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Wednesday, FEBRUARY 27, 2002

Contest B
The MATHEMATICAL ASSOCIATION OF AMERICA American Mathematics Competitions
Presented by the Akamai Foundation

3rd Annual American Mathematics Contest 10

1. DO NOT OPEN THIS BOOKLET UNTIL TOLD TO DO SO BY YOUR PROCTOR. 2. This is a twenty-five question, multiple choice test. Each question is followed by answers marked A,B,C,D and E. Only one of these is correct. 3. The answers to the problems are to be marked on the AMC 10 Answer Form with a #2 pencil. Check the blackened circles for accuracy and erase errors and stray marks completely. Only answers properly marked on the answer form will be graded. 4. SCORING: You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered, and 0 points for each incorrect answer. 5. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor, erasers and calculators that are accepted for use on the SAT. No problems on the test will require the use of a calculator. 6. Figures are not necessarily drawn to scale. 7. Before beginning the test, your proctor will ask you to record certain information on the answer form. When your proctor gives the signal, begin working the problems. You will have 75 MINUTES working time to complete the test. 8. When you finish the exam, sign your name in the space provided on the Answer Form.
Students who score in the top 1% on this AMC 10 will be invited to take the 20th annual American Invitational Mathematics Examination (AIME) on Tuesday, March 26, 2002 or on Tuesday, April 9, 2002. More details about the AIME and other information are on the back page of this test booklet.
The Committee on the American Mathematics Competitions (CAMC) reserves the right to re-examine students before deciding whether to grant official status to their scores. The CAMC also reserves the right to disqualify all scores from a school if it is determined that the required security procedures were not followed. The publication, reproduction, or communication of the problems or solutions of the AMC 10 during the period when students are eligible to participate seriously jeopardizes the integrity of the results. Duplication at any time via copier, telephone, eMail, World Wide Web or media of any type is a violation of the copyright law. Copyright 2002, Committee on the American Mathematics Competitions, Mathematical Association of America

AMC 10

3rd AMC 10 B 1. The ratio (A) 1 6 22001 32003 is 62002 (B) 1 3 (C) 1 2 (D) 2 3

2002

(E)

3 2

2. For the nonzero numbers a, b, and c, dene (a, b, c) = Find (2, 4, 6). (A) 1 (B) 2 (C) 4 (D) 6 (E) 24 abc . a+b+c

3. The arithmetic mean of the nine numbers in the set {9,99,999,9999, . . . ,999999999} is a 9-digit number M , all of whose digits are distinct. The number M does not contain the digit (A) 0 (B) 2 (C) 4 (D) 6 (E) 8

4. What is the value of (3x 2)(4x + 1) (3x 2)4x + 1 when x = 4? (A) 0 (B) 1 (C) 10 (D) 11 (E) 12

5. Circles of radius 2 and 3 are externally tangent and are circumscribed by a third circle, as shown in the gure. Find the area of the shaded region.

(A) 3

(B) 4

(C) 6

(D) 9

(E) 12

3rd AMC 10 B

2002

6. For how many positive integers n is n2 3n + 2 a prime number? (A) none (B) one (C) two (D) more than two, but nitely many (E) innitely many 7. Let n be a positive integer such that following statements is not true: (A) 2 divides n (E) n > 84 8. Suppose July of year N has ve Mondays. Which of the following must occur ve times in August of year N ? (Note: Both months have 31 days.) (A) Monday (B) Tuesday (C) Wednesday (D) Thursday (E) Friday (B) 3 divides n
1 2

1 3

1 7

1 n

is an integer. Which of the (D) 7 divides n

(C) 6 divides n

9. Using the letters A, M, O, S, and U, we can form 120 ve-letter words. If these words are arranged in alphabetical order, then the word USAMO occupies position (A) 112 (B) 113 (C) 114 (D) 115 (E) 116

10. Suppose that a and b are nonzero real numbers, and that the equation x2 + ax + b = 0 has solutions a and b. Then the pair (a, b) is (A) (2, 1) (B) (1, 2) (C) (1, 2) (D) (2, 1) (E) (4, 4)

11. The product of three consecutive positive integers is 8 times their sum. What is the sum of their squares? (A) 50 (B) 77 (C) 110 (D) 149 (E) 194 x1 xk = have no x2 x6

12. For which of the following values of k does the equation solution for x? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5

13. Find the value(s) of x such that 8xy 12y + 2x 3 = 0 is true for all values of y. (A) 2 3 (B) 3 1 or 2 4 (C) 2 1 or 3 4 (D) 3 2 (E) 3 1 or 2 4

3rd AMC 10 B

2002

14. The number 2564 6425 is the square of a positive integer N . In decimal representation, the sum of the digits of N is (A) 7 (B) 14 (C) 21 (D) 28 (E) 35

15. The positive integers A, B, A B, and A + B are all prime numbers. The sum of these four primes is (A) even (E) prime (B) divisible by 3 (C) divisible by 5 (D) divisible by 7

16. For how many integers n is (A) 1 (B) 2 (C) 3

n 20n

the square of an integer? (E) 10

(D) 4

17. A regular octagon ABCDEF GH has sides of length two. Find the area of ADG. (A) 4 + 2 2 (B) 6 + 2 (C) 4 + 3 2 (D) 3 + 4 2 (E) 8 + 2

18. Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect? (A) 8 (B) 9 (C) 10 (D) 12 (E) 16

19. Suppose that {an } is an arithmetic sequence with a1 + a2 + + a100 = 100 and a101 + a102 + + a200 = 200. What is the value of a2 a1 ? (A) 0.0001 (B) 0.001 (C) 0.01 (D) 0.1 (E) 1

20. Let a, b, and c be real numbers such that a 7b + 8c = 4 and 8a + 4b c = 7. Then a2 b2 + c2 is (A) 0 (B) 1 (C) 4 (D) 7 (E) 8

3rd AMC 10 B

2002

21. Andys lawn has twice as much area as Beths lawn and three times as much area as Carlos lawn. Carlos lawn mower cuts half as fast as Beths mower and one third as fast as Andys mower. If they all start to mow their lawns at the same time, who will nish rst? (A) Andy (B) Beth (C) Carlos (D) Andy and Carlos tie for rst.

(E) All three tie.

22. Let XOY be a right-angled triangle with m XOY = 90 . Let M and N be the midpoints of legs OX and OY , respectively. Given that XN = 19 and Y M = 22, nd XY . (A) 24 (B) 26 (C) 28 (D) 30 (E) 32

23. Let {ak } be a sequence of integers such that a1 = 1 and am+n = am + an + mn, for all positive integers m and n. Then a12 is (A) 45 (B) 56 (C) 67 (D) 78 (E) 89

24. Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius 20 feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point 10 vertical feet above the bottom? (A) 5 (B) 6 (C) 7.5 (D) 10 (E) 15

25. When 15 is appended to a list of integers, the mean is increased by 2. When 1 is appended to the enlarged list, the mean of the enlarged list is decreased by 1. How many integers were in the original list? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8

3rd AMC 10 B

2002

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WRITE TO US!
Correspondence about the problems and solutions for this AMC 10 should be addressed to:

Prof. Douglas Faires, Department of Mathematics Youngstown State University, Youngstown, OH 44555-0001 Phone:330-742-1805; Fax: 330-742-3170; email: faires@math.ysu.edu
Orders for any of the publications listed below should be addressed to:

Titu Andreescu, Director American Mathematics Competitions University of Nebraska, P.O. Box 81606 Lincoln, NE 68501-1606
Phone: 402-472-2257; Fax: 402-472-6087; email: titu@amc.unl.edu;

2002 AIME
The AIME will be held on Tuesday, March 26, 2002 with the alternate on April 9,2002. It is a 15-question, 3-hour, integer-answer exam. You will be invited to participate only if you score in the top 1% of this AMC 10 or receive a score of 100 or above on the AMC 12. Alternately, you must be in the top 5% of the AMC 12. Top-scoring students on the AMC 10/12/AIME will be selected to take the USA Mathematical Olympiad (USAMO) on Thursday through Sunday, May 9-12, 2002 in Cambridge, MA. The best way to prepare for the AIME and USAMO is to study previous years of these exams. Copies may be ordered as indicated below.

PUBLICATIONS
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Examinations: Each price is for one copy of an exam and its solutions for one year. Specify the years you want and how many copies of each. All prices effective to September 1, 2002. AMC 10 2000-2002/AHSME (AMC 12) 1989-2002, $1 per copy per year. AIME 1989-2002, $2 per copy per year. USA and International Math Olympiads, 1989-1999, $5 per copy per year, 2000, $14.00 per copy. National Summary of Results and Awards, 1989-2002, $10 per copy per year. Problem Book I, AHSMEs 1950-60, Problem Book II, AHSMEs 1961-65, $10/ea Problem Book III, AHSMEs 1966-72, Problem Book IV, AHSMEs 1973-82, $13/ea Problem Book V, AHSMEs and AIMEs 1983-88, $30/ea Problem Book VI, AHSMEs 1989-1994, $24/ea USA Mathematical Olympiad Book 1972-86, $18/ea International Mathematical Olympiad Book I, 1959-77, $20/ea International Mathematical Olympiad Book II, 1978-85, $20/ea World Olympiad Problems/Solutions 1995-96, 1996-97, 1997-98, $15/ea Mathematical Olympiads Problems & Solutions from around the World 1998-1999, $25/ea The Arbelos, Volumes I-V, and a Special Geometry Issue, $8/ea

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2002

AMC 10 - Contest B
DO NOT OPEN UNTIL Wednesday, FEBRUARY 27, 2002
**Administration On An Earlier Date Will Disqualify Your Schools Results**
1. All information (Rules and Instructions) needed to administer this exam is contained in the TEACHERS MANUAL, which is outside of this package. PLEASE READ THE MANUAL BEFORE FEBRUARY 27. Nothing is needed from inside this package until February 27. 2. Your PRINCIPAL or VICE PRINCIPAL must sign the Certification Form A found in the Teachers Manual. 3. The Answer Forms must be mailed by First Class mail to the AMC Director, Titu Andreescu, no later than 24 hours following the examination. 4. Please Note: All Problems and Solutions are copyrighted; it is illegal to make copies or transmit them on the internet or world wide web without permission. 5. The publication, reproduction or communication of the problems or solutions of this test during the period when students are eligible to participate seriously jeopardizes the integrity of the results. Duplication at any time via copier, telephone, eMail, World Wide Web or media of any type is a violation of the copyright law.

Contributors American Statistical Association Casualty Actuarial Society Society of Actuaries National Council of Teachers of Mathematics American Society of Pension Actuaries American Mathematical Society American Mathematical Association of Two Year Colleges Pi Mu Epsilon Consortium for Mathematics and its Applications Mu Alpha Theta National Association of Mathematicians Kappa Mu Epsilon School Science and Mathematics Association Clay Mathematics Institute Institute for Operations Research and the Management Sciences

Sponsored by The MATHEMATICAL ASSOCIATION OF AMERICA The Akamai Foundation University of Nebraska Lincoln