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American Mathematics Competitions

64th Annual

AMC 12 A

American Mathematics Contest 12 A

Tuesday, February 5, 2013 INSTRUCTIONS

1. DO NOT OPEN THIS 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 120 or above or finish in the top 2.5% on this AMC 10 will be invited to take the 31st annual American Invitational Mathematics Examination (AIME) on Thursday, March 14, 2013 or Wednesday, April 3, 2013. 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. ? 2013 Mathematical Association of America

AMC 12 A

DO NOT OPEN UNTIL Tuesday, february 5, 2013

**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 5, 2013. Nothing is needed from inside this package until February 5. 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.

2013

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 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 Awesome Math ........................................................................................................................... www.awesomemath.org Casualty Actuarial Society – CAS........................................................................................................... www.casact.org D.E. Shaw & Co. ............................................................................................................................ www.deshaw.com Delta Airlines................................................................................................................................... www.delta.com Jane Street Capital..................................................................................................................www.janestreet.com Math For America............................................................................................................. www.mathforamerica.org Mu Alpha Theta – MAT................................................................................................................. www.mualphatheta.org National Council of Teachers of Mathematics – NCTM............................................................................. www.nctm.org Pi Mu Epsilon – PME.......................................................................................................................... www.pme-math.org Society for Industrial and Applied Math (SIAM)..................................................................................www.siam.org

2013 AMC12A Problems 1. Square ABCD has side length 10. Point E is on BC , and the area of is 40. What is BE ? (A) 4 (B) 5 (C) 6

B

2 ABE

(D) 7

E

(E) 8

C

A

D

2. A softball team played ten games, scoring 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 runs. They lost by one run in exactly ?ve games. In each of their other games, they scored twice as many runs as their opponent. How many total runs did their opponents score? (A) 35 (B) 40 (C) 45 (D) 50 (E) 55

3. A ?ower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink ?owers are roses, three fourths of the red ?owers are carnations, and six tenths of the ?owers are pink. What percent of the ?owers are carnations? (A) 15 (B) 30 (C) 40 (D) 60 (E) 70

4. What is the value of

22014 + 22012 ? 22014 ? 22012 (C) 5 3 (D) 2013 (E) 24024

(A) ?1

(B) 1

5. Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $105, Dorothy paid $125, and Sammy paid $175. In order to share the costs equally, Tom gave Sammy t dollars, and Dorothy gave Sammy d dollars. What is t ? d ? (A) 15 (B) 20 (C) 25 (D) 30 (E) 35

2013 AMC12A Problems

3

6. In a recent basketball game, Shenille attempted only three-point shots and twopoint shots. She was successful on 20% of her three-point shots and 30% of her two-point shots. Shenille attempted 30 shots. How many points did she score? (A) 12 (B) 18 (C) 24 (D) 30 (E) 36

7. The sequence S1 , S2 , S3 , . . . , S10 has the property that every term beginning with the third is the sum of the previous two. That is, Sn = Sn?2 + Sn?1 for n ≥ 3. Suppose that S9 = 110 and S7 = 42. What is S4 ? (A) 4 (B) 6 (C) 10 (D) 12 (E) 16

2 x 2 =y+ y ,

8. Given that x and y are distinct nonzero real numbers such that x + what is xy ? (A) 1 4 (B) 1 2 (C) 1 (D) 2 (E) 4

9. In ABC , AB = AC = 28 and BC = 20. Points D, E , and F are on sides AB , BC , and AC , respectively, such that DE and EF are parallel to AC and AB , respectively. What is the perimeter of parallelogram ADEF ?

A

D

F

B

E

C

(A) 48

(B) 52

(C) 56

(D) 60

(E) 72

1 10. Let S be the set of positive integers n for which n has the repeating decimal representation 0.ab = 0.ababab . . ., with a and b di?erent digits. What is the sum of the elements of S ?

(A) 11

(B) 44

(C) 110

(D) 143

(E) 155

2013 AMC12A Problems

4

11. Triangle ABC is equilateral with AB = 1. Points E and G are on AC and points D and F are on AB such that both DE and F G are parallel to BC . Furthermore, triangle ADE and trapezoids DF GE and F BCG all have the same perimeter. What is DE + F G ?

(A) 1

(B)

3 2

(C)

21 13

(D)

13 8

(E)

5 3

12. The angles in a particular triangle are in arithmetic progression, and√ the side √ lengths are 4, 5, and x. The sum of the possible values of x equals a + b + c, where a, b, and c are positive integers. What is a + b + c ? (A) 36 (B) 38 (C) 40 (D) 42 (E) 44

13. Let points A = (0, 0), B = (1, 2), C = (3, 3), and D = (4, 0). Quadrilateral ABCD is cut into equal area pieces by a line passing through A. This line r intersects CD at point ( p q , s ), where these fractions are in lowest terms. What is p + q + r + s ? (A) 54 (B) 58 (C) 62 (D) 70 (E) 75

14. The sequence log12 162, log12 x, log12 y, log12 z, log12 1250 is an arithmetic progression. What is x ? √ √ (A) 125 3 (B) 270 (C) 162 5 √ (E) 225 6

(D) 434

15. Rabbits Peter and Pauline have three o?spring—Flopsie, Mopsie, and Cottontail. These ?ve rabbits are to be distributed to four di?erent pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many di?erent ways can this be done? (A) 96 (B) 108 (C) 156 (D) 204 (E) 372

2013 AMC12A Problems

5

16. A, B , and C are three piles of rocks. The mean weight of the rocks in A is 40 pounds, the mean weight of the rocks in B is 50 pounds, the mean weight of the rocks in the combined piles A and B is 43 pounds, and the mean weight of the rocks in the combined piles A and C is 44 pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles B and C ? (A) 55 (B) 56 (C) 57 (D) 58 (E) 59

17. A group of 12 pirates agree to divide a treasure chest of gold coins among k of the coins that themselves as follows. The kth pirate to take a share takes 12 remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the 12th pirate receive? (A) 720 (B) 1296 (C) 1728 (D) 1925 (E) 3850

18. Six spheres of radius 1 are positioned so that their centers are at the vertices of a regular hexagon of side length 2. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere? (A) √ 2 (B) 3 2 (C) 5 3 (D) √ 3 (E) 2

19. In ABC , AB = 86, and AC = 97. A circle with center A and radius AB intersects BC at points B and X . Moreover BX and CX have integer lengths. What is BC ? (A) 11 (B) 28 (C) 33 (D) 61 (E) 72

20. Let S be the set {1, 2, 3, . . . , 19}. For a, b ∈ S , de?ne a b to mean that either 0 < a ? b ≤ 9 or b ? a > 9. How many ordered triples (x, y, z ) of elements of S have the property that x y , y z , and z x ? (A) 810 (B) 855 (C) 900 (D) 950 (E) 988

2013 AMC12A Problems 21. Consider A = log (2013 + log (2012 + log (2011 + log(· · · + log (3 + log 2) · · ·)))). Which of the following intervals contains A ? (A) (log 2016, log 2017) (B) (log 2017, log 2018) (C) (log 2018, log 2019) (D) (log 2019, log 2020) (E) (log 2020, log 2021)

6

22. A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome n is also a n is chosen uniformly at random. What is the probability that 11 palindrome? (A) 8 25 (B) 33 100 (C) 7 20 (D) 9 25 (E) 11 30

√ √ 23. ABCD is a square of side length 3 + 1. Point P is on AC such that AP = 2. The square region bounded by ABCD is rotated 90? counterclockwise with center P , sweeping out a region whose area is 1 c (aπ + b), where a, b, and c are positive integers and gcd(a, b, c) = 1. What is a + b + c ? (A) 15 (B) 17 (C) 19 (D) 21 (E) 23

24. Three distinct segments are chosen at random among the segments whose endpoints are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area? (A) 553 715 (B) 443 572 (C) 111 143 (D) 81 104 (E) 223 286

25. Let f : C → C be de?ned by f (z ) = z 2 + iz + 1. How many complex numbers z are there such that Im(z ) > 0 and both the real and the imaginary parts of f (z ) are integers with absolute value at most 10 ? (A) 399 (B) 401 (C) 413 (D) 431 (E) 441

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 University of Nebraska, P.O. Box 81606 Lincoln, NE 68501-1606

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 2013 AIME

The 31st annual AIME will be held on Thursday, March 14, with the alternate on Wednesday, April 3. 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 42nd Annual USA Mathematical Olympiad (USAMO) on April 30-May 1, 2013. 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: amc.maa.org