American Mathematics Competitions
65th Annual
AMC 12 B
American Mathematics Contest 12 B
Wednesday, February 19, 2014 INSTRUCTIONS
1. DO NOT OPEN T
HIS BOOKLET UNTIL YOUR PROCTOR TELLS YOU. 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. Mark your answer to each problem on the AMC 12 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, 1.5 points for each problem left unanswered, and 0 points for each incorrect answer. 5. No aids are permitted other than scratch paper, graph paper, rulers, compass, protractors, and erasers. No calculators are allowed. 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. 8. When your proctor gives the signal, begin working on the problems. You will have 75 minutes to complete the test. 9. When you finish the exam, sign your name in the space provided on the Answer Form.
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.
Students who score 100 or above or finish in the top 5% on this AMC 12 will be invited to take the 32nd annual American Invitational Mathematics Examination (AIME) on Thursday, March 13, 2014 or Wednesday, March 26, 2014. More details about the AIME and other information are on the back page of this test booklet.
The publication, reproduction or communication of the problems or solutions of the AMC 12 during the period when students are eligible to participate seriously jeopardizes the integrity of the results. Dissemination via copier, telephone, e-mail, World Wide Web or media of any type during this period is a violation of the competition rules. After the contest period, permission to make copies of problems in paper or electronic form including posting on web-pages for educational use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear the copyright notice. ? 2014 Mathematical Association of America
AMC 12 B
DO NOT OPEN UNTIL wednesday, february 19, 2014
**Administration On An Earlier Date Will Disqualify Your School’s 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 19, 2014. Nothing is needed from inside this package until February 19. 2. Your PRINCIPAL or VICE-PRINCIPAL must verify on the AMC 12 CERTIFICATION FORM (found in the Teachers’ Manual) that you followed all rules associated with the conduct of the exam. 3. The Answer Forms must be mailed by trackable mail to the AMC office no later than 24 hours following the exam. 4. 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. Dissemination at any time via copier, telephone, email, internet or media of any type is a violation of the competition rules.
2014
The American Mathematics Competitions
are Sponsored by
The Mathematical Association of America – MAA .................................................... www.maa.org The Akamai Foundation ........................................................................................www.akamai.com
Contributors
Academy of Applied Sciences – AAS........................................................................................................... www.aas-world.org American Institute of Mathematics - AIM........................................................................................................www.aimath.org American Mathematical Association of Two-Year Colleges – AMATYC............................................................ www.amatyc.org American Mathematical Society – AMS.............................................................................................................. www.ams.org American Statistical Association – ASA.......................................................................................................... www.amstat.org Art of Problem Solving – AoPS................................................................................................ www.artofproblemsolving.com Association of Symbolic Logic - ASL............................................................................................................ www.alsonline.org Awesome Math ...................................................................................................................................www.awesomemath.org Casualty Actuarial Society – CAS. ................................................................................................................... www.casact.org Conference Board of the Mathematical Sciences - CBMS............................................................................ www.cbmsweb.org The D.E. Shaw Group ................................................................................................................................... www.deshaw.com IDEA Math...................................................................................................................................................www.ideamath.org Jane Street Capital..................................................................................................................................www.janestreet.com Math For America............................................................................................................................. www.mathforamerica.org Math Training Center..................................................................................................................www.mathtrainingcenter.com Mu Alpha Theta – MAT......................................................................................................................... www.mualphatheta.org Pi Mu Epsilon – PME. ................................................................................................................................. www.pme-math.org Society for Industrial and Applied Math – SIAM................................................................................................. www.siam.org W.H. Freeman..........................................................................................................................................www.whfreeman.com
2014 AMC12B Problems
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1. Leah has 13 coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah’s coins worth? (A) 33 (B) 35 (C) 37 (D) 39 (E) 41
2. Orvin went to the store with just enough money to buy 30 balloons. When he arrived he discovered that the store had a special sale on balloons: buy 1 balloon at the regular price and get a second at 1 3 o? the regular price. What is the greatest number of balloons Orvin could buy? (A) 33 (B) 34 (C) 36 (D) 38 (E) 39
3. Randy drove the ?rst third of his trip on a gravel road, the next 20 miles on pavement, and the remaining one-?fth on a dirt road. In miles, how long was Randy’s trip? (A) 30 (B) 400 11 (C) 75 2 (D) 40 (E) 300 7
4. Susie pays for 4 mu?ns and 3 bananas. Calvin spends twice as much paying for 2 mu?ns and 16 bananas. A mu?n is how many times as expensive as a banana? (A) 3 2 (B) 5 3 (C) 7 4 (D) 2 (E) 13 4
5. Doug constructs a square window using 8 equal-size panes of glass, as shown. The ratio of the height to width for each pane is 5 : 2, and the borders around and between the panes are 2 inches wide. In inches, what is the side length of the square window?
(A) 26
(B) 28
(C) 30
(D) 32
(E) 34
2014 AMC12B Problems
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6. Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is 50% more than the regular. After both consume 3 4 of their drinks, Ann gives Ed a third of what she has left, and 2 additional ounces. When they ?nish their lemonades they realize that they both drank the same amount. How many ounces of lemonade did they drink together? (A) 30 (B) 32 (C) 36 (D) 40
n 30?n
(E) 50 also a positive integer?
7. For how many positive integers n is (A) 4 (B) 5 (C) 6
(D) 7
(E) 8
8. In the addition shown below A, B , C , and D are distinct digits. How many di?erent values are possible for D ? ABBCB BCADA DBDDD (E) 9
+
(A) 2
(B) 4
(C) 7
(D) 8
9. Convex quadrilateral ABCD has AB = 3, BC = 4, CD = 13, AD = 12, and ∠ABC = 90? , as shown. What is the area of the quadrilateral?
(A) 30
(B) 36
(C) 40
(D) 48
(E) 58.5
2014 AMC12B Problems
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10. Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, abc miles was displayed on the odometer, where abc is a 3-digit number with a ≥ 1 and a + b + c ≤ 7. At the end of the trip, the odometer showed cba miles. What is a2 + b2 + c2 ? (A) 26 (B) 27 (C) 36 (D) 37 (E) 41
11. A list of 11 positive integers has a mean of 10, a median of 9, and a unique mode of 8. What is the largest possible value of an integer in the list? (A) 24 (B) 30 (C) 31 (D) 33 (E) 35
12. A set S consists of triangles whose sides have integer lengths less than 5, and no two elements of S are congruent or similar. What is the largest number of elements that S can have? (A) 8 (B) 9 (C) 10 (D) 11 (E) 12
13. Real numbers a and b are chosen with 1 < a < b such that no triangle with 1 positive area has side lengths 1, a, and b or 1 b , a , and 1. What is the smallest possible value of b ? √ √ √ 3+ 3 5 3+ 5 3+ 6 (A) (B) (C) (D) (E) 3 2 2 2 2 14. A rectangular box has a total surface area of 94 square inches. The sum of the lengths of all its edges is 48 inches. What is the sum of the lengths in inches of all of its interior diagonals? √ √ √ √ √ (B) 10 2 (C) 16 3 (D) 20 2 (E) 40 2 (A) 8 3 15. When p = k=1 k ln k , the number ep is an integer. What is the largest power of 2 that is a factor of ep ? (A) 212 (B) 214 (C) 216 (D) 218 (E) 220
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16. Let P be a cubic polynomial with P (0) = k , P (1) = 2k , and P (?1) = 3k . What is P (2) + P (?2) ? (A) 0 (B) k (C) 6k (D) 7k (E) 14k
2014 AMC12B Problems
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17. Let P be the parabola with equation y = x2 and let Q = (20, 14). There are real numbers r and s such that the line through Q with slope m does not intersect P if and only if r < m < s. What is r + s ? (A) 1 (B) 26 (C) 40 (D) 52 (E) 80
18. The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is bad if it is not true that for every n from 1 to 15 one can ?nd a subset of the numbers that appear consecutively on the circle that sum to n. Arrangements that di?er only by a rotation or a re?ection are considered the same. How many di?erent bad arrangements are there? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5
19. A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?
(A)
3 2
(B)
√ 1+ 5 2
(C)
√
3
(D) 2
(E)
√ 3+ 5 2
20. For how many positive integers x is log10 (x ? 40) + log10 (60 ? x) < 2 ? (A) 10 (B) 18 (C) 19 (D) 20 (E) in?nitely many
2014 AMC12B Problems
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21. In the ?gure, ABCD is a square of side length 1. The rectangles JKHG and EBCF are congruent. What is BE ?
1 √ (A) ( 6 ? 2) 2
1 (B) 4
√ (C) 2 ? 3
3 (D) 6
√
√ 2 (E) 1 ? 2
22. In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad N , 0 < N < 10, it will jump to N N and to pad N + 1 with probability 1 ? 10 . Each pad N ? 1 with probability 10 jump is independent of the previous jumps. If the frog reaches pad 0 it will be eaten by a patiently waiting snake. If the frog reaches pad 10 it will exit the pond, never to return. What is the probability that the frog will escape being eaten by the snake? (A) 32 79 (B) 161 384 (C) 63 146 (D) 7 16 (E) 1 2
23. The number 2017 is prime. Let S = S is divided by 2017 ? (A) 32 (B) 684 (C) 1024
62 2014 k=0 k
. What is the remainder when (E) 2016
(D) 1576
24. Let ABCDE be a pentagon inscribed in a circle such that AB = CD = 3, BC = DE = 10, and AE = 14. The sum of the lengths of all diagonals of ABCDE is equal to m n , where m and n are relatively prime positive integers. What is m + n ? (A) 129 (B) 247 (C) 353 (D) 391 (E) 421
2014 AMC12B Problems 25. What is the sum of all positive real solutions x to the equation 2 cos(2x) cos(2x) ? cos (A) π (B) 810π (C) 1008π 2014π 2 x (D) 1080π = cos(4x) ? 1 ? (E) 1800π
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American Mathematics Competitions
WRITE TO US!
Correspondence about the problems and solutions for this AMC 12 and orders for publications should be addressed to:
American Mathematics Competitions 1740 Vine Street Lincoln, NE 68508-1228
Phone 402-472-2257 | Fax 402-472-6087 | amcinfo@maa.org The problems and solutions for this AMC 12 were prepared by the MAA’s Committee on the AMC 10 and AMC 12 under the direction of AMC 12 Subcommittee Chair:
Prof. Bernardo M. Abrego 2014 AIME
The 32nd annual AIME will be held on Thursday, March 13, with the alternate on Wednesday, March 26. It is a 15-question, 3-hour, integer-answer exam. You will be invited to participate only if you score 120 or above or finish in the top 2.5% of the AMC 10, or if you score 100 or above or finish in the top 5% of the AMC 12. Top-scoring students on the AMC 10/12/AIME will be selected to take the 43rd Annual USA Mathematical Olympiad (USAMO) on April 29-30, 2014. The best way to prepare for the AIME and USAMO is to study previous exams. Copies may be ordered as indicated below.
PUBLICATIONS
A complete listing of current publications, with ordering instructions, is at our web site: maa.org/math-competitions