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2014AMC12-B

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.? 2014 Mathematical Association of America

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.

**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

AMC 12 B DO NOT OPEN UNTIL wEDNEsDAy, fEbrUAry 19, 2014The American Mathematics Competitions

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1.Leah has13coins,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 buy30balloons.When

he arrived he discovered that the store had a special sale on balloons:buy1

balloon at the regular price and get a second at1

3o?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 next20miles on

pavement,and the remaining one-?fth on a dirt road.In miles,how long was Randy’s trip?

(A)30(B)400

(C)

(D)40(E)

300

4.Susie pays for4mu?ns and3bananas.Calvin spends twice as much paying

for2mu?ns and16bananas.A mu?n is how many times as expensive as a banana?

(A)3

(B)

(C)

(D)2(E)

5.Doug constructs a square window using8equal-size panes of glass,as shown.

The ratio of the height to width for each pane is5:2,and the borders around and between the panes are2inches wide.In inches,what is the side length of the square window?

(A)26(B)28(C)30(D)32(E)34

6.Ed and Ann both have lemonade with their lunch.Ed orders the regular size.

Ann gets the large lemonade,which is50%more than the regular.After both

consume3

4of their drinks,Ann gives Ed a third of what she has left,and2

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(E)50

7.For how many positive integers n is n

30?n

also a positive integer?

(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

(A)2(B)4(C)7(D)8(E)9

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

10.Danica drove her new car on a trip for a whole number of hours,averaging55

miles per hour.At the beginning of the trip,abc miles was displayed on the odometer,where abc is a3-digit number with a≥1and 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 of11positive integers has a mean of10,a median of9,and a unique mode

of8.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 than5,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 with1

positive area has side lengths1,a,and b or1

b ,1

,and1.What is the smallest

possible value of b?

(A)3+

√3

(B)

(C)

√5

(D)

√6

(E)3

14.A rectangular box has a total surface area of94square inches.The sum of the

lengths of all its edges is48inches.What is the sum of the lengths in inches of all of its interior diagonals?

(A)8

√3(B)10√2(C)16√3(D)20√2(E)40√2

15.When p= 6

k=1

k ln k,the number e p is an integer.What is the largest power

of2that is a factor of e p?

(A)212(B)214(C)216(D)218(E)220

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

17.Let P be the parabola with equation y =x 2and 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

(A)1(B)26(C)40(D)52(E)80

18.The numbers 1,2,3,4,5are to be arranged in a circle.An arrangement is

bad if it is not true that for every n from 1to 15one 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)32(B)1+√52(C)√3(D)2(E)3+√52

20.For how many positive integers x is log 10(x ?40)+log 10(60?x )<2?

(A)10(B)18(C)19(D)20(E)in?nitely many

21.In the?gure,ABCD is a square of side length1.The rectangles JKHG and

EBCF are congruent.What is BE?

(A)1

(

√6?2)(B)1

√3(D)√3

(E)1?

√2

22.In a small pond there are eleven lily pads in a row labeled0through10.A frog

is sitting on pad1.When the frog is on pad N,0

pad N?1with probability N

10and to pad N+1with probability1?N

.Each

jump is independent of the previous jumps.If the frog reaches pad0it will be eaten by a patiently waiting snake.If the frog reaches pad10it will exit the pond,never to return.What is the probability that the frog will escape being eaten by the snake?

(A)32

(B)

161

384

(C)

146

(D)

(E)

23.The number2017is prime.Let S= 62

k=0

2014

.What is the remainder when

S is divided by2017?

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

25.What is the sum of all positive real solutions x to the equation

2cos(2x)

cos(2x)?cos

2014π2

=cos(4x)?1?

(A)π(B)810π(C)1008π(D)1080π(E)1800π

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@https://www.wendangku.net/doc/0714297172.html,

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. T op-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

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