwhere k is a positive integer.) (b) There is exactly one value of n for which T n is a positive power of 2. (c) There are exactly two values of n for which T n is a positive power of 2. (d) There are more than two but finitely many values of n for which T n is a positive power of 2. (e) There are infinitely many values of n for which T n is a positive power of 2.
5. A, B and C are the vertices of a triangle whose perimeter is 73 inches. AB is the longest side of the triangle, and the product of the lengths of sides AB and BC is 750 square inches. Side BC is 7 inches longer than side AC. The length of side AB
(a) cannot be determined uniquely from the given information. (b) must be 15 inches. (c) must be 25 inches. (d) must be 30 inches. (e) must be 50 inches.
6. Define the function
. Which of the following numbers is the maximum
value of the function?
(a) 1/2 (b) 5/8 (c) 3/4 (d) 1 (e) 5/2
7. Consider the equation
. The number of rational
numbers x that solve the equation is
(a) 0 (b) 1 (c) 2 (d) 3 (e) 4 or more.
8. A contestant travels from Charlotte to Durham on the interstate highway with an average velocity that is 1.5 times his average velocity on the return trip from Durham back to Charlotte. The average velocity for the round trip is 60 mph. What was the average velocity of the contestant on the initial leg from Charlotte to Durham? (a) 40 mph (b) 50 mph (c) 60 mph (d) 75 mph (e) 90 mph
9. If
and 
, then cos(2y) is equal to
(a) 10x (b) 25x (c) 2(5a)x (d) 25x + a2x. (e) 25x - a2x
10. If n is a positive integer, let S(n) denote the sum of the positive divisors of n, including n itself. Thus, for example, S(6) = 1 + 2 + 3 + 6 = 12. Which of the following five integers is the smallest? (a) S(1998) (b) S(1999) (c) S(2000) (d) S(2001) (e) S(2002)
11. C1 and C2 are concentric circles with radii r1 and r2, respectively, with r1 < r2. Points A and B are chosen on C2 in such a way that the chord AB is tangent to C1 and has length 16 centimeters. Then the area of the annulus (the ring shaped region) bounded by C1 and C2 (a) cannot be determined uniquely from the given information. (b) is less than 200 square centimeters. (c) is at least 200 square centimeters but is less than 500 square centimeters. (d) is at least 500 square centimeters but is less than 800 square centimeters. (e) is at least 800 square centimeters.
12. One hundred positive integers are arranged in a row. The sum of any six adjacent integers is the same as the sum of any other six adjacent integers. The value of the eleventh integer is 11, the value of the twenty-second integer is 22, the value of the thirty-third integer is 33, the value of the forty-fourth integer is 44, and the value of the fifty-fifth integer is 55. The sum of the nineteenth and the ninety-ninth integer (a) cannot be determined uniquely from the given information. (b) must be less than 60. (c) must be greater than or equal to 60 and must be less than 75. (d) must be greater than or equal to 75 and must be less than 90. (e) must be greater than 90.
13. P, Q, R and S are the vertices of a square of side length s inches. A point Mis chosen on side QRso that the length of the segment QMis twice the length of the segment MR.A point X is chosen on PQand a point Y is chosen on RSso that angle XMY is a right angle. If the product of the lengths of the segments XQand YR is 144 square inches, what is the value of s, the length of a side of the square? (a) The value of s cannot be determined uniquely from the given information. (b) s must equal 9 inches. (c) s must equal
inches.
(d) s must equal 18 inches. (e) s must equal 
inches.
14. Let C be the circle described by (a - x)2 + y2 = r2 where 0 < r < a. Let m be the slope of
the line through the origin that is tangent to C at a point in the first quadrant. Then (a)
(b)
(c)
(d)
(e) 
15. A box contains b red, 2b white and 3b blue balls, where b is a positive integer. Three balls are selected at random and without replacement from the box. Let p(b) denote the probability that no two of the selected balls have the same color. Then (a) There is no value of b for which
(b) There is exactly one value of b for which , and this value is less than 10.
(c) There is exactly one value of b for which , and this value is greater or equal to 10
but is less than 100.
(d) There is exactly one value of b for which , and this value is greater than 100.
(e) There is more than one value of b for which .
16. The positive integers a1,a2, ..., a20 satisfy the following properties (i) a1 = 20, (ii) a20 = 5, and (iii) for 1 < i < 19, a i+1 - ai > -1. Then, which of the following is valid? (a) The value of a10 is uniquely determined. (b) a10 can take on any one of 5 distinct values. (c) a10 can take on any one of 9 distinct values. (d) a10 can take on any one of 10 distinct values. (e) a10 can take on any one of 15 distinct values.
17. Let
for 
. Let N be the number of complex numbers z
for which f(z)= 0. Then
(a) N = 0. (b) 1 < N < 2. (c) 3 < N < 4. (d) 5 < N < 6. (e) 7 < N.
18. P, Q and R are the vertices of a triangle and the length of the segment PQ is 34 inches. Points S and T are chosen as follows. Extend segment PR through R to S so that R is the bisector of the segment PS. Choose T on PQ so that if U is the point in which ST and RQ meet, then U is the bisector of the segment QR. What is the length of PT to the nearest inch? (a) The length cannot be determined uniquely from the given information. (b) 20 inches (c) 21 inches (d) 22 inches (e) 23 inches
19. A pair (M, N) of two digit positive integers, with M < N, is said to be reversible provided: (i) if the digits of M are reversed, the resulting integer is twice as large as N, and (ii) if the digits of N are reversed, the resulting integer is twice as large as M. Which of the following statements is correct? (a) There are no reversible pairs (M, N) of two digit integers. (b) There is exactly one reversible pair (M, N) of two digit integers. (c) There are exactly two reversible pairs (M, N) of two digit integers. (d) There are exactly three reversible pairs (M, N) of two digit integers. (e) There are more than three reversible pairs (M, N) of two digit integers.
20. A sequence consists of 4n (complex) numbers, a1, a2, . . . , a4n, which satisfy the following three properties. (i) For each j,
where 
, and each element of S appears exactly n times in the sequence.
(ii) For each value of k = 1, 2,..., 4n, the real and the imaginary parts of the sum of the first k terms of the sequence fall in the interval [0, 2].
(iii) If 1 < k < m < 4n, then 
.
Determine the maximum possible value of n.
(a) The maximum value of n is 1.
(b) The maximum value of n is 2.
(c) The maximum value of n is 3.
(d) The maximum value of n is 4.
(e) The maximum value of n is greater than 4.
PART II: INTEGER ANSWER (10 PROBLEMS) 1. Let A be one vertex of a regular hexagon whose side length is one meter. To the nearest centimeter, what is the sum of the distances from A to the other five vertices of the hexagon?
2. How many points of intersection do the graphs of y = cos(x) and y = tan(x) have over the interval 0 < x < 100, where x is measured in radians.
3. A line of people has gathered in front of a theater six hours before a matinee showing of the blockbuster sci-fi adventure, The Menacing Phantom. The ratio of females to males in the line is 6 to 7. After 15 minutes, another 21 people have joined the line and the ratio of females to males has become 7 to 8. What is the fewest number of people that could have originally been in the line?
4. The total number of edges in two regular polygons is 2002, and the total number of diagonals is 1,995,002. (A diagonal of a polygon is a segment joining two non-adjacent vertices.) How many edges does the polygon with the larger number of edges have?
5. If b is a positive integer greater than 1, the base b expansion of a positive integer is denoted (dkdk-1 ... d1d0)b where the di are base b digits. Thus, the base 5 expansion of (54)10 is (204)5 since in base 10, 54 = 2 (52) +0(51) + 4(50). Similarly, the base 5 expansion of (37)10 is (122)5. How many of the integers between one and one thousand, inclusive, have a base 5 expansion that contains at least one zero?
6. Let A, B, C and D be the vertices of a square, and let E be the center of the square. Assign, randomly, the numbers 1, 2, 3, 4 and 5 to the five points A through E so that each point is assigned a different number. Let p be the probability that the sum of the numbers assigned to the points A, E and C will equal the sum of the numbers assigned to the points B, E and D. If p = m/n, in lowest terms, with n positive, what is the value of m + n?
7. Let
. Define a sequence of functions by setting f1(z) = f(z) and then setting
f n+1 (z) = f(f n (z)). If the value of f1999(1999) is written as P/Q , a fraction in reduced form
with Q positive, what is the number of digits in the integer Q?
8. Consider the infinite series 1 + 3x + 7x2 + . . . + (1 + 2 + 22 + . . . + 2n)xn + ... whose sum is defined if -1/2 < x < 1/2. If the value of the sum is 2 when x = x0 and |x0| < 1/2, what is the value of 10,000x0, rounded to the nearest integer?
9. Given A0 and B0,define An, and Bn for n > 1 using A n+1, = r(An - Bn,) and B n+1, = sBn. If r = 1.1, s = 1.05 and A0 = 500, find, to the nearest integer, the value of B0 such that A50 = B50.
10. ABCD is a rectangle. Points E and F are chosen as follows. Segment BA is extended through A to a point E , and F is the intersection of segment EC with edge AD. Suppose that the measure of the angle ACD is 60 degrees and that the length of segment EF is twice the length of the diagonal AC. To the nearest degree, what is the measure of angle ECD ?