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AMC 8 试题汇编(1999-2012)


USA
AMC 8 1999

1 (6?3) + 4 ? (2 ? 1) = 5. To make this statement true, the question mark between the 6 and the 3 should be replaced by (A) ÷ (B) × (C) + (D) ? (E) None of thes

e

2 What is the degree measure of the smaller angle formed by the hands of a clock at 10 o’clock? (A) 30 (B) 45 (C) 60 (D) 75 (E) 90

3 Which triplet of numbers has a sum NOT equal to 1? (A) (1/2, 1/3, 1/6) (B) (2, ?2, 1) (C) (0.1, 0.3, 0.6) (D) (1.1, ?2.1, 1.0) (E) (?3/2, ?5/2, 5)

4 The diagram shows the miles traveled by bikers Alberto and Bjorn. After four hours, about how many more miles has Alberto biked than Bjorn?

75 M 60 I L 45 E 30 S 15 0

o ert b Al rn Bjo
0 1 2 3 4 HOUR S 5

(A) 15

(B) 20

(C) 25

(D) 30

(E) 35

5 A rectangular garden 50 feet long and 10 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden? (A) 100 (B) 200 (C) 300 (D) 400 (E) 500

6 Bo, Coe, Flo, Joe, and Moe have di?erent amounts of money. Neither Jo nor Bo has as much money as Flo. Both Bo and Coe have more than Moe. Jo has more than Moe, but less than Bo. Who has the least amount of money? (A) Bo (B) Coe (C) Flo (D) Joe (E) Moe

7 The third exit on a highway is located at milepost 40 and the tenth exit is at milepost 160. There is a service center on the highway located three-fourths of the way from the third exit to the tenth exit. At what milepost would you expect to ?nd this service center? (A) 90 (B) 100 (C) 110 (D) 120 (E) 130

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USA
AMC 8 1999

8 Six squares are colored, front and back, (R = red, B = blue, O = orange, Y = yellow, G = green, and W = white). They are hinged together as shown, then folded to form a cube. The face opposite the white face is

R

B G Y W O

(A) B

(B) G

(C) O

(D) R

(E) Y

9 Three ?ower beds overlap as shown. Bed A has 500 plants, bed B has 450 plants, and bed C has 350 plants. Beds A and B share 50 plants, while beds A and C share 100. The total number of plants is

A C B

(A) 850

(B) 1000

(C) 1150

(D) 1300

(E) 1450

10 A complete cycle of a tra?c light takes 60 seconds. During each cycle the light is green for 25 seconds, yellow for 5 seconds, and red for 30 seconds. At a randomly chosen time, what is the probability that the light will NOT be green? (A)
1 4

(B)

1 3

(C)

5 12

(D)

1 2

(E)

7 12

11 Each of the ?ve numbers 1, 4, 7, 10, and 13 is placed in one of the ?ve squares so that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column. The largest possible value for the horizontal or vertical sum is

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USA
AMC 8 1999

(A) 20

(B) 21

(C) 22

(D) 24

(E) 30

12 The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is 11 : 4 . To the nearest whole percent, what percent of its games did the team lose? (A) 24 (B) 27 (C) 36 (D) 45 (E) 73

13 The average age of the 40 members of a computer science camp is 17 years. There are 20 girls, 15 boys, and 5 adults. If the average age of the girls is 15 and the average age of the boys is 16, what is the average age of the adults? (A) 26 (B) 27 (C) 28 (D) 29 (E) 30

14 In trapezoid ABCD , the sides AB and CD are equal. The perimeter of ABCD is

B 3 A

8 16

C D

(A) 27

(B) 30

(C) 32

(D) 34

(E) 48

15 Bicycle license plates in Flatville each contain three letters. The ?rst is chosen from the set {C, H, L, P, R}, the second from {A, I, O}, and the third from {D, M, N, T }. When Flatville needed more license plates, they added two new letters. The new letters may both be added to one set or one letter may be added to one set and one to another set. What

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USA
AMC 8 1999

is the largest possible number of ADDITIONAL license plates that can be made by adding two letters? (A) 24 (B) 30 (C) 36 (D) 40 (E) 60

16 Tori’s mathematics test had 75 problems: 10 arithmetic, 30 algebra, and 35 geometry problems. Although she answered 70% of the arithmetic, 40% of the algebra, and 60% of the geometry problems correctly, she did not pass the test because she got less than 60% of the problems right. How many more problems would she have needed to answer correctly to earn a 60% passing grade? (A) 1 (B) 5 (C) 7 (D) 9 (E) 11

17 Problems 17, 18, and 19 refer to the following: At Central Middle School the 108 students who take the AMC8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie’s Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: 1.5 cups ?our, 2 eggs, 3 tablespoons butter, 3/4 cups sugar, and 1 package of chocolate drops. They will make only full recipes, not partial recipes. Walter can buy eggs by the half-dozen. How many half-dozens should he buy to make enough cookies? (Some eggs and some cookies may be left over.) (A) 1 (B) 2 (C) 5 (D) 7 (E) 15

18 Problems 17, 18, and 19 refer to the following: At Central Middle School the 108 students who take the AMC8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie’s Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: 1.5 cups ?our, 2 eggs, 3 tablespoons butter, 3/4 cups sugar, and 1 package of chocolate drops. They will make only full recipes, not partial recipes. They learn that a big concert is scheduled for the same night and attendance will be down 25%. How many recipes of cookies should they make for their smaller party? (A) 6 (B) 8 (C) 9 (D) 10 (E) 11

19 Problems 17, 18, and 19 refer to the following: At Central Middle School the 108 students who take the AMC8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie’s Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: 1.5 cups ?our, 2 eggs, 3 tablespoons butter, 3/4 cups sugar, and 1 package of chocolate drops. They will make only full recipes, not partial recipes.

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

Page 4

USA
AMC 8 1999

The drummer gets sick. The concert is cancelled. Walter and Gretel must make enough pans of cookies to supply 216 cookies. There are 8 tablespoons in a stick of butter. How many sticks of butter will be needed? (Some butter may be left over, of course.) (A) 5 (B) 6 (C) 7 (D) 8 (E) 9

20 Figure 1 is called a ”stack map.” The numbers tell how many cubes are stacked in each position. Fig. 2 shows these cubes, and Fig. 3 shows the view of the stacked cubes as seen from the front. Which of the following is the front view for the stack map in Fig. 4?

3 2

4 1 Figure 2 Figure 3

2 1

2 3

4 1

Figure 1

Figure 4

(A)

(B)

(C)

(D)

(E)

21 The degree measure of angle A is

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Page 5

USA
AMC 8 1999

A 100? 110?

40

?

(A) 20

(B) 30

(C) 35

(D) 40

(E) 45

22 In a far-o? land three ?sh can be traded for two loaves of bread and a loaf of bread can be traded for four bags of rice. How many bags of rice is one ?sh worth? (A)
3 8

(B)

1 2

(C)

3 4

(D) 2 2 3

1 (E) 3 3

23 Square ABCD has sides of length 3. Segments CM and CN divide the square’s area into three equal parts. How long is segment CM ?

B

C

M

A
√ √ √

N


D
√ 15

(A)

10

(B)

12

(C)

13

(D)

14

(E)

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Page 6

USA
AMC 8 1999

24 When 19992000 is divided by 5 , the remainder is (A) 4 (B) 3 (C) 2 (D) 1 (E) 0

25 Points B ,D , and J are midpoints of the sides of right triangle ACG . Points K , E , I are midpoints of the sides of triangle , etc. If the dividing and shading process is done 100 times (the ?rst three are shown) and AC = CG = 6, then the total area of the shaded triangles is nearest

G H I L J K F E D

A

B

C

(A) 6

(B) 7

(C) 8

(D) 9

(E) 10

The problems on this page are copyrighted by the Mathematical Association of America’s American Mathematics Competitions.

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Page 7

USA
AMC 8 2000

1 Aunt Anna is 42 years old. Caitlin is 5 years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin? (A) 15 (B) 16 (C) 17 (D) 21 (E) 37

2 Which of these numbers is less than its reciprocal? (A) ? 2 (B) ? 1 (C) 0 (D) 1 (E) 2
5 3

3 How many whole numbers lie in the interval between (A) 2 (B) 3 (C) 4 (D) 5

and 2π ?

(E) in?nitely many

4 In 1960 only 5% of the working adults in Carlin City worked at home. By 1970 the ”at-home” work force increased to 8%. In 1980 there were approximately 15% working at home, and in 1990 there were 30%. The graph that best illustrates this is

(A)

% 30 20 10 1960 1970 1980 1990 % 30 20 10 1960 1970 1980 1990

(B)

% 30 20 10 1960 1970 1980 1990 % 30 20 10 1960 1970 1980 1990

(C)

% 30 20 10 1960 1970 1980 1990

(D)

(E)

5 Each principal of Lincoln High School serves exactly one 3-year term. What is the maximum number of principals this school could have during an 8-year period? (A) 2 (B) 3 (C) 4 (D) 5 (E) 8

6 Figure ABCD is a square. Inside this square three smaller squares are drawn with the side lengths as labeled. The area of the shaded L-shaped region is

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Page 1

USA
AMC 8 2000

D

1 1

A

3 3 1 C B 1

(A) 7

(B) 10

(C) 12.5

(D) 14

(E) 15

7 What is the minimum possible product of three di?erent numbers of the set {?8, ?6, ?4, 0, 3, 5, 7}? (A) ? 336 (B) ? 280 (C) ? 210 (D) ? 192 (E) 0

8 Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). The total number of dots NOT visible in this view is

(A) 21

(B) 22

(C) 31

(D) 41

(E) 53

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USA
AMC 8 2000

9 Three-digit powers of 2 and 5 are used in this ”cross-number” puzzle. What is the only possible digit for the outlined square? ACROSS 2. 2 m DOWN 1. 5n

1

2

(A) 0

(B) 2

(C) 4

(D) 6

(E) 8

10 Ara and Shea were once the same height. Since then Shea has grown 20% while Ara has grow half as many inches as Shea. Shea is now 60 inches tall. How tall, in inches, is Ara now? (A) 48 (B) 51 (C) 52 (D) 54 (E) 55

11 The number 64 has the property that it is divisible by its units digit. How many whole numbers between 10 and 50 have this property? (A) 15 (B) 16 (C) 17 (D) 18 (E) 20

12 A block wall 100 feet long and 7 feet high will be constructed using blocks that are 1 foot high and either 2 feet long or 1 foot long (no blocks may be cut). The vertical joins in the blocks must be staggered as shown, and the wall must be even on the ends. What is the smallest number of blocks needed to build this wall?

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Page 3

USA
AMC 8 2000

(A) 344

(B) 347

(C) 350

(D) 353

(E) 356

13 In triangle CAT , we have ∠ACT = ∠AT C and ∠CAT = 36? . If T R bisects ∠AT C , then ∠CRT =

A

R

C

T

(A) 36?

(B) 54?

(C) 72?

(D) 90?

(E) 108?

14 What is the units digit of 1919 + 9999 ? (A) 0 (B) 1 (C) 2 (D) 8 (E) 9

15 Triangles ABC , ADE , and EF G are all equilateral. Points D and G are midpoints of AC and AE , respectively. If AB = 4, what is the perimeter of ?gure ABCDEF G?

C D G B A E F

(A) 12

(B) 13

(C) 15

(D) 18

(E) 21

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Page 4

USA
AMC 8 2000

16 In order for Mateen to walk a kilometer (1000m) in his rectangular backyard, he must walk the length 25 times or walk its perimeter 10 times. What is the area of Mateen’s backyard in square meters? (A) 40 (B) 200 (C) 400 (D) 500 (E) 1000 a2 . Determine [(1 ? 2) ? 3] ? b

17 The operation ? is de?ned for all nonzero numbers by a ? b = [1 ? (2 ? 3)]. 1 2 (B) ? (A) ? 3 4 1 4 2 3

(C) 0

(D)

(E)

18 Consider these two geoboard quadrilaterals. Which of the following statements is true?

I

II

(A) The area of quadrilateral I is more than the area of quadrilateral II. (B) The area of quadrilateral I is less than the area of quadrilateral II. (C) The quadrilaterals have the same area and the same perimeter. (D) The quadrilaterals have the same area, but the perimeter of I is more than the perimeter of II. (E) The quadrilaterals have the same area, but the perimeter of I is less than the perimeter of II. 19 Three circular arcs of radius 5 units bound the region shown. Arcs AB and AD are quartercircles, and arc BCD is a semicircle. What is the area, in square units, of the region?

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Page 5

USA
AMC 8 2000

C

B

D

A

(A) 25

(B) 10 + 5π

(C) 50

(D) 50 + 5π

(E) 25π

20 You have nine coins: a collection of pennies, nickels, dimes, and quarters having a total value of $1.02, with at least one coin of each type. How many dimes must you have? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5

21 Keiko tosses one penny and Ephraim tosses two pennies. The probability that Ephraim gets the same number of heads that Keiko gets is 1 3 1 2 3 (A) (B) (C) (D) (E) 4 8 2 3 4 22 A cube has edge length 2. Suppose that we glue a cube of edge length 1 on top of the big cube so that one of its faces rests entirely on the top face of the larger cube. The percent increase in the surface area (sides, top, and bottom) from the original cube to the new solid formed is closest to

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Page 6

USA
AMC 8 2000

(A) 10

(B) 15

(C) 17

(D) 21

(E) 25

23 There is a list of seven numbers. The average of the ?rst four numbers is 5, and the average of the last four numbers is 8. If the average of all seven numbers is 6 4 7 , then the number common to both sets of four numbers is
3 (A) 5 7

(B) 6

4 (C) 6 7

(D) 7

(E) 7 3 7

24 If ∠A = 20? and ∠AF G = ∠AGF , then ∠B + ∠D =

B A F D E G C

(A) 48?

(B) 60?

(C) 72?

(D) 80?

(E) 90?

25 The area of rectangle ABCD is 72. If point A and the midpoints of BC and CD are joined to form a triangle, the area of that triangle is

A

B

D

C

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USA
AMC 8 2000

(A) 21

(B) 27

(C) 30

(D) 36

(E) 40

The problems on this page are copyrighted by the Mathematical Association of America’s American Mathematics Competitions.

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

Page 8

USA
AMC 8 2001

1 Casey’s shop class is making a golf trophy. He has to paint 300 dimples on a golf ball. If it takes him 2 seconds to paint one dimple, how many minutes will he need to do his job? (A) 4 (B) 6 (C) 8 (D) 10 (E) 12

2 I’m thinking of two whole numbers. Their product is 24 and their sum is 11. What is the larger number? (A) 3 (B) 4 (C) 6 (D) 8 (E) 12

3 Granny Smith has $63. Elberta has $2 more than Anjou and Anjou has one-third as much as Granny Smith. How many dollars does Elberta have? (A) 17 (B) 18 (C) 19 (D) 21 (E) 23

4 The digits 1, 2, 3, 4 and 9 are each used once to form the smallest possible even ?ve-digit number. The digit in the tens place is (A) 1 (B) 2 (C) 3 (D) 4 (E) 9

5 On a dark and stormy night Snoopy suddenly saw a ?ash of lightning. Ten seconds later he heard the sound of thunder. The speed of sound is 1088 feet per second and one mile is 5280 feet. Estimate, to the nearest half-mile, how far Snoopy was from the ?ash of lightning. (A) 1 (B) 1 1 2 (C) 2 (D) 2 1 2 (E) 3

6 Six trees are equally spaced along one side of a straight road. The distance from the ?rst tree to the fourth is 60 feet. What is the distance in feet between the ?rst and last trees? (A) 90 (B) 100 (C) 105 (D) 120 (E) 140

7 Problems 7, 8 and 9 are about these kites. To promote her school’s annual Kite Olympics, Genevieve makes a small kite and a large kite for a bulletin board display. The kites look like the one in the diagram. For her small kite Genevieve draws the kite on a one-inch grid. For the large kite she triples both the height and width of the entire grid.

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USA
AMC 8 2001

What is the number of square inches in the area of the small kite? (A) 21 (B) 22 (C) 23 (D) 24 (E) 25

8 Problems 7, 8 and 9 are about these kites.

Genevieve puts bracing on her large kite in the form of a cross connecting opposite corners of the kite. How many inches of bracing material does she need? (A) 30 (B) 32 (C) 35 (D) 38 (E) 39

9 Problems 7, 8 and 9 are about these kites.

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USA
AMC 8 2001

The large kite is covered with gold foil. The foil is cut from a rectangular piece that just covers the entire grid. How many square inches of waste material are cut o? from the four corners? (A) 63 (B) 72 (C) 180 (D) 189 (E) 264

10 A collector o?ers to buy state quarters for 2000% of their face value. At that rate how much will Bryden get for his four state quarters? (A) 20 dollars (B) 50 dollars (C) 200 dollars (D) 500 dollars (E) 2000 dollars

11 Points A, B, C and D have these coordinates: A(3, 2), B (3, ?2), C (?3, ?2) and D(?3, 0). The area of quadrilateral ABCD is

(A) 12

(B) 15

(C) 18

(D) 21

(E) 24

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USA
AMC 8 2001

12 If a ? b = (A) 4

a+b a?b

, then (6 ? 4) ? 3 = = (C) 15 (D) 30 (E) 72

(B) 13

13 Of the 36 students in Richelle’s class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle’s pie graph showing this data, how many degrees should she use for cherry pie? (A) 10 (B) 20 (C) 30 (D) 50 (E) 72

14 Tyler has entered a bu?et line in which he chooses one kind of meat, two di?erent vegetables and one dessert. If the order of food items is not important, how many di?erent meals might he choose? - Meat: beef, chicken, pork - Vegetables: baked beans, corn, potatoes, tomatoes - Dessert: brownies, chocolate cake, chocolate pudding, ice cream (A) 4 (B) 24 (C) 72 (D) 80 (E) 144

15 Homer began peeling a pile of 44 potatoes at the rate of 3 potatoes per minute. Four minutes later Christen joined him and peeled at the rate of 5 potatoes per minute. When they ?nished, how many potatoes had Christen peeled? (A) 20 (B) 24 (C) 32 (D) 33 (E) 40

16 A square piece of paper, 4 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle?

(A)

1 3

(B)

1 2

(C)

3 4

(D)

4 5

(E)

5 6

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Page 4

USA
AMC 8 2001

17 For the game show Who Wants To Be A Millionaire?, the dollar values of each question are shown in the following table (where K = 1000).

Question Value

1 100

2 200

3 300

4 500

5 1K

6 2K

7 4K

8 8K

9 16K

10 32K

11 64K

12 125K

13 250K

14 500K

15 1000K

Between which two questions is the percent increase of the value the smallest? (A) From 1 to 2 (B) From 2 to 3 (C) From 3 to 4 (D) From 11 to 12 (E) From 14 to 15

18 Two dice are thrown. What is the probability that the product of the two numbers is a multiple of 5? (A)
1 36

(B)

1 18

(C)

1 6

(D)

11 36

(E)

1 3

19 Car M traveled at a constant speed for a given time. This is shown by the dashed line. Car N traveled at twice the speed for the same distance. If Car N’s speed and time are shown as solid line, which graph illustrates this?

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USA
AMC 8 2001

(A) s p e e d N

(B) s p e e d N

(C) s p e e d

M

M

M N

time

time

time

(D) s p e e d N

(E) s p e e d

M

M N

time

time

20 Kaleana shows her test score to Quay, Marty and Shana, but the others keep theirs hidden. Quay thinks, ”At least two of us have the same score.” Marty thinks, ”I didn’t get the lowest score.” Shana thinks, ”I didn’t get the highest score.” List the scores from lowest to highest for Marty (M), Quay (Q) and Shana (S). (A) S,Q,M (B) Q,M,S (C) Q,S,M (D) M,S,Q (E) S,M,Q

21 The mean of a set of ?ve di?erent positive integers is 15. The median is 18. The maximum possible value of the largest of these ?ve integers is (A) 19 (B) 24 (C) 32 (D) 35 (E) 40

22 On a twenty-question test, each correct answer is worth 5 points, each unanswered question is worth 1 point and each incorrect answer is worth 0 points. Which of the following scores is NOT possible? (A) 90 (B) 91 (C) 92 (D) 95 (E) 97

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Page 6

USA
AMC 8 2001

23 Points R, S and T are vertices of an equilateral triangle, and points X, Y and Z are midpoints of its sides. How many noncongruent triangles can be drawn using any three of these six points as vertices?

S

Y

Z

R

X

T

(A) 1

(B) 2

(C) 3

(D) 4

(E) 20

24 Each half of this ?gure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. There are 2 red-white pairs. How many white pairs coincide?

(A) 4

(B) 5

(C) 6

(D) 7

(E) 9

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Page 7

USA
AMC 8 2001

25 There are 24 four-digit whole numbers that use each of the four digits 2, 4, 5 and 7 exactly once. Only one of these four-digit numbers is a multiple of another one. Which of the following is it? (A) 5724 (B) 7245 (C) 7254 (D) 7425 (E) 7542

The problems on this page are copyrighted by the Mathematical Association of America’s American Mathematics Competitions.

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Page 8

USA
AMC 8 2002

1 A circle and two distinct lines are drawn on a sheet of paper. What is the largest possible number of points of intersection of these ?gures? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6

2 How many di?erent combinations of 5 bills and 2 bills can be used to make a total of 17? Order does not matter in this problem. (A) 2 (B) 3 (C) 4 (D) 5 (E) 6

3 What is the smallest possible average of four distinct positive even integers? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7

4 The year 2002 is a palindrome (a number that reads the same from left to right as it does from right to left). What is the product of the digits of the next year after 2002 that is a palindrome? (A) 0 (B) 4 (C) 9 (D) 16 (E) 25

5 Carlos Montado was born on Saturday, November 9, 2002. On what day of the week will Carlos be 706 days old? (A) Monday (B) Wednesday (C) Friday (D) Saturday (E) Sunday

6 A birdbath is designed to over?ow so that it will be self-cleaning. Water ?ows in at the rate of 20 milliliters per minute and drains at the rate of 18 milliliters per minute. One of these graphs shows the volume of water in the birdbath during the ?lling time and continuing into the over?ow time. Which one is it?

Volume

Volume

Volume

Volume

Time A

Time B

Time C

Time D

Volume Time E
Page 1

(A) A

(B) B

(C) C

(D) D

(E) E

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USA
AMC 8 2002

7 The students in Mrs. Sawyer’s class were asked to do a taste test of ?ve kinds of candy. Each student chose one kind of candy. A bar graph of their preferences is shown. What percent of her class chose candy E?

SWEET TOOTH 8 7 6 5 4 3 2 1 0 Number of students

A

B C D E Kinds of candy

(A) 5

(B) 12

(C) 15

(D) 16

(E) 20

8 Juan’s Old Stamping Grounds Juan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and France, 6 cents each, Peru 4 cents each, and Spain 5 cents each. (Brazil and Peru are South American countries and France and Spain are in Europe.)

Number of Stamps by Decade
Country Brazil France Peru Spain 50s 4 8 6 3 60s 7 4 4 9 70s 12 12 6 13 80s 8 15 10 9

Juan’s Stamp Collection

How many of his European stamps were issued in the ’80s? (A) 9 (B) 15 (C) 18 (D) 24 (E) 42

9 Juan’s Old Stamping Grounds Juan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop

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were: Brazil and France, 6 cents each, Peru 4 cents each, and Spain 5 cents each. (Brazil and Peru are South American countries and France and Spain are in Europe.)

Number of Stamps by Decade
Country Brazil France Peru Spain 50s 4 8 6 3 60s 7 4 4 9 70s 12 12 6 13 80s 8 15 10 9

Juan’s Stamp Collection

In dollars and cents, how much did his South American stampes issued before the ’70s cost him? (A) $0.40 (B) $1.06 (C) $1.80 (D) $2.38 (E) $2.64

10 Juan’s Old Stamping Grounds Juan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and France, 6 cents each, Peru 4 cents each, and Spain 5 cents each. (Brazil and Peru are South American countries and France and Spain are in Europe.)

Number of Stamps by Decade
Country Brazil France Peru Spain 50s 4 8 6 3 60s 7 4 4 9 70s 12 12 6 13 80s 8 15 10 9

Juan’s Stamp Collection

The average price of his ’70s stamps is closest to (A) 3.5 cents (B) 4 cents (C) 4.5 cents (D) 5 cents (E) 5.5 cents

11 A sequence of squares is made of identical square tiles. The edge of each square is one tile length longer than the edge of the previous square. The ?rst three squares are shown. How many more tiles does the seventh square require than the sixth?

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(A) 11

(B) 12

(C) 13

(D) 14

(E) 15

12 A board game spinner is divided into three regions labeled A, B and C . The probability 1 of the arrow stopping on region A is 1 3 and on region B is 2 . The probability of the arrow stopping on region C is: (A)
1 12

(B)

1 6

(C)

1 5

(D)

1 3

(E)

2 5

13 For his birthday, Bert gets a box that holds 125 jellybeans when ?lled to capacity. A few weeks later, Carrie gets a larger box full of jellybeans. Her box is twice as high, twice as wide and twice as long as Bert’s. Approximately, how many jellybeans did Carrie get? (A) 250 (B) 500 (C) 625 (D) 750 (E) 1000

14 A merchant o?ers a large group of items at 30% o?. Later, the merchant takes 20% o? these sale prices and claims that the ?nal price of these items is 50% o? the original price. The total discount is (A) 35% (B) 44% (C) 50% (D) 56% (E) 60%

15 Which of the following polygons has the largest area?

A

B

C

D

E

(A)A

(B) B

(C) C

(D) D

(E) E

16 Right isosceles triangles are constructed on the sides of a 3-4-5 right triangle, as shown. A capital letter represents the area of each triangle. Which one of the following is true?

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Z

5

X W
4 3

Y

(A) X + Z = W + Y (B) W + X = Z 1 ( Y + Z ) (E) X + Y = Z 2

(C) 3X + 4Y = 5Z

(D) X + W =

17 In a mathematics contest with ten problems, a student gains 5 points for a correct answer and loses 2 points for an incorrect answer. If Olivia answered every problem and her score was 29, how many correct answers did she have? (A) 5 (B) 6 (C) 7 (D) 8 (E) 9

18 Gage skated 1 hr 15 min each day for 5 days and 1 hr 30 min each day for 3 days. How long would he have to skate the ninth day in order to average 85 minutes of skating each day for the entire time? (A) 1 hr (B) 1 hr 10 min (C) 1 hr 20 min (D) 1 hr 40 min (E) 2 hr

19 How many whole numbers between 99 and 999 contain exactly one 0? (A) 72 (B) 90 (C) 144 (D) 162 (E) 180

20 The area of triangle XY Z is 8 square inches. Points A and B are midpoints of congruent segments XY and XZ . Altitude XC bisects Y Z . What is the area (in square inches) of the shaded region?

X A Y B Z

C

(A) 1 1 2

(B) 2

1 (C) 2 2

(D) 3

(E) 3 1 2

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21 Harold tosses a nickel four times. The probability that he gets at least as many heads as tails is (A)
5 16

(B)

3 8

(C)

1 2

(D)

5 8

(E)

11 16

22 Six cubes, each an inch on an edge, are fastened together, as shown. Find the total surface area in square inches. Include the top, bottom and sides.

(A) 18

(B) 24

(C) 26

(D) 30

(E) 36

23 A portion of a corner of a tiled ?oor is shown. If the entire ?oor is tiled in this way and each of the four corners looks like this one, then what fraction of the tiled ?oor is made of darker tiles?

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(A)

1 3

(B)

4 9

(C)

1 2

(D)

5 9

(E)

5 8

24 Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percent of the blend is pear juice? (A) 30 (B) 40 (C) 50 (D) 60 (E) 70

25 Loki, Moe, Nick and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-?fth of his money, Loki gave Ott one-fourth of his money and Nick gave Ott one-third of his money. Each gave Ott the same amount of money. What fractional part of the group’s money does Ott now have? (A)
1 10

(B)

1 4

(C)

1 3

(D)

2 5

(E)

1 2

The problems on this page are copyrighted by the Mathematical Association of America’s American Mathematics Competitions.

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

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1 Jamie counted the number of edges of a cube, Jimmy counted the number of corners, and Judy counted the number of faces. They then added the three numbers. What was the resulting sum? (A) 12 (B) 16 (C) 20 (D) 22 (E) 26

2 Which of the following numbers has the smallest prime factor? (A) 55 (B) 57 (C) 58 (D) 59 (E) 61

3 A burger at Ricky C’s weighs 120 grams, of which 30 grams are ? ller. What percent of the burger is not ? ller? (A) 60% (B) 65% (C) 70% (D) 75% (E) 90%

4 A group of children riding on bicycles and tricycles rode past Billy Bob’s house. Billy Bob counted 7 children and 19 wheels. How many tricycles were there? (A) 2 (B) 4 (C) 5 (D) 6 (E) 7

5 If 20% of a number is 12, what is 30% of the same number? (A) 15 (B) 18 (C) 20 (D) 24 (E) 30

6 Given the areas of the three squares in the ?gure, what is the area of the interior triangle?

169 25

144

(A) 13

(B) 30

(C) 60

(D) 300

(E) 1800

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7 Blake and Jenny each took four 100 point tests. Blake averaged 78 on the four tests. Jenny scored 10 points higher than Blake on the ?rst test, 10 points lower on the second test, and 20 points higher on both the third and fourth test. What is the di?erence between Blake’s average on the four tests and Jenny’s average on the four tests? (A) 10 (B) 15 (C) 20 (D) 25 (E) 40

8 Bake Sale Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies di?er, as shown. ? Art’s cookies are trapezoids:

3 in 3 in 5 in

? Roger’s cookies are rectangles:

2 in 4 in

? Paul’s cookies are parallelograms:

2 in 3 in

? Trisha’s cookies are triangles:

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4 in

3 in

Each friend uses the same amount of dough, and Art makes exactly 12 cookies. Who gets the fewest cookies from one batch of cookie dough? (A) Art (B) Roger (C) Paul (D) Trisha (E) There is a tie for fewest.

9 Bake Sale Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies di?er, as shown. ? Art’s cookies are trapezoids:

3 in 3 in 5 in

? Roger’s cookies are rectangles:

2 in 4 in

? Paul’s cookies are parallelograms:

2 in 3 in

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? Trisha’s cookies are triangles:

4 in

3 in

Each friend uses the same amount of dough, and Art makes exactly 12 cookies. Art’s cookies sell for 60 cents each. To earn the same amount from a single batch, how much should one of Roger’s cookies cost in cents? (A) 18 (B) 25 (C) 40 (D) 75 (E) 90

10 Bake Sale Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies di?er, as shown. ? Art’s cookies are trapezoids:

3 in 3 in 5 in

? Roger’s cookies are rectangles:

2 in 4 in

? Paul’s cookies are parallelograms:

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2 in 3 in

? Trisha’s cookies are triangles:

4 in

3 in

Each friend uses the same amount of dough, and Art makes exactly 12 cookies. How many cookies will be in one batch of Trisha’s cookies? (A) 10 (B) 12 (C) 16 (D) 18 (E) 24

11 Business is a little slow at Lou’s Fine Shoes, so Lou decides to have a sale. On Friday, Lou increases all of Thursday’s prices by 10 percent. Over the weekend, Lou advertises the sale: ”Ten percent o? the listed price. Sale starts Monday.” How much does a pair of shoes cost on Monday that cost 40 dollars on Thursday? (A) 36 (B) 39.60 (C) 40 (D) 40.40 (E) 44

12 When a fair six-sided dice is tossed on a table top, the bottom face cannot be seen. What is the probability that the product of the 5 faces than can be seen is divisible by 6? (A) 1/3 (B) 1/2 (C) 2/3 (D) 5/6 (E) 1

13 Fourteen white cubes are put together to form the ? gure on the right. The complete surface of the ?gure, including the bottom, is painted red. The ?gure is then separated into individual cubes. How many of the individual cubes have exactly four red faces?

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(A) 4

(B) 6

(C) 8

(D) 10

(E) 12

T W O 14 In this addition problem, each letter stands for a di?erent digit. + T W O If T = 7 F O U R and the letter O represents an even number, what is the only possible value for W? (A) 0 15 A ? gure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a ? gure with the front and side views shown? (B) 1 (C) 2 (D) 3 (E) 4

FRONT

SIDE

(A) 3

(B) 4

(C) 5

(D) 6

(E) 7

16 Ali, Bonnie, Carlo, and Dianna are going to drive together to a nearby theme park. The car they are using has 4 seats: 1 Driver seat, 1 front passenger seat, and 2 back passenger seat. Bonnie and Carlo are the only ones who know how to drive the car. How many possible seating arrangements are there? (A) 2 (B) 4 (C) 6 (D) 12 (E) 24

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17 The six children listed below are from two families of three siblings each. Each child has blue or brown eyes and black or blond hair. Children from the same family have at least one of these characteristics in common. Which two children are Jim’s siblings? Child Eye Color Hair Color Benjamin Blue Black Jim Brown Blonde Nadeen Brown Black Austin Blue Blonde Tevyn Blue Black Sue Blue Blonde (A) Nadeen and Austin (B) Benjamin and Sue (C) Benjamin and Austin

(D) Nadeen and Tev

18 Each of the twenty dots on the graph below represents one of Sarah’s classmates. Classmates who are friends are connected with a line segment. For her birthday party, Sarah is inviting only the following: all of her friends and all of those classmates who are friends with at least one of her friends. How many classmates will not be invited to Sarah’s party?

Sarah

(A) 1

(B) 4

(C) 5

(D) 6

(E) 7

19 How many integers between 1000 and 2000 have all three of the numbers 15, 20, and 25 as factors? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5

20 What is the measure of the acute angle formed by the hands of the clock at 4 : 20 PM? (A) 0 (B) 5 (C) 8 (D) 10 (E) 12

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21 The area of trapezoid ABCD is 164cm2 . The altitude is 8cm, AB is 10cm, and CD is 17cm. What is BC , in centimeters?

B 10 8

C 17

A

D

(A) 9

(B) 10

(C) 12

(D) 15

(E) 20

22 The following ?gures are composed of squares and circles. Which ?gure has a shaded region with largest area?

A

B

C

2 cm

2 cm

2 cm

(A) A only

(B) B only

(C) C only

(D) both A and B

(E) all are equal

23 In the pattern below, the cat (denoted as a large circle in the ?gures below) moves clockwise through the four squares and the mouse (denoted as a dot in the ?gures below) moves counterclockwise through the eight exterior segments of the four squares.

1

2

3

4

5

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If the pattern is continued, where would the cat and mouse be after the 247th move? (A)

(B)

(C)

(D)

(E)

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24 A ship travels from point A to point B along a semicircular path, centered at Island X. Then it travels along a straight path from B to C. Which of these graphs best shows the ship’s distance from Island X as it moves along its course?

C

A

X

B

(A)

distance to X distance traveled

(B)

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distance to X distance traveled

(C)

distance to X distance traveled

(D)

distance to X distance traveled

(E)

distance to X distance traveled

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25 In the ?gure, the area of square WXYZ is 25cm2 . The four smaller squares have sides 1 cm long, either parallel to or coinciding with the sides of the large square. In ?ABC , AB = AC , and when ?ABC is folded over side BC, point A coincides with O, the center of square WXYZ. What is the area of ?ABC , in square centimeters?

W B A C O

X

Y Z

(A)

15 4

(B)

21 4

(C)

27 4

(D)

21 2

(E)

27 2

The problems on this page are copyrighted by the Mathematical Association of America’s American Mathematics Competitions.

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

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1 On a map, a 12-centimeter length represents 72 kilometers. How many kilometers does a 17-centimeter length represent? (A) 6 (B) 102 (C) 204 (D) 864 (E) 1224

2 How many di?erent four-digit numbers can be formed by rearranging the four digits in 2004? (A) 4 (B) 6 (C) 16 (D) 24 (E) 81

3 Twelve friends met for dinner at Oscar’s Overstu?ed Oyster House, and each ordered one meal. The portions were so large, there was enough food for 18 people. If they share, how many meals should they have ordered to have just enough food for the 12 of them? (A) 8 (B) 9 (C) 10 (D) 15 (E) 18

4 Lance, Sally, Joy and Fred are chosen for the team. In how many ways can the three starters be chosen? (A)2 (B)4 (C)6 (D)8 (E)10

5 The losing team of each game is eliminated from the tournament. If sixteen teams compete, how many games will be played to determine the winner? (A)4 (B)7 (C)8 (D)15 (E)16

6 After Sally takes 20 shots, she has made 55% of her shots. After she takes 5 more shots, she raises her percentage to 56%. How many of the last 5 shots did she make? (A)1 (B)2 (C)3 (D)4 (E)5

7 An athlete’s target heart rate, in beats per minute, is 80% of the theoretical maximum heart rate. The maximum heart rate is found by subtracting the athlete’s age, in years, from 220. To the nearest whole number, what is the target heart rate of an athlete who is 26 years old? (A) 134 (B) 155 (C) 176 (D) 194 (E) 243

8 Find the number of two-digit positive integers whose digits total 7. (A) 6 (B) 7 (C) 8 (D) 9 (E) 10

9 The average of the ?ve numbers in a list is 54. The average of the ?rst two numbers is 48. What is the average of the last three numbers? (A) 55 (B) 56 (C) 57 (D) 58 (E) 59

10 Handy Aaron helped a neighbor 1 1 4 hours on Monday, 50 minutes on Tuesday, from 8 : 20 to 10 : 45 on Wednesday morning, and a half-hour on Friday. He is paid $3 per hour. How much did he earn for the week? (A) 8 (B) 9 (C) 10 (D) 12 (E) 15

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11 The numbers -2, 4, 6, 9 and 12 are rearranged according to these rules: 1. The largest isn’t ?rst, but it is in one of the ?rst three places. 2. The smallest isn’t last, but it is in one of the last three places. 3. The median isn’t ?rst or last. What is the average of the ?rst and last numbers? (A) 3.5 (B) 5 (C) 6.5 (D) 7.5 (E) 8

12 Niki usually leaves her cell phone on. If her cell phone is on but she is not actually using it, the battery will last for 24 hours. If she is using it constantly, the battery will last for only 3 hours. Since the last recharge, her phone has been on 9 hours, and during that time she has used it for 60 minutes. If she doesn’t talk any more but leaves the phone on, how many more hours will the battery last? (A) 7 (B) 8 (C) 11 (D) 14 (E) 15

13 Amy, Bill and Celine are friends with di?erent ages. Exactly one of the following statements is true. Rank the friends from the oldest to the youngest. (A) Bill, Amy, Celine (D) Celine, Bill, Amy (B) Amy, Bill, Celine (E) Amy, Celine, Bill (C) Celine, Amy, Bill

14 What is the area enclosed by the geoboard quadrilateral below?

(A)15

(B)18 1 2

1 (C)22 2

(D)27

(E)41

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15 Thirteen black and six white hexagonal tiles were used to create the ?gure below. If a new ?gure is created by attaching a border of white tiles with the same size and shape as the others, what will be the di?erence between the total number of white tiles and the total number of black tiles in the new ?gure?

(A) 5

(B) 7

(C) 11

(D) 12

(E) 18

1 full and the other pitcher is 2 16 Two 600 ml pitchers contain orange juice. One pitcher is 3 5 full. Water is added to ?ll each pitcher completely, then both pitchers are poured into one large container. What fraction of the mixture in the large container is orange juice?

(A)

1 8

(B)

3 16

(C)

11 30

(D)

11 19

(E)

11 15

17 Three friends have a total of 6 identical pencils, and each one has at least one pencil. In how many ways can this happen? (A) 1 (B) 3 (C) 6 (D) 10 (E) 12

18 Five friends compete in a dart-throwing contest. Each one has two darts to throw at the same circular target, and each individual’s score is the sum of the scores in the target regions that are hit. The scores for the target regions are the whole numbers 1 through 10. Each throw hits the target in a region with a di?erent value. The scores are: Alice 16 points, Ben 4 points, Cindy 7 points, Dave 11 points, and Ellen 17 points. Who hits the region worth 6 points? (A) Alice (B) Ben (C) Cindy (D) Dave (E) Ellen

19 A whole number larger than 2 leaves a remainder of 2 when divided by each of the numbers 3, 4, 5 and 6. The smallest such number lies between which two numbers? (A) 40 and 49 (B) 60 and 79 (C) 100 and 129 (D) 210 and 249 (E) 320 and 369

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20 Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If there are 6 empty chairs, how many people are in the room? (A) 12 (B) 18 (C) 24 (D) 27 (E) 36

21 Spinners A and B are spun. On each spinner, the arrow is equally likely to land on each number. What is the probability that the product of the two spinners’ numbers is even?

1 4 A

2 3

1 3 B

2

(A)

1 4

(B)

1 3

(C)

1 2

(D)

2 3

(E)

3 4

22 At a party there are only single women and married men with their wives. The probability 2 that a randomly selected woman is single is 5 . What fraction of the people in the room are married men? (A)
1 3

(B)

3 8

(C)

2 5

(D)

5 12

(E)

3 5

23 Tess runs counterclockwise around rectangular block JKLM. She lives at corner J. Which graph could represent her straight-line distance from home?

J

M

K

L

(A)

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distance time
(B)

distance time
(C)

distance time
(D)

distance time
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(E)

distance time
24 In the ?gure, ABCD is a rectangle and EF GH is a parallelogram. Using the measurements given in the ?gure, what is the length d of the segment that is perpendicular to HE and F G?

A 3 H

4

E

6

B

5 d

5

F 3

D

6

G

4

C

(A) 6.8

(B) 7.1

(C) 7.6

(D) 7.8

(E) 8.1

25 Two 4 × 4 squares intersect at right angles, bisecting their intersecting sides, as shown. The circle’s diameter is the segment between the two points of intersection. What is the area of the shaded region created by removing the circle from the squares?

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(A) 16 ? 4π

(B) 16 ? 2π

(C) 28 ? 4π

(D) 28 ? 2π

(E) 32 ? 2π

The problems on this page are copyrighted by the Mathematical Association of America’s American Mathematics Competitions.

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

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1 Connie multiplies a number by 2 and gets 60 as her answer. However, she should have divided the number by 2 to get the correct answer. What is the correct answer? (A) 7.5 (B) 15 (C) 30 (D) 120 (E) 240

2 Karl bought ?ve folders from Pay-A-Lot at a cost of $2.50 each. Pay-A-Lot had a 20%-o? sale the following day. How much could Karl have saved on the purchase by waiting a day? (A) $1.00 (B) $2.00 (C) $2.50 (D) $2.75 (E) $5.00

3 What is the minimum number of small squares that must be colored black so that a line of symmetry lies on the diagonal BD of square ABCD?

A

B

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

C

4 A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are 6.1 cm, 8.2 cm and 9.7 cm. What is the area of the square in square centimeters? (A) 24 (B) 25 (C) 36 (D) 48 (E)4

5 Soda is sold in packs of 6, 12 and 24 cans. What is the minimum number of packs needed to buy exactly 90 cans of soda? (A) 4 (B) 5 (C) 6 (D) 8 (E) 15

6 Suppose d is a digit. For how many values of d is 2.00d5 > 2.005? (A) 0 (B) 4 (C) 5 (D) 6 (E) 10
1 2 3 7 Bill walks 1 2 mile south, then 4 mile east, and ?nally a direct line, from his starting point?

mile south. How many miles is he, in

(A) 1

1 (B) 1 4

1 (C) 1 2

(D) 1 3 4

(E) 2

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8 Suppose m and n are positive odd integers. Which of the following must also be an odd integer? (A) m + 3n (B) 3m ? n (C) 3m2 + 3n2 (D) (nm + 3)2 (E) 3mn

9 In quadrilateral ABCD, sides AB and BC both have length 10, sides CD and DA both have length 17, and the measure of angle ADC is 60? . What is the length of diagonal AC ?

A B

C

D

(A) 13.5

(B) 14

(C) 15.5

(D) 17

(E) 18.5

10 Joe had walked half way from home to school when he realized he was late. He ran the rest of the way to school. He ran 3 times as fast as he walked. Joe took 6 minutes to walk half way to school. How many minutes did it take Joe to get from home to school? (A) 7 (B) 7.3 (C) 7.7 (D) 8 (E) 8.3

11 The sales tax rate in Bergville is 6%. During a sale at the Bergville Coat Closet, the price of a coat is discounted 20% from its $90.00 price. Two clerks, Jack and Jill, calculate the bill independently. Jack rings up $90.00 and adds 6% sales tax, then subtracts 20% from this total. Jill rings up $90.00, subtracts 20% of the price, then adds 6% of the discounted price for sales tax. What is Jack’s total minus Jill’s total? (A) ? $1.06 (B) ? $0.53 (C) 0 (D) $0.53 (E) $1.06

12 Big Al, the ape, ate 100 bananas from May 1 through May 5. Each day he ate six more bananas than on the previous day. How many bananas did Big Al eat on May 5? (A) 20 (B) 22 (C) 30 (D) 32 (E) 34

13 The area of polygon ABCDEF is 52 with AB = 8, BC = 9 and F A = 5. What is DE + EF ?

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A 5

8

B

9 F E

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

C

14 The Little Twelve Basketball Conference has two divisions, with six teams in each division. Each team plays each of the other teams in its own division twice and every team in the other division once. How many conference games are scheduled? (A) 80 (B) 96 (C) 100 (D) 108 (E) 192

15 How many di?erent isosceles triangles have integer side lengths and perimeter 23? (A) 2 (B) 4 (C) 6 (D) 9 (E) 11

16 A ?ve-legged Martian has a drawer full of socks, each of which is red, white or blue, and there are at least ?ve socks of each color. The Martian pulls out one sock at a time without looking. How many socks must the Martian remove from the drawer to be certain there will be 5 socks of the same color? (A) 6 (B) 9 (C) 12 (D) 13 (E) 15

17 The results of a cross-country team’s training run are graphed below. Which student has the greatest average speed?

Evelyn distance

Carla Debra

Briana time

Angela

O

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(A) Angela

(B) Briana

(C) Carla

(D) Debra

(E) Evelyn

18 How many three-digit numbers are divisible by 13? (A) 7 (B) 67 (C) 69 (D) 76 (E) 77

19 What is the perimeter of trapezoid ABCD?

B 30 A 24 E

50

C 25 D

(A) 180

(B) 188

(C) 196

(D) 200

(E) 204

20 Alice and Bob play a game involving a circle whose circumference is divided by 12 equallyspaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. The game ends when they stop on the same point. How many turns will this take? (A) 6 (B) 8 (C) 12 (D) 14 (E) 24

21 How many distinct triangles can be drawn using three of the dots below as vertices?

(A) 9

(B) 12

(C) 18

(D) 20

(E) 24

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22 A company sells detergent in three di?erent sized boxes: small (S), medium (M) and large (L). The medium size costs 50% more than the small size and contains 20% less detergent than the large size. The large size contains twice as much detergent as the small size and costs 30% more than the medium size. Rank the three sizes from best to worst buy. (A) SML (B) LMS (C) MSL (D) LSM (E) MLS

23 Isosceles right triangle ABC encloses a semicircle of area 2π . The circle has its center O on hypotenuse AB and is tangent to sides AC and BC . What is the area of triangle ABC ?

C

A

O

B

(A) 6

(B) 8

(C) 3π

(D) 10

(E) 4π

24 A certain calculator has only two keys [+1] and [x2]. When you press one of the keys, the calculator automatically displays the result. For instance, if the calculator originally displayed ”9” and you pressed [+1], it would display ”10.” If you then pressed [x2], it would display ”20.” Starting with the display ”1,” what is the fewest number of keystrokes you would need to reach ”200”? (A) 8 (B) 9 (C) 10 (D) 11 (E) 12

25 A square with side length 2 and a circle share the same center. The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square. What is the radius of the circle?

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(A)

2 √ π

(B)

√ 1+ 2 2

(C)

3 2

(D)

√ 3

(E)



π

The problems on this page are copyrighted by the Mathematical Association of America’s American Mathematics Competitions.

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

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1 Mindy made three purchases for $1.98, $5.04 and $9.89. What was her total, to the nearest dollar? (A) $10 (B) $15 (C) $16 (D) $17 (E) $18

2 On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions incorrectly and doesn’t answer the last 5. What is his score? (A) 1 (B) 6 (C) 13 (D) 19 (E) 26

3 Elisa swims laps in the pool. When she ?rst started, she completed 10 laps in 25 minutes. Now she can ?nish 12 laps in 24 minutes. By how many minutes has she improved her lap time? 1 3 (A) (B) (C) 1 (D) 2 (E) 3 2 4 1 4 Initially, a spinner points west. Chenille moves it clockwise 2 revolutions and then counter4 3 clockwise 3 revolutions. In what direction does the spinner point after the two moves? 4

N

W

E

S
(A) north (B) east (C) south (D) west (E) northwest

5 Points A, B, C and D are midpoints of the sides of the larger square. If the larger square has area 60, what is the area of the smaller square?

A

D

B

C

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(A) 15

(B) 20

(C) 24

(D) 30

(E) 40

6 The letter T is formed by placing two 2 × 4 inch rectangles next to each other, as shown. What is the perimeter of the T, in inches?

(A) 12

(B) 16

(C) 20

(D) 22

(E) 24

7 Circle X has a radius of π . Circle Y has a circumference of 8π . Circle Z has an area of 9π . List the circles in order from smallest to largest radius. (A) X, Y, Z (B) Z, X, Y (C) Y, X, Z (D) Z, Y, X (E) X, Z, Y

8 The table shows some of the results of a survey by radiostation KAMC. What percentage of the males surveyed listen to the station? Males Females Total (A) 39 Listen ? 58 136 (B) 48 Don’t Listen 26 ? 64 (C) 52 Total ? 96 200 (D) 55 (E) 75

9 What is the product of (A) 1 (B) 1002

3 4 5 2006 × × × ··· × ? 2 3 4 2005 (C) 1003 (D) 2005

(E) 2006

10 Jorge’s teacher asks him to plot all the ordered pairs (w, l) of positive integers for which w is the width and l is the length of a rectangle with area 12. What should his graph look like? (A)

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l w

(B)

l w

(C)

l w

(D)

l w

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(E)

l w

11 How many two-digit numbers have digits whose sum is a perfect square? (A) 13 (B) 16 (C) 17 (D) 18 (E) 19

12 Antonette gets 70% on a 10-problem test, 80% on a 20-problem test and 90% on a 30-problem test. If the three tests are combined into one 60-problem test, which percent is closest to her overall score? (A) 40 (B) 77 (C) 80 (D) 83 (E) 87

13 Cassie leaves Escanaba at 8:30 AM heading for Marquette on her bike. She bikes at a uniform rate of 12 miles per hour. Brian leaves Marquette at 9:00 AM heading for Escanaba on his bike. He bikes at a uniform rate of 16 miles per hour. They both bike on the same 62-mile route between Escanaba and Marquette. At what time in the morning do they meet? (A) 10 : 00 (B) 10 : 15 (C) 10 : 30 (D) 11 : 00 (E) 11 : 30

14 Problems 14, 15 and 16 involve Mrs. Reed’s English assignment. A Novel Assignment The students in Mrs. Reed’s English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds. If Bob and Chandra both read the whole book, Bob will spend how many more seconds reading than Chandra? (A) 7, 600 (B) 11, 400 (C) 12, 500 (D) 15, 200 (E) 22, 800

15 Problems 14, 15 and 16 involve Mrs. Reed’s English assignment. A Novel Assignment The students in Mrs. Reed’s English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds.

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Chandra and Bob, who each have a copy of the book, decide that they can save time by ”team reading” the novel. In this scheme, Chandra will read from page 1 to a certain page and Bob will read from the next page through page 760, ?nishing the book. When they are through they will tell each other about the part they read. What is the last page that Chandra should read so that she and Bob spend the same amount of time reading the novel? (A) 425 (B) 444 (C) 456 (D) 484 (E) 506

16 Problems 14, 15 and 16 involve Mrs. Reed’s English assignment. A Novel Assignment The students in Mrs. Reed’s English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds. Before Chandra and Bob start reading, Alice says she would like to team read with them. If they divide the book into three sections so that each reads for the same length of time, how many seconds will each have to read? (A) 6400 (B) 6600 (C) 6800 (D) 7000 (E) 7200

17 Je? rotates spinners P , Q and R and adds the resulting numbers. What is the probability that his sum is an odd number?

P 1 3 2

Q 2 8 4 6

R 1 3 11 5 9 7

(A)

1 4

(B)

1 3

(C)

1 2

(D)

2 3

(E)

3 4

18 A cube with 3-inch edges is made using 27 cubes with 1-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white? 1 4 5 19 1 (A) (B) (C) (D) (E) 9 4 9 9 27 19 Triangle ABC is an isosceles triangle with AB = BC . Point D is the midpoint of both BC and AE , and CE is 11 units long. Triangle ABD is congruent to triangle ECD. What is the length of BD?

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B D A C

E

(A) 4

(B) 4.5

(C) 5

(D) 5.5

(E) 6

20 A singles tournament had six players. Each player played every other player only once, with no ties. If Helen won 4 games, Ines won 3 games, Janet won 2 games, Kendra won 2 games and Lara won 2 games, how many games did Monica win? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4

21 An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. The aquarium is tilled with water to a depth of 37 cm. A rock with volume 1000cm3 is then placed in the aquarium and completely submerged. By how many centimeters does the water level rise? (A) 0.25 (B) 0.5 (C) 1 (D) 1.25 (E) 2.5

22 Three di?erent one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell. What is the di?erence between the largest and smallest numbers possible in the top cell?

+

+

+

(A) 16

(B) 24

(C) 25

(D) 26

(E) 35

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23 A box contains gold coins. If the coins are equally divided among six people, four coins are left over. If the coins are equally divided among ?ve people, three coins are left over. If the box holds the smallest number of coins that meets these two conditions, how many coins are left when equally divided among seven people? (A) 0 (B) 1 (C) 2 (D) 3 (E) 5

24 In the multiplication problem below, A, B , C and D are di?erent digits. What is A + B ? A × C (A) 1 (B) 2 (C) 3 (D) 4 D B C C A D D

(E) 9

25 Barry wrote 6 di?erent numbers, one on each side of 3 cards, and laid the cards on a table, as shown. The sums of the two numbers on each of the three cards are equal. The three numbers on the hidden sides are prime numbers. What is the average of the hidden prime numbers?

44

59

38

(A) 13

(B) 14

(C) 15

(D) 16

(E) 17

The problems on this page are copyrighted by the Mathematical Association of America’s American Mathematics Competitions.

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

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1 Theresa’s parents have agreed to buy her tickets to see her favorite band if she spends an average of 10 hours per week helping around the house for 6 weeks. For the ?rst 5 weeks, she helps around the house for 8, 11, 7, 12 and 10 hours. How many hours must she work during the ?nal week to earn the tickets? (A) 9 (B) 10 (C) 11 (D) 12 (E) 13

2 Six-hundred ?fty students were surveyed about their pasta preferences. The choices were lasagna, manicotti, ravioli and spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti?

250

200 Number of People

150

100

50

Manicotti

Lasagna

Ravioli

(A)

2 5

(B)

1 2

(C)

5 4

(D)

5 3

(E)

5 2

3 What is the sum of the two smallest prime factors of 250? (A) 2 (B) 5 (C) 7 (D) 10 (E) 12

4 A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a di?erent window? (A) 12 (B) 15 (C) 18 (D) 30 (E) 36

5 Chandler wants to buy a $500 dollar mountain bike. For his birthday, his grandparents send him $50, his aunt sends him $35 and his cousin gives him $15. He earns $16 per week for his

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Spaghetti

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paper route. He will use all of his birthday money and all of the money he earns from his paper route. In how many weeks will he be able to buy the mountain bike? (A) 24 (B) 25 (C) 26 (D) 27 (E) 28

6 The average cost of a long-distance call in the USA in 1985 was 41 cents per minute, and the average cost of a long-distance call in the USA in 2005 was 7 cents per minute. Find the approximate percent decrease in the cost per minute of a long-distance call. (A) 7 (B) 17 (C) 34 (D) 41 (E) 80

7 The average age of 5 people in a room is 30 years. An 18-year-old person leaves the room. What is the average age of the four remaining people? (A) 25 (B) 26 (C) 29 (D) 33 (E) 36

8 In trapezoid ABCD, AD is perpendicular to DC , AD = AB = 3, and DC = 6. In addition, E is on DC , and BE is parallel to AD. Find the area of ?BEC .

A 3

3

B

E D 6 C

(A) 3

(B) 4.5

(C) 6

(D) 9

(E) 18

9 To complete the grid below, each of the digits 1 through 4 must occur once in each row and once in each column. What number will occupy the lower right-hand square? 1 2 2 3 4 (A) 1 (B) 2 (C) 3 (D) 4 (E) cannot be determined

10 For any positive integer n, de?ne n to be the sum of the positive factors of n. For example, 6 = 1 + 2 + 3 + 6 = 12. Find 11 . (A) 13 (B) 20 (C) 24 (D) 28 (E) 30

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11 Tiles I, II, III and IV are translated so one tile coincides with each of the rectangles A, B, C and D. In the ?nal arrangement, the two numbers on any side common to two adjacent tiles must be the same. Which of the tiles is translated to Rectangle C ?

I 8 3 7 9 4

II 6 3 2 A B

III 7 1 0 5 9

IV 2 1 6

C

D

(A) I

(B) II

(C) III

(D) IV

(E) cannot be determined

12 A unit hexagon is composed of a regular haxagon of side length 1 and its equilateral triangular extensions, as shown in the diagram. What is the ratio of the area of the extensions to the area of the original hexagon?

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(A) 1 : 1

(B) 6 : 5

(C) 3 : 2

(D) 2 : 1

(E) 3 : 1

13 Sets A and B, shown in the venn diagram, have the same number of elements. Thier union has 2007 elements and their intersection has 1001 elements. Find the number of elements in A.

A

B

1001

(A) 503

(B) 1006

(C) 1504

(D) 1507

(E) 1510

14 The base of isosceles congruent sides? (A) 5 (B) 8

ABC is 24 and its area is 60. What is the length of one of the (D) 14 (E) 18

(C) 13

15 Let a, b and c be numbers with 0 < a < b < c. Which of the following is impossible? (A) a + c < b (B) a · b < c (C) a + b < c (D) a · c < b (E)
b c

=a

16 Amanda Reckonwith draws ?ve circles with radii 1, 2, 3, 4 and 5. Then for each circle she plots the point (C; A), where C is its circumference and A is its area. Which of the following could be her graph? (A)

A C

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(B)

A C
(C)

A C
(D)

A C
(E)

A C

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17 A mixture of 30 liters of paint is 25% red tint, 30% yellow tint, and 45% water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint that is the mixture? (A) 25 (B) 35 (C) 40 (D) 45 (E) 50 18 The product of the two 99-digit numbers 303, 030, 303, ..., 030, 303 and 505, 050, 505, ..., 050, 505 has thousands digit A and units digit B . What is the sum of A and B ? (A) 3 (B) 5 (C) 6 (D) 8 (E) 10

19 Pick two consecutive positive integers whose sum is less than 100. Square both of those integers and then ?nd the di?erence of the squares. Which of the following could be the di?erence? (A) 2 (B) 64 (C) 79 (D) 96 (E) 131

20 Before district play, the Unicorns had won 45% of their basketball games. During district play, they won six more games and lost two, to ?nish the season having won half their games. How many games did the Unicorns play in all? (A) 48 (B) 50 (C) 52 (D) 54 (E) 60

21 Two cards are dealt from a deck of four red cards labeled A, B , C , D and four green cards labeled A, B , C , D. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair? (A)
2 7

(B)

3 8

(C)

1 2

(D)

4 7

(E)

5 8

22 A lemming sits at a corner of a square with side length 10 meters. The lemming runs 6.2 meters along a diagonal toward the opposite corner. It stops, makes a 90? right turn and runs 2 more meters. A scientist measures the shortest distance between the lemming and each side of the square. What is the average of these four distances in meters? (A) 2 (B) 4.5 (C) 5 (D) 6.2 (E) 7

23 What is the area of the shaded pinwheel shown in the 5 × 5 grid?

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(A) 4

(B) 6

(C) 8

(D) 10

(E) 12

24 A bag contains four pieces of paper, each labeled with one of the digits 1, 2, 3 or 4, with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. What is the probability that the three-digit number is a multiple of 3? (A)
1 4

(B)

1 3

(C)

1 2

(D)

2 3

(E)

3 4

25 On the dart board shown in the ?gure, the outer circle has radius 6 and the inner circle has radius 3. Three radii divide each circle into the three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to to the area of the region. What two darts hit this board, the score is the sum of the point values in the regions. What is the probability that the score is odd?

2 1 2 1 2

1

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(A)

17 36

(B)

35 72

(C)

1 2

(D)

37 72

(E)

19 36

The problems on this page are copyrighted by the Mathematical Association of America’s American Mathematics Competitions.

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1 Susan had $50 to spend at the carnival. She spent $12 on food and twice as much on rides. How many dollars did she have left to spend? (A) 12 (B) 14 (C) 26 (D) 38 (E) 50

2 The ten-letter code BEST OF LUCK represents the ten digits 0 ? 9, in order. What 4-digit number is represented by the code word CLUE? (A) 8671 (B) 8672 (C) 9781 (D) 9782 (E) 9872

3 If February is a month that contains Friday the 13th , what day of the week is February 1? (A) Sunday (B) Monday (C) Wednesday (D) Thursday (E) Saturday

4 In the ?gure, the outer equilateral triangle has area 16, the inner equilateral triangle has area 1, and the three trapezoids are congruent. What is the area of one of the trapezoids?

(A) 3

(B) 4

(C) 5

(D) 6

(E) 7

5 Barney Schwinn notices that the odometer on his bicycle reads 1441, a palindrome, because it reads the same forward and backward. After riding 4 more hours that day and 6 the next, he notices that the odometer shows another palindrome, 1661. What was his average speed in miles per hour? (A) 15 (B) 16 (C) 18 (D) 20 (E) 22

6 In the ?gure, what is the ratio of the area of the gray squares to the area of the white squares?

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(A) 3 : 10 7 If
3 5

(B) 3 : 8
60 N,

(C) 3 : 7

(D) 3 : 5

(E) 1 : 1

=

M 45

=

what is M + N ? (C) 45 (D) 105 (E) 127

(A) 27

(B) 29

8 Candy sales from the Boosters Club from January through April are shown. What were the average sales per month in dollars?

$120 $80 $40 Jan Feb Mar Apr

(A) 60

(B) 70

(C) 75

(D) 80

(E) 85

9 In 2005 Tycoon Tammy invested $100 for two years. During the the ?rst year her investment su?ered a 15% loss, but during the second year the remaining investment showed a 20% gain. Over the two-year period, what was the change in Tammy’s investment? (A) 5% loss (B) 2% loss (C) 1% gain (D) 2% gain (E) 5% gain

10 The average age of the 6 people in Room A is 40. The average age of the 4 people in Room B is 25. If the two groups are combined, what is the average age of all the people? (A) 32.5 (B) 33 (C) 33.5 (D) 34 (E) 35

11 Each of the 39 students in the eighth grade at Lincoln Middle School has one dog or one cat or both a dog and a cat. Twenty students have a dog and 26 students have a cat. How many students have both a dog and a cat? (A) 7 (B) 13 (C) 19 (D) 39 (E) 46

12 A ball is dropped from a height of 3 meters. On its ?rst bounce it rises to a height of 2 meters. It keeps falling and bouncing to 2 3 of the height it reached in the previous bounce. On which bounce will it not rise to a height of 0.5 meters? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 13 Mr. Harman needs to know the combined weight in pounds of three boxes he wants to mail. However, the only available scale is not accurate for weights less than 100 pounds or more

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than 150 pounds. So the boxes are weighed in pairs in every possible way. The results are 122, 125 and 127 pounds. What is the combined weight in pounds of the three boxes? (A) 160 (B) 170 (C) 187 (D) 195 (E) 354

14 Three A’s, three B’s, and three C’s are placed in the nine spaces so that each row and column contain one of each letter. If A is placed in the upper left corner, how many arrangements are possible?

A

(A) 2

(B) 3

(C) 4

(D) 5

(E) 6

15 In Theresa’s ?rst 8 basketball games, she scored 7, 4, 3, 6, 8, 3, 1 and 5 points. In her ninth game, she scored fewer than 10 points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than 10 points and her pointsper-game average for the 10 games was also an integer. What is the product of the number of points she scored in the ninth and tenth games? (A) 35 (B) 40 (C) 48 (D) 56 (E) 72

16 A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units?

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(A) 1 : 6

(B) 7 : 36

(C) 1 : 5

(D) 7 : 30

(E) 6 : 25

17 Ms.Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of 50 units. All of her students calculate the area of the rectangle they draw. What is the di?erence between the largest and smallest possible areas of the rectangles? (A) 76 (B) 120 (C) 128 (D) 132 (E) 136

18 Two circles that share the same center have radii 10 meters and 20 meters. An aardvark runs along the path shown, starting at A and ending at K . How many meters does the aardvark run?

A

K

(A) 10π + 20 (E) 20π + 40

(B) 10π + 30

(C) 10π + 40

(D) 20π + 20

19 Eight points are spaced around at intervals of one unit around a 2 × 2 square, as shown. Two of the 8 points are chosen at random. What is the probability that the two points are one unit apart?

(A)

1 4

(B)

2 7

(C)

4 11

(D)

1 2

(E)

4 7

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USA
AMC 8 2008

20 The students in Mr. Neatkin’s class took a penmanship test. Two-thirds of the boys and 3 4 of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class? (A) 12 (B) 17 (C) 24 (D) 27 (E) 36

21 Jerry cuts a wedge from a 6-cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters?

6 cm

8 cm

(A)48

(B)75

(C)151

(D)192

(E)603
n 3

22 For how many positive integer values of n are both (A) 12 (B) 21 (C) 27 (D) 33

and 3n three-digit whole numbers?

(E) 34 BF D to

23 In square ABCE , AF = 2F E and CD = 2DE . What is the ratio of the area of the area of square ABCE ?

A

B

F E D
1 3 7 20

C

(A)

1 6

(B)

2 9

(C)

5 18

(D)

(E)

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USA
AMC 8 2008

24 Ten tiles numbered 1 through 10 are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square? (A)
1 10

(B)

1 6

(C)

11 60

(D)

1 5

(E)

7 30

25 Margie’s winning art design is shown. The smallest circle has radius 2 inches, with each successive circle’s radius increasing by 2 inches. Approximately what percent of the design is black?

(A) 42

(B) 44

(C) 45

(D) 46

(E) 48

The problems on this page are copyrighted by the Mathematical Association of America’s American Mathematics Competitions.

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

Page 6

USA
AMC 8 2009

1 Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie 3 apples, keeping 4 apples for herself. How many apples did Bridget buy? (A) 3 (B) 4 (C) 7 (D) 11 (E) 14

2 On average, for every 4 sports cars sold at the local dealership, 7 sedans are sold. The dealership predicts that it will sell 28 sports cars next month. How many sedans does it expect to sell? (A) 7 (B) 32 (C) 35 (D) 49 (E) 112

3 The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden?

4 MILES 0 0 5 10 15 20 MINUTES
(A)5 (B)5.5 (C)6

3 2 1

(D)6.5

(E)7

4 The ?ve pieces shown below can be arranged to form four of the ?ve ?gures shown in the choices. Which ?gure cannot be formed?

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AMC 8 2009

(A)

(B)

(C)

(D)

(E)

5 A sequence of numbers starts with 1, 2, and 3. The fourth number of the sequence is the sum of the previous three numbers in the sequence: 1 + 2 + 3 = 6. In the same way, every number after the fourth is the sum of the previous three numbers. What is the eighth number in the sequence? (A) 11 (B) 20 (C) 37 (D) 68 (E) 99

6 Steve’s empty swimming pool will hold 24, 000 gallons of water when full. It will be ?lled by 4 hoses, each of which supplies 2.5 gallons of water per minute. How many hours will it take to ?ll Steve’s pool? (A) 40 (B) 42 (C) 44 (D) 46 (E) 48

7 The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of land ACD?

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USA
AMC 8 2009

D 3

Bro wn

C 3

A

Main 3

A sp en
(A) 2 (B) 3 (C) 4.5 (D) 6 (E) 9 (A) 90 (B) 99 (C) 100 (D) 101 (E) 110

B

8 The length of a rectangle is increased by 10% and the width is decreased by 10%. What percent of the old area is the new area?

9 Construct a square on one side of an equilateral triangle. One on non-adjacent side of the square, construct a regular pentagon, as shown. One a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have?

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AMC 8 2009

(A)21

(B)23

(C)25

(D)27

(E)29

10 On a checkerboard composed of 64 unit squares, what is the probability that a randomly chosen unit square does not touch the outer edge of the board?

1 (A) 16

7 (B) 16

(C) 1 2

9 (D) 16

(E) 49 64

11 The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $1.43. Some of the 30 sixth graders each bought a pencil, and they paid a total of $1.95. How many more sixth graders than seventh graders bought a pencil? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5

12 The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime?

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USA
AMC 8 2009

1 5 3

2 6 4

(A)

1 2

(B)

2 3

(C)

3 4

(D)

7 9

(E)

5 6

13 A three-digit integer contains one of each of the digits 1, 3, and 5. What is the probability that the integer is divisible by 5? (A)
1 6

(B)

1 3

(C)

1 2

(D)

2 3

(E)

5 6

14 Austin and Temple are 50 miles apart along Interstate 35. Bonnie drove from Austin to her daughter’s house in Temple, averaging 60 miles per hour. Leaving the car with her daughter, Bonnie rod a bus back to Austin along the same route and averaged 40 miles per hour on the return trip. What was the average speed for the round trip, in miles per hour? (A) 46 (B) 48 (C) 50 (D) 52 (E) 54

1 15 A recipe that makes 5 servings of hot chocolate requires 2 squares of chocolate, 4 cup sugar, 1 cup water and 4 cups milk. Jordan has 5 squares of chocolate, 2 cups of sugar, lots of water and 7 cups of milk. If she maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate she can make?

(A) 5 1 8

1 (B) 6 4

(C) 7 1 2

(D) 8 3 4

(E) 9 7 8

16 How many 3-digit positive integers have digits whose product equals 24? (A) 12 (B) 15 (C) 18 (D) 21 (E) 24

17 The positive integers x and y are the two smallest positive integers for which the product of 360 and x is a square and the product of 360 and y is a cube. What is the sum of x and y ? (A) 80 (B) 85 (C) 115 (D) 165 (E) 610

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AMC 8 2009

18 The diagram represents a 7-foot-by-7-foot ?oor that is tiled with 1-square-foot black tiles and white tiles. Notice that the corners have white tiles. If a 15-foot-by-15-foot ?oor is to be tiled in the same manner, how many white tiles will be needed?

(A) 49

(B) 57

(C) 64

(D) 96

(E) 126

19 Two angles of an isosceles triangle measure 70? and x? . What is the sum of the three possible values of x? (A) 95 (B) 125 (C) 140 (D) 165 (E) 180

20 How many non-congruent triangles have vertices at three of the eight points in the array shown below?

(A) 5

(B) 6

(C) 7

(D) 8

(E) 9

21 Andy and Bethany have a rectangular array of numbers with 40 rows and 75 columns. Andy adds the numbers in each row. The average of his 40 sums is A. Bethany adds the numbers A ? in each column. The average of her 75 sums is B . What is the value of B (A)
64 225

(B)

8 15

(C) 1

(D)

15 8

(E)

225 64

22 How many whole numbers between 1 and 1000 do not contain the digit 1? (A) 512 (B) 648 (C) 720 (D) 728 (E) 800

23 On the last day of school, Mrs. Wonderful gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as

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USA
AMC 8 2009

there were girls in the class. She brought 400 jelly beans, and when she ?nished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class? (A) 26 (B) 28 (C) 30 (D) 32 (E) 34 A C D B A and A A C B A , what digit does A

24 The letters A, B , C and D represent digits. If + D represent? (A) 5 (B) 6 (C) 7 (D) 8 (E) 9

25 A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. 1 The ?rst cub is 1 2 foot from the top face. The second cut is 3 foot below the ?rst cut, and 1 foot below the second cut. From the top to the bottom the pieces are the third cut is 17 labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet?

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AMC 8 2009

(A) 6

(B) 7

(C)

419 51

(D)

158 17

(E) 11

The problems on this page are copyrighted by the Mathematical Association of America’s American Mathematics Competitions.

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

Page 8

USA
AMC 8 2010

1 At Euclid High School, the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are 11 students in Mrs. Germain’s class, 8 in Mr. Newton, and 9 in Mrs. Young’s class are taking the AMC 8 this year. How many mathematics students at Euclid High School are taking the contest? (A) 26 2 If a@b = (A)
3 10

(B) 27
a×b a+b ,

(C) 28

(D) 29

(E) 30

for a, b positive integers, then what is 5@10? (C) 2 (D)
10 3

(B) 1

(E) 50

3 The graph shows the price of ?ve gallons of gasoline during the ?rst ten months of the year. By what percent is the highest price more than the lowest price?

$ 20

$ 15 Price $ 10

$5

0

1

2

3

4

5

6

7 Month

8

9

10

(A) 50

(B) 62

(C) 70

(D) 89

(E) 100

4 What is the sum of the mean, median, and mode of the numbers, 2, 3, 0, 3, 1, 4, 0, 3? (A) 6.5 (B) 7 (C) 7.5 (D) 8.5 (E) 9

5 Alice needs to replace a light bulb located 10 centimeters below the ceiling of her kitchen. The ceiling is 2.4 meters above the ?oor. Alice is 1.5 meters tall and can reach 46 centimeters above her head. Standing on a stool, she can just reach the light bulb. What is the height of the stool, in centimeters?

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AMC 8 2010

(A) 32

(B) 34

(C) 36

(D) 38

(E) 40

6 Which of the following has the greatest number of line of symmetry? (A) Equilateral Triangle (B) Non-square rhombus (C) Non-square rectangle (D) Isosceles Triangle (E) Square 7 Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than one dollar? (A) 6 (B) 10 (C) 15 (D) 25 (E) 99

8 As Emily is riding her bike on a long straight road, she spots Ermenson skating in the same direction 1/2 mile in front of her. After she passes him, she can see him in her rear mirror until he is 1/2 mile behind her. Emily rides at a constant rate of 12 miles per hour. Ermenson skates at a constant rate of 8 miles per hour. For how many minutes can Emily see Ermenson? (A) 6 (B) 8 (C) 12 (D) 15 (E) 16

9 Ryan got 80% of the problems on a 25-problem test, 90% on a 40-problem test, and 70% on a 10-problem test. What percent of all problems did Ryan answer correctly? (A) 64 (B) 75 (C) 80 (D) 84 (E) 86

10 6 pepperoni circles will exactly ?t across the diameter of a 12-inch pizza when placed. If a total of 24 circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered with pepperoni? (A)
1 2

(B)

2 3

(C)

3 4

(D)

5 6

(E)

7 8

11 The top of one tree is 16 feet higher than the top of another tree. The height of the 2 trees are at a ratio of 3 : 4. In feet, how tall is the taller tree? (A) 48 (B) 64 (C) 80 (D) 96 (E) 112

12 Of the 500 balls in a large bag, 80% are red and the rest are blue. How many of the red balls must be removed so that 75% of the remaining balls are red? (A) 25 (B) 50 (C) 75 (D) 100 (E) 150

13 The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shorter side is 30% of the perimeter. What is the length of the longest side? (A) 7 (B) 8 (C) 9 (D) 10 (E) 11

14 What is the sum of the prime factors of 2010? (A) 67 (B) 75 (C) 77 (D) 201 (E) 210

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USA
AMC 8 2010

15 A jar contains 5 di?erent colors of gumdrops. 30% are blue, 20% are brown, 15% red, 10% yellow, and the other 30 gumdrops are green. If half of the blue gumdrops are replaced with brown gumdrops, how many gumdrops will be brown? (A) 35 (B) 36 (C) 42 (D) 48 (E) 64

16 A square and a circle have the same area. What is the ratio of the side length of the square to the radius of the circle? √ √ (A) 2π (B) π (C) π (D) 2π (E) π 2 17 The diagram shows an octagon consisting of 10 unit squares. The portion below P Q is a unit square and a triangle with base 5. If P Q bisects the area of the octagon, what is the ratio XQ QY ?

X Q Y

P

(A)

2 5

(B)

1 2

(C)

3 5

(D)

2 3

(E)

3 4

18 A decorative window is made up of a rectangle with semicircles at either end. The ratio of AD to AB is 3 : 2. And AB is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircle.

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AMC 8 2010

D

C

A

B

(A) 2 : 3

(B) 3 : 2

(C) 6 : π

(D) 9 : π

(E) 30 : π

19 The two circles pictured have the same center C . Chord AD is tangent to the inner circle at B , AC is 10, and chord AD has length 16. What is the area between the two circles?

C

A

B

D

(A) 36π

(B) 49π

(C) 64π

(D) 81π

(E) 100π

20 In a room, 2/5 of the people are wearing gloves, and 3/4 of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove? (A) 3 (B) 5 (C) 8 (D) 15 (E) 20

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AMC 8 2010

21 Hui is an avid reader. She bought a copy of the best seller Math is Beautiful. On the ?rst day, she read 1/5 of the pages plus 12 more, and on the second day she read 1/4 of the remaining pages plus 15 more. On the third day she read 1/3 of the remaining pages plus 18 more. She then realizes she has 62 pages left, which she ?nishes the next day. How many pages are in this book? (A) 120 (B) 180 (C) 240 (D) 300 (E) 360

22 The hundreds digit of a three-digit number is 2 more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result? (A) 0 (B) 2 (C) 4 (D) 6 (E) 8

23 Semicircles P OQ and ROS pass through the center of circle O. What is the ratio of the combined areas of the two semicircles to the area of circle O?

P (?1, 1)

Q(1, 1)

O

R(?1, ?1)

S (1, ?1)



(A)

2 4

(B)

1 2

(C)

2 π

(D)

2 3



(E)

2 2

24 What is the correct ordering of the three numbers, 108 , 512 , and 224 ? (A) 224 < 108 < 512 (B) 224 < 512 < 108 (C) 512 < 224 < 108 (D) 108 < 512 < 224 (E) 108 < 224 < 512

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AMC 8 2010

25 Everyday at school, Jo climbs a ?ight of 6 stairs. Joe can take the stairs 1, 2, or 3 at a time. For example, Jo could climb 3, then 1, then 2. In how many ways can Jo climb the stairs? (A) 13 (B) 18 (C) 20 (D) 22 (E) 24

The problems on this page are copyrighted by the Mathematical Association of America’s American Mathematics Competitions.

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

Page 6

USA
AMC 8 2011

1 Margie bought 3 apples at a cost of 50 cents each. She paid with a 5-dollar bill. How much change did Margie receive? (A)$1.50 (B)$2.00 (C)$2.50 (D)$3.00 (E)$3.50

2 Karl’s rectangular vegetable garden is 20 by 45 feet, and Makenna’s is 25 by 40 feet. Which garden is larger in area? (A) Karl’s garden is larger by 100 square feet. (B) Karl’s garden is larger by 25 square feet. (C) The gardens are the same size. (D) Makenna’s garden is larger by 25 square feet. (E) Makenna’s garden is larger by 100 square feet. 3 Extend the square pattern of 8 black and 17 white square tiles by attaching a border of black tiles around the square. What is the ratio of black tiles to white tiles in the extended pattern?

(A) 8 : 17

(B) 25 : 49

(C) 36 : 25

(D) 32 : 17

(E)6 : 17

4 Here is a list of the numbers of ?sh that Tyler caught in nine outings last summer: 2, 0, 1, 3, 0, 3, 3, 1, 2. Which statement about the mean, median, and mode is true? (A)median < mean < mode (C)mean < median < mode (E)mode < median < mean (B)mean < mode < median (D)median < mode < mean

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AMC 8 2011

5 What time was it 2011 minutes after midnight on January 1, 2011? (A)January 1 at 9:31PM (B)January 1 at 11:51PM (C)January 2 at 3:11AM (D)January 2 at 9:31AM (E)January 2 at 6:01PM 6 In a town of 351 adults, every adult owns a car, motorcycle, or both. If 331 adults own cars and 45 adults own motorcycles, how many of the car owners do not own a motorcycle? (A)20 (B)25 (C)45 (D)306 (E)351

7 Each of the following four large congruent squares is subdivided into combinations of congruent triangles or rectangles and is partially bolded. What percent of the total area is partially bolded?

(A)12 1 2

(B)20

(C)25

(D)33 1 3

(E)37 1 2

8 Bag A has three chips labeled 1, 3, and 5. Bag B has three chips labeled 2, 4, and 6. If one chip is drawn from each bag, how many di?erent values are possible for the sum of the two numbers on the chips? (A)4 (B)5 (C)6 (D)7 (E)9

9 Carmen takes a long bike ride on a hilly highway. The graph indicates the miles traveled during the time of her ride. What is Carmen’s average speed for her entire ride in miles per hour?

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USA
AMC 8 2011

35 30 M I L E S 25 20 15 10 5

1

2

3

4 HOURS

5

6

7

(A)2

(B)2.5

(C)4

(D)4.5

(E)5

10 The taxi fare in Gotham City is $2.40 for the ?rst 1 2 mile and additional mileage charged at the rate $0.20 for each additional 0.1 mile. You plan to give the driver a $2 tip. How many miles can you ride for $10? (A)3.0 (B)3.25 (C)3.3 (D)3.5 (E)3.75

11 The graph shows the number of minutes studied by both Asha (black bar) and Sasha (grey bar) in one week. On the average, how many more minutes per day did Sasha study than Asha?

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AMC 8 2011

120 100 M I N U T E S 80 60 40 20 0 M Tu W Th F

(A) 6

(B) 8

(C) 9

(D) 10

(E) 12

12 Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other? (A) 1 4 (B) 1 3 (C) 1 2 (D) 2 3 (E) 3 4

13 Two congruent squares, ABCD and P QRS , have side length 15. They overlap to form the 15 by 25 rectangle AQRD shown. What percent of the area of rectangle AQRD is shaded?

A

P

B

Q

D

S

C

R

(A) 15

(B) 18

(C) 20

(D) 24

(E) 25

14 There are 270 students at Colfax Middle School, where the ratio of boys to girls is 5 : 4. There are 180 students at Winthrop Middle School, where the ratio of boys to girls is 4 : 5.

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USA
AMC 8 2011

The two schools hold a dance and all students from both schools attend. What fraction of the students at the dance are girls? 7 7 22 1 23 (A) (B) (C) (D) (E) 18 15 45 2 45 15 How many digits are in the product 45 · 510 ? (A)8 (B)9 (C)10 (D)11 (E)12

16 Let A be the area of the triangle with sides of length 25, 25, and 30. Let B be the area of the triangle with sides of length 25, 25, and 40. What is the relationship between A and B ? 3 4 9 (B)A = B (C)A = B (D)A = B (A)A = B 16 4 3 16 (E)A = B 9 17 Let w, x, y , and z be whole numbers. If 2w · 3x · 5y · 7z = 588, then what does 2w + 3x + 5y + 7z equal? (A)21 (B)25 (C)27 (D)35 (E)56

18 A fair 6-sided die is rolled twice. What is the probability that the ?rst number that comes up is greater than or equal to the second number? 1 5 1 7 5 (A) (B) (C) (D) (E) 6 12 2 12 6 19 How many rectangles are in this ?gure?

(A) 8

(B) 9

(C) 10

(D) 11

(E) 12

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AMC 8 2011

20 Quadrilateral ABCD is a trapezoid, AD = 15, AB = 50, BC = 20, and the altitude is 12. What is the area of the trapeziod?

A 15 D

50 12

B 20 C

(A)600

(B)650

(C)700

(D)750

(E)800

21 Students guess that Norb’s age is 24, 28, 30, 32, 36, 38, 41, 44, 47, and 49. Norb says, ”At least half of you guessed too low, two of you are o? by one, and my age is a prime number.” How old is Norb? (A)29 (B)31 (C)37 (D)43 (E)48

22 What is the tens digit of 72011 ? (A)0 (B)1 (C)3 (D)4 (E)7

23 How many 4-digit positive integers have four di?erent digits, where the leading digit is not zero, the integer is a multiple of 5, and 5 is the largest digit? (A)24 (B)48 (C)60 (D)84 (E)108

24 In how many ways can 10001 be written as the sum of two primes? (A)0 (B)1 (C)2 (D)3 (E)4

25 A circle with radius 1 is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle’s shaded area to the area between the two squares?

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AMC 8 2011

(A)

1 2

(B) 1

(C)

3 2

(D) 2

(E)

5 2

The problems on this page are copyrighted by the Mathematical Association of America’s American Mathematics Competitions.

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

Page 7

USA
AMC 8 2012

1 Rachelle uses 3 pounds of meat to make 8 hamburgers for her family. How many pounds of meat does she need to make 24 hamburgers for a neighborhood picnic? 2 1 (A) 6 (B) 6 (C) 7 (D) 8 (E) 9 3 2 2 In the country of East Westmore, statisticians estimate there is a baby born every 8 hours and a death every day. To the nearest hundred, how many people are added to the population of East Westmore each year? (A) 600 (B) 700 (C) 800 (D) 900 (E) 1000

3 On February 13 The Oshkosh Northwester listed the length of daylight as 10 hours and 24 minutes, the sunrise was 6 : 57am, and the sunset as 8 : 15pm. The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun really set? (A) 5 : 10pm (B) 5 : 21pm (C) 5 : 41pm (D) 5 : 57pm (E) 6 : 03pm 4 Peter’s family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice equally with his brother Paul. What fraction of the pizza did Peter eat? 1 1 1 1 1 (A) (B) (C) (D) (E) 24 12 8 6 4 5 In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is X , in centimeters?

11 4 1 2 2 2 X 1

2

1 3 2 1 2 1 6

(A) 1

(B) 2

(C) 3

(D) 4

(E) 5

6 A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures 8 inches high and 10 inches wide. What is the area of the border, in square inches? (A) 36 (B) 40 (C) 64 (D) 72 (E) 88

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

Page 1

USA
AMC 8 2012

7 Isabella must take four 100-point tests in her math class. Her goal is to achieve an average grade of 95 on the tests. Her ?rst two test scores were 97 and 91. After seeing her score on the third test, she realized she can still reach her goal. What is the lowest possible score she could have made on the third test? (A) 90 (B) 92 (C) 95 (D) 96 (E) 97

8 A shop advertises everything is ”half price in today’s sale.” In addition, a coupon gives a 20 (A) 10 (B) 33 (C) 40 (D) 60 (E) 70

9 The Fort Worth Zoo has a number of two-legged birds and a number of four-legged mammals. On one visit to the zoo, Margie counted 200 heads and 522 legs. How many of the animals that Margie counted were two-legged birds? (A) 61 (B) 122 (C) 139 (D) 150 (E) 161

10 How many 4-digit numbers greater than 1000 are there that use the four digits of 2012? (A) 6 (B) 7 (C) 8 (D) 9 (E) 12

11 The mean, median, and unique mode of the positive integers 3, 4, 5, 6, 6, 7, x are all equal. What is the value of x? (A) 5 (B) 6 (C) 7 (D) 11 (E) 12

12 What is the units digit of 132012 ? (A) 1 (B) 3 (C) 5 (D) 7 (E) 9

13 Jamar bought some pencils costing more than a penny each at the school bookstore and paid $1.43. Sharona bought some of the same pencils and paid $1.87. How many more pencils did Sharona buy than Jamar? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6

14 In the BIG N, a middle school football conference, each team plays every other team exactly once. If a total of 21 conference games were played during the 2012 season, how many teams were members of the BIG N conference? (A) 6 (B) 7 (C) 8 (D) 9 (E) 10

15 The smallest number greater than 2 that leaves a remainder of 2 when divided by 3, 4, 5, or 6 lies between what numbers? (A) 40 and 50 (B) 51 and 55 (C) 56 and 60 (D) 61 and 65 (E) 66 and 99

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

Page 2

USA
AMC 8 2012

16 Each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 is used only once to make two ?ve-digit numbers so that they have the largest possible sum. Which of the following could be one of the numbers? (A) 76531 (B) 86724 (C) 87431 (D) 96240 (E) 97403

17 A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7

18 What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50? (A) 3127 (B) 3133 (C) 3137 (D) 3139 (E) 3149

19 In a jar of red, green, and blue marbles, all but 6 are red marbles, all but 8 are green, and all but 4 are blue. How many marbles are in the jar? (A) 6 (B) 8 (C) 9 (D) 10 (E) 12

5 7 9 20 What is the correct ordering of the three numbers 19 , 21 , and 23 , in increasing order? 9 7 9 5 7 9 9 5 7 5 9 7 7 (A) < < (B) < < (C) < < (D) < < (E) < 23 21 23 19 21 23 23 19 21 19 23 21 21 5 9 < 19 23

21 Marla has a large white cube that has an edge of 10 feet. She also has enough green paint to cover 300 square feet. Marla uses all the paint to create a white square centered on each face, surrounded by a green border. What is the area of one of the white squares, in square feet? √ √ √ (A) 5 2 (B) 10 (C) 10 2 (D) 50 (E) 50 2 22 Let R be a set of nine distinct integers. Six of the elements are 2, 3, 4, 6, 9, and 14. What is the number of possible values of the median of R ? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8

23 An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is 4, what is the area of the hexagon? √ √ (A) 4 (B) 5 (C) 6 (D) 4 3 (E) 6 3 24 A circle of radius 2 is cut into four congruent arcs. The four arcs are joined to form the star ?gure shown. What is the ratio of the area of the star ?gure to the area of the original circle?

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

Page 3

USA
AMC 8 2012

4?π (A) π

1 (B) π

√ (C)

2 π

(D)

π?1 π

(E)

3 π

25 A square with area 4 is inscribed in a square with area 5, with one vertex of the smaller square on each side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length a, and the other of length b. What is the value of ab ?

b

a

(A)

1 5

(B)

2 5

(C)

1 2

(D) 1

(E) 4

The problems on this page are copyrighted by the Mathematical Association of America’s American Mathematics Competitions.

This ?le was downloaded from the AoPS Math Olympiad Resources Page http://www.artofproblemsolving.com/

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