Problem 1 - 5
Compute the sum of all the roots of (2x + 3)(x − 4) + (2x + 3)(x − 6) = 0.
(A) 7/2 (B) 4 (C) 5 (D) 7 (E) 13
(A) 7/2
For how many positive integers m does there exist at least one positive integer n such that m · n ≤ m + n?
(A) 4 (B) 6 (C) 9 (D) 12 (E) infinitely many
(E) infinitely many
Mr. Earl E. Bird leaves his house for work at exactly 8:00 A.M. every morning. When he averages 40 miles per hour, he arrives at his workplace three minutes late. When he averages 60 miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace precisely on time?
(A) 45 (B) 48 (C) 50 (D) 55 (E) 58
(B) 48
Tina randomly selects two distinct numbers from the set {1, 2, 3, 4, 5} and Sergio randomly selects a number from the set {1, 2, ..., 10}. The probability that Sergio’s number is larger than the sum of the two numbers chosen by Tina is:
(A) 2/5 (B) 9/20 (C) 1/2 (D) 11/20 (E) 24/25
(A) 2/5
Consider the sequence of numbers: 4, 7, 1, 8, 9, 7, 6, . . . . For n > 2, the nth term of the sequence is the units digit of the sum of the two previous terms. Let Sn denote the sum of the first n terms of this sequence. The smallest value of n for which Sn > 10, 000 is:
(A) 1992 (B) 1999 (C) 2001 (D) 2002 (E) 2004
(B) 1999
Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly?
(A) 15 (B) 34 (C) 43 (D) 51 (E) 138
(A) 15
If an arc of 45◦ on circle A has the same length as an arc of 30◦ on circle B, then the ratio of the area of circle A to the area of circle B is
(A) 4/9 (B) 2/3 (C) 5/6 (D) 3/2 (E) 9/4
(A) 4/9
Both roots of the quadratic equation x^2−63x+k = 0 are prime numbers. The number of possible values of k is:
(A) 0 (B) 1 (C) 2 (D) 3 (E) more than four
(B) 1
Several sets of prime numbers, such as {7, 83, 421, 659} use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?
(A) 193 (B) 207 (C) 225 (D) 252 (E) 447
(B) 207
A square is drawn in the Cartesian coordinate plane with vertices at (2, 2), (−2, 2), (−2, −2), and (2, −2). A particle starts at (0, 0). Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is 1 8 that the particle will move from (x, y) to each of (x, y + 1), (x + 1, y + 1), (x + 1, y), (x + 1, y − 1), (x, y − 1), (x − 1, y − 1), (x − 1, y), (x − 1, y + 1). The particle will eventually hit the square for the first time, either at one of the 4 corners of the square or one of the 12 lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is m n , where m and n are relatively prime positive integers. What is m + n?
(A) 4 (B) 5 (C) 7 (D) 15 (E) 39
(C) 7
Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which of these statements necessarily follows logically?
(A) If Lewis did not receive an A, then he got all of the multiple choice questions wrong.
(B) If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.
(C) If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A.
(D) If Lewis received an A, then he got all of the multiple choice questions right.
(E) If Lewis received an A, then he got at least one of the multiple choice questions right.
(B) If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.
The region consisting of all points in three-dimensional space within 3 units of line segment AB has volume 216π. What is the length AB?
(A) 6 (B) 12 (C) 18 (D) 20 (E) 24
(D) 20
Two different positive numbers a and b each differ from their reciprocals by 1. What is a + b?
(A) 1 (B) 2 (C) √5 (D) √6 (E) 3
(Yes, those are square root symbols - sorry)
(C) √5
Let C1 and C2 be circles defined by
(x − 10)^2 + y^2 = 36 and (x + 15)^2 + y^2 = 81, respectively. What is the length of the shortest line segment P Q that is tangent to C1 at P and to C2 at Q?
(A) 15 (B) 18 (C) 20 (D) 21 (E) 24
(C) 20
In triangle ABC, side AC and the perpendicular bisector of BC meet in point D, and BD bisects ∠ABC. If AD = 9 and DC = 7, what is the area of triangle ABD?
(A) 14 (B) 21 (C) 28 (D) 14√ 5 (E) 28√ 5
(D) 14√ 5
Find the degree measure of an angle whose complement is 25% of its supplement.
(A) 48 (B) 60 (C) 75 (D) 120 (E) 150
(B) 60
Jamal wants to store 30 computer files on floppy disks, each of which has a capacity of 1.44 megabytes (MB). Three of his files require 0.8 MB of memory each, 12 more require 0.7 MB each, and the remaining 15 require 0.4 MB each. No file can be split between floppy disks. What is the minimal number of floppy disks that will hold all the files?
(A) 12 (B) 13 (C) 14 (D) 15 (E) 16
(B) 13
For all positive integers n, let f(n) = log(n^2)/log(2002). Let N = f(11) + f(13) + f(14) Which of the following relations is true?
(A) N < 1 (B) N = 1 (C) 1 < N < 2 (D) N = 2 (E) N > 2
(D) N = 2
A square with side length x is inscribed in a right triangle with sides of length 3, 4, and 5 so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length y is inscribed so that one side of the square lies on the hypotenuse of the triangle. What is x/y ?
(A) 12/13 (B) 35/37 (C) 1 (D) 37/35 (E) 13/12
(D) 37/35
Find the number of ordered pairs of real numbers
(a, b) such that (a + bi)^2002 = a − bi.
(A) 1001 (B) 1002 (C) 2001 (D) 2002 (E) 2004
(E) 2004
At a gathering of 30 people, there are 20 people who all know each other and 10 people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?
(A) 240 (B) 245 (C) 290 (D) 480 (E) 490
(B) 245
Sarah pours four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then transfers half the coffee from the first cup to the second and, after stirring thoroughly, transfers half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream?
(A) 1/4 (B) 1/3 (C) 3/8 (D) 2/5 (E) 1/2
(D) 2/5
The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is:
(A) 11 (B) 12 (C) 13 (D) 14 (E) 15
(D) 14
Suppose that a and b are digits, not both nine and not both zero, and the repeating decimal 0.ab is expressed as a fraction in lowest terms. How many different denominators are possible?
(A) 3 (B) 4 (C) 5 (D) 8 (E) 9
(C) 5
Let a and b be real numbers such that sin a + sin b = √2/2 and cos a + cos b = √6/2.
Find sin(a + b).
(A) 1/2 (B) √2/2 (C) √3/2 (D) √6/2 (E) 1
(E) 1