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AMC B

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Question 1Not yet answered

Points out of 6


Points out of 6

Each third-grade classroom at Pearl Creek Elementary has students and pet rabbits. Howmany more students than rabbits are there in all of the third-grade classrooms?

Select one:

A

B

C

D

E

Leave blank (1.5 points)

A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle toits width is 2:1.

What is the area of the rectangle?

Select one:

A

B

C

D

E


18 2

4

(A) 48 (B) 56 (C) 64 (D) 72 (E) 80

(A) 50 (B) 100 (C) 125 (D) 150 (E) 200


Points out of 6


Points out of 6


Points out of 6

For a science project, Sammy observed a chipmunk and squirrel stashing acorns in holes. Thechipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes itdug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. Howmany acorns did the chipmunk hide?

Select one:

A

B

C

D

E


Suppose that the euro is worth 1.3 dollars. If Diana has 500 dollars and Etienne has 400 euros, bywhat percent is the value of Etienne's money greater that the value of Diana's money?

Select one:

A

B

C

D

E


Two integers have a sum of 26. when two more integers are added to the first two, the sum is 41.Finally, when two more integers are added to the sum of the previous 4 integers, the sum is 57.What is the minimum number of even integers among the 6 integers?

Select one:

A

B

C

D

E


(A) 30 (B) 36 (C) 42 (D) 48 (E) 54

(A) 2 (B) 4 (C) 6.5 (D) 8 (E) 13

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5


Points out of 6


Points out of 6

In order to estimate the value of where and are real numbers with , Xiaolirounded up by a small amount, rounded down by the same amount, and then subtracted herrounded values. Which of the following statements is necessarily correct?

(A) Her estimate is larger than .

(B) Her estimate is smaller than .

(C) Her estimate equals .

(D) Her estimate equals .

(E) Her estimate is .

Select one:

A

B

C

D

E


Small lights are hung on a string 6 inches apart in the order red, red, green, green, green, red, red,green, green, green, and so on continuing this pattern of 2 red lights followed by 3 green lights.How many feet separate the 3rd red light and the 21st red light?

Note: 1 foot is equal to 12 inches.

Select one:

A

B

C

D

E


x − y x y x > y > 0

x y

x − y

x − y

x − y

y − x

0

(A) 18 (B) 18.5 (C) 20 (D) 20.5 (E) 22.5


Points out of 6


Points out of 6


Points out of 6

A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert eachday is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in arow. There must be cake on Friday because of a birthday. How many different dessert menus forthe week are possible?

Select one:

A

B

C

D

E


It takes Clea 60 seconds to walk down an escalator when it is not moving, and 24 seconds when itis moving. How seconds would it take Clea to ride the escalator down when she is not walking?

Select one:

A

B

C

D

E


What is the area of the polygon whose vertices are the points of intersection of the curves and

Select one:

A

B

C

D

E


(A) 729 (B) 972 (C) 1024 (D) 2187 (E) 2304

(A) 36 (B) 40 (C) 42 (D) 48 (E) 52

+ = 25

x

2

y

2

(x − 4 + 9 = 81?

)

2

y

2

(A) 24 (B) 27 (C) 36 (D) 37.5 (E) 42


Points out of 6


Points out of 6


Points out of 6

In the equation below, and are consecutive positive integers, and , , and represent number bases:

What is ?

Select one:

A

B

C

D

E


How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all theones consecutive, or both?

Select one:

A

B

C

D

E


Two parabolas have equations and , where and areintegers, each chosen independently by rolling a fair six-sided die. What is the probability that theparabolas will have a least one point in common?

Select one:

A

B

C

D

E


A B A B A + B

+ = .132

A

43

B

69

A+B

A + B

(A) 9 (B) 11 (C) 13 (D) 15 (E) 17

(A) 190 (B) 192 (C) 211 (D) 380 (E) 382

y = + ax + b

x

2

y = + cx + d

x

2

a, b, c, d

(A) (B) (C) (D) (E) 1

1

2

25

36

5

6

31

36


Points out of 6


Points out of 6


Points out of 6

Bernardo and Silvia play the following game. An integer between and inclusive is selectedand given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the resultto Silvia. Whenever Silvia receives a number, she adds to it and passes the result to Bernardo.The winner is the last person who produces a number less than . Let be the smallest initialnumber that results in a win for Bernardo. What is the sum of the digits of ?

Select one:

A

B

C

D

E


Jesse cuts a circular disk of radius 12, along 2 radii to form 2 sectors, one with a central angle of120. He makes two circular cones using each sector to form the lateral surface of each cone. Whatis the ratio of the volume of the smaller cone to the larger cone?

Select one:

A

B

C

D

E


Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is likedby all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked bythose two girls but disliked by the third. In how many different ways is this possible?

Select one:

A

B

C

D

E


0 999

50

1000 N

N

(A) 7 (B) 8 (C) 9 (D) 10 (E) 11

(A) (B) (C) (D) (E)

1

8

1

4

10

−−

√

10

5

–

√

6

5

–

√

5

(A) 108 (B) 132 (C) 671 (D) 846 (E) 1105


Points out of 6


Points out of 6

Square lies in the first quadrant. Points and lie on lines , and , respectively. What is the sum of the coordinates of the center of the

square ?

Select one:

A

B

C

D

E


Let be a list of the first 10 positive integers such that for each either or or both appear somewhere before in the list. How many such lists are there?

Select one:

A

B

C

D

E


PQRS (3, 0), (5, 0), (7, 0), (13, 0)

SP ,RQ,PQ SR

PQRS

(A) 6 (B) 6.2 (C) 6.4 (D) 6.6 (E) 6.8

( , ,… , )a

1

a

2

a

10

2 ≤ i ≤ 10

+ 1a

i

− 1a

i

a

i

(A) 120 (B) 512 (C) 1024 (D) 181, 440 (E) 362, 880


Points out of 6


Points out of 6

A unit cube has vertices and . Vertices , , and areadjacent to , and for vertices and are opposite to each other. A regularoctahedron has one vertex in each of the segments , , , , , and

.

What is the octahedron's side length?

Select one:

A

B

C

D

E


A trapezoid has side lengths 3, 5, 7, and 11. The sums of all the possible areas of the trapezoid canbe written in the form of , where , , and are rational numbers and

and are positive integers not divisible by the square of any prime. What is the greatestinteger less than or equal to ?

Select one:

A

B

C

D

E


, , , , , , ,P

1

P

2

P

3

P

4

P

′

1

P

′

2

P

′

3

P

′

4

P

2

P

3

P

4

P

1

1 ≤ i ≤ 4, P

i

P

′

i

P

1

P

2

P

1

P

3

P

1

P

4

P

′

1

P

′

2

P

′

1

P

′

3

P

′

1

P

′

4

(A) (B) (C) (D) (E)

3 2

–

√

4

7 6

–

√

16

5

–

√

2

2 3

–

√

3

6

–

√

2

+ +

r

1

n

1

−−

√

r

2

n

2

−−

√

r

3

r

1

r

2

r

3

n

1

n

2

+ + + +r

1

r

2

r

3

n

1

n

2

(A) 57 (B) 59 (C) 61 (D) 63 (E) 65


Points out of 6


Points out of 6

Square is inscribed in equiangular hexagon with on , on , and on . Suppose that , and .

What is the side-length of the square?

Select one:

A

B

C

D

E


A bug travels from A to B along the segments in the hexagonal lattice pictured below. Thesegments marked with an arrow can be traveled only in the direction of the arrow, and the bugnever travels the same segment more than once.

How many different paths are there?

Select one:

A

B

C

D

E


AXYZ ABCDEF X BC

¯ ¯¯̄¯̄ ¯̄

Y DE

¯ ¯¯̄¯̄¯̄

Z EF

¯ ¯¯̄¯̄¯̄

AB = 40 EF = 41( − 1)3

–

√

(A) 29 (B) + (C) 20 + 16 (D) 20 + 13 (E) 213

–

√

21

2

2

–

√

41

2

3

–

√ 3

–

√ 2

–

√

3

–

√ 6

–

√

(A) 2112 (B) 2304 (C) 2368 (D) 2384 (E) 2400


Points out of 6


Points out of 6

Consider all polynomials of a complex variable, , where and are integers, , and the polynomial has a zero with

What is the sum of all values over all the polynomials with these properties?

Select one:

A

B

C

D

E


Define the function on the positive integers by setting and if isthe prime factorization of , then

For every , let . For how many in the range isthe sequence unbounded?

Note: A sequence of positive numbers is unbounded if for every integer , there is a member ofthe sequence greater than .

Select one:

A

B

C

D

E


P (z) = 4 + a + b + cz + dz

4

z

3

z

2

a, b, c, d 0 ≤ d ≤ c ≤ b ≤ a ≤ 4 z

0

| | = 1.z

0

P (1)

(A) 84 (B) 92 (C) 100 (D) 108 (E) 120

f

1

(1) = 1f

1

n = ⋯p

e

1

1

p

e

2

2

p

e

k

k

n > 1

(n) = ( + 1 ( + 1 ⋯( + 1 .f

1

p

1

)

−1e

1

p

2

)

−1e

2

p

k

)

−1e

k

m ≥ 2 (n) = ( (n))f

m

f

1

f

m−1

N 1 ≤ N ≤ 400

( (N), (N), (N),…)f

1

f

2

f

3

B

B

(A) 15 (B) 16 (C) 17 (D) 18 (E) 19


Points out of 6

Let . Let bethe set of all right triangles whose vertices are in . For every right triangle withvertices , , and in counter-clockwise order and right angle at , let .

What is

Select one:

A

B

C

D

E


S = {(x, y) : x ∈ {0, 1, 2, 3, 4}, y ∈ {0, 1, 2, 3, 4, 5}, and (x, y) ≠ (0, 0)} T

S t = △ABC

A B C A f(t) = tan(∠CBA)

f(t)?∏

t∈T

(A) 1 (B) (C) (D) 6 (E)

625

144

125

24

625

24

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AMC B - ziml.areteem.org