AMC 12A&B 2011 Problems & Answers

USA AMC 12/AHSME 2011 • Art of Problem SolvingUSA AMC 12/AHSME 2011

Page 1of 5A

1

A cell phone plan costs dollars each month, plus cents per text message sent, plus cents for each minute used over

hours. In January Michelle sent text messages and talked for hours. How much did she have to pay?

2

There are 5 coins placed flat on a table according to the figure. What is the order of the coins from top to bottom?

3

A small bottle of shampoo can hold 35 milliliters of shampoo, whereas a large bottle can hold 500 milliliters of shampoo. Jasmine

wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?

4

At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of 12, 15, and 10 minutes per

day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders.

What is the average number of minutes run per day by these students?

5

Last summer of the birds living on Town Lake were geese,

percent of the birds that were not swans were geese?

6

The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They

scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than

their number of successful two-point shots. The team's total score was 61 points. How many free throws did they make?

7

A majority of the students in Ms. Demeanor's class bought penciles at the school bookstore. Each of these students bought

the same number of pencils, and this number was greater than . The cost of a pencil in cents was greater than the number of

pencils each student bought, and the total cost of all the pencils was . What was the cost of a pencil in cents?

8

In the eight-term sequence

is ?

9

At a twins and triplets convention, there were sets of twins and sets of triplets, all from different families. Each twin shook

hands with all the twins except his/her sibling and with half the triplets. Each triplet shook hands with all the triplets except

his/her siblings and half the twins. How many handshakes took place?

10

A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the

probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?

11

Circles , , and each have radius . Circles and share one point of tangency. Circle

midpoint of . What is the area inside circle but outside circle and circle ?

12

A power boat and a raft both left dock on a river and headed downstream. The raft drifted at the speed of the river current.

has a point of tangency with the

, the value of

is and the sum of any three consecutive terms is . What

were swans, were herons, and were ducks. What

USA AMC 12/AHSME 2011 • Art of Problem SolvingPage 2of 5The power boat maintained a constant speed with respect to the river. The power boat reached dock downriver, then

immediately turned and traveled back upriver. It eventually met the raft on the river hours after leaving dock How many

hours did it take the power boat to go from to ?

13

Triangle

intersects

14

Suppose and are single-digit positive integers chosen independently and at random. What is the probability that the point

lies above the parabola ?

15

The circular base of a hemisphere of radius rests on the base of a square pyramid of height . The hemisphere is tangent to

the other four faces of the pyramid. What is the edge-length of the base of the pyramid?

is to be assigned a color. There are colors to choose from, and the ends of each

16

Each vertex of convex pentagon

diagonal must have different colors. How many different colorings are possible?

17

Circles with radii

tangency?

, and are mutually externally tangent. What is the area of the triangle determined by the points of

has side-lengths

at and at

, , and

. What is the perimeter of

. The line through the incenter of

?

parallel to

18

Suppose that

19

At a competition with players, the number of players given elite status is equal to

. What is the maximum possible value of

?

Suppose that

20

Let

players are given elite status. What is the sum of the two smallest possible values of

, where , , and are integers. Suppose that

for some integer . What is ?

, ,

?

, and

21

Let , and for integers , let

of is nonempty, the domain of is . What is

22

Let be a square region and

emanating from that divide

partitional?

23

Let

which

such that , ,

24

Consider all quadrilaterals

circle that fits inside or on the boundary of such a quadrilateral?

has

25

Triangle

circumcenter of

?

, , , and . Let

, respectively. Assume that the area of the pentagon

, , and be the orthocenter, incenter, and

is the maximum possible. What is

, . What is the radius of the largest possible

and , where and are complex numbers. Suppose that and

?

for all for

an integer. A point in the interior of is called n-ray partitional if there are rays

into triangles of equal area. How many points are 100-ray partitional but not 60-ray

. If

?

is the largest value of for which the domain

is defined. What is the difference between the largest and smallest possible values of

USA AMC 12/AHSME 2011 • Art of Problem Solving

B

1

What is

Page 3of 5

2

Josanna's test scores to date are 90, 80, 70, 60, and 85. Her goal is to raise her test average at least 3 points with her next test.

What is the minimum test score she would need to accomplish this goal?

3

LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid

for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid dollars and

Bernardo had paid dollars, where . How many dollars must LeRoy give to Bernardo so that they share the costs

equally?

4

In multiplying two positive integers and , Ron reversed the digits of the two-digit number . His errorneous product was

What is the correct value of the product of and ?

5

Let be the second smallest positive integer that is divisible by every positive integer less than 7. What is the sum of the digits

of ?

6

Two tangents to a circle are drawn from a point . The points of contact

ratio . What is the degree measure of ?

7

Let and be two-digit positive integers with mean 60. What is the maximum value of the ratio ?

8

Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends

are semicircles. The track has width 6 meters, and it takes her 36 seconds longer to walk around the outside edge of the track

than around the inside edge. What is Keiko's speed in meters per second?

9

Two real numbers are selected independently at random from the interval [-20, 10]. What is the probability that the product of

those numbers is greater than zero?

10

Rectangle

measure of

11

A frog located at , with both and integers, makes successive jumps of length and always lands on points with integer

coordinates. Suppose that the frog starts at and ends at . What is the smallest possible number of jumps the frog

makes?

12

A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land

anywhere on the board. What is probability that the dart lands within the center square?

has

?

and . Point is chosen on side

so that . What is the degree

and divide the circle into arcs with lengths in the

.

USA AMC 12/AHSME 2011 • Art of Problem SolvingPage 4of 5

whose sum is . The pairwise positive differences of these numbers are

13

Brian writes down four integers

and . What is the sum of the possible values for ?

14

A segment through the focus

. What is ?

of a parabola with vertex is perpendicular to and intersects the parabola in points and

15

How many positive two-digit integers are factors of

has side length and . Region consists of all points inside the rhombus that are closer to vertex

16

Rhombus

than any of the other three vertices. What is the area of ?

17

Let

of the digits of

18

A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within

the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the

pyramid. What is the volume of this cube?

19

A lattice point in an -coordinate system is any point

for all

where both and are integers. The graph of

. What is the maximum possible value of ? passes

through no lattice point with

20

Triangle has

respectively. Let

, , and . The points

be the intersection of the circumcircles of

21

The arithmetic mean of two distinct positive integers and is a two-digit integer. The geometric mean of and is obtained

by reversing the digits of the arithmetic mean. What is ?

and . For , if

22

Let be a triangle with sides

the incircle of to the sides and , respectively, then

if it exists. What is the perimeter of the last triangle in the sequence ?

23

A bug travels in the coordinate plane, moving only along the lines that are parallel to the x-axis or y-axis. Let and

and and are the points of tangency of

is a triangle with side lengths and ,

and are the midpoints of

and . What is

, , and

?

such that

?

and for integers . What is the sum

?

USA AMC 12/AHSME 2011 • Art of Problem Solving. Consider all possible paths of the bug from to

lie on at least one of these paths?

of length at most

Page 5of 5. How many points with integer coordinates

24

Let . What is the minimum perimeter among all the 8-sided polygons in the complex

plane whose vertices are precisely the zeros of ?

25

For every and integers with odd, denote by the integer closest to . For every odd integer , let be the

probability that

for an integer randomly chosen from the interval

integers in the interval ?

. What is the minimum possible value of over the odd

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AMC 10/12Answers for the 2011 AMC 10 A&B / AMC 12 A&BThe AIME qualifying score for the 12A is 93.0;The AIME qualifying score for the 10A is 117.0The AIME qualifying score for the 12B is 97.5;The AIME qualifying score for the 10B is 117.02010 High School Directory 2010 Answers AMC 12 Esoterica Archive Administration HomeSchool Sliffe AwardsAAMC 10ADEDACCBCABBAABCBCCEDDCCDCMarch 4, 2011Q# 12ADEECCABCBBCDBEACDDCCACCCD

AMC 10BCEACEABBDBDADCEACEACBADBDBQ# 12BCECEACBADEBABDDCBABCDDCBD