math competition practice

7/27/2019 Math competition practice

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Please print the below document or follow along on screen. Good luck.

#H34N1 of the Art of Problem SolvingAmerican Mathematics Competitions

MOCK EXAMINATION

AMC 10(American Mathematics Contest 10)

July 1, 2008

INSTRUCTIONS1. DO NOT OPEN THIS BOOKLET UNTIL YOUR PROCTOR TELLS

YOU.

2. This is a sample twenty-five question MCQ test. Each question has answerchoices (A), (B), (C), (D), and (E). Only one of these is correct.

3. There is a penalty for guessing. You will receive 6 points for a correct answer,1.5 points for a blank answer, and 0 points for an incorrect answer. It is not

to your benefit to answer all of the questions.4. No aids are permitted other than scratch paper, graph paper, rulers, and

erasers. NO CALCULATORS ARE PERMITTED.

5. Figures are not necessarily drawn to scale.6. When your proctor gives the signal, begin working on the problems. You will

have 40 minutes to complete this test.

7. When you finish your exam, please go to http://mersenne.zapto.org/amc/amcto submit your answers. Once your answers have been received, you will be

given an electronic participation certificate.

8. The statistics will be computed and you will be notified of your score.Once you have read the above directions, you may turn the page. You have 75

minutes to finish this examination. While you are completing this test, a timer is

available online. Please refer to this timer. If you are unable to input your answers

within 75 minutes, there will be no reprieve and the score that you will receive

will be based only on the answers received by the server.

Good luck.


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Mock AMC 10 2008 2

1. The AMC 10 has 25 questions. If there are an equal number of correct answers being (A),(B), (C), (D), and (E), what is the number of questions will I get correct that if I select (C)

for each question?(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

2. A rhombus is given with diagonals of which one is double the length of the other. If thelength of the longer diagonal is x, what is the area of the rhombus?

(A) x2

2(B) x2 (C) x24 (D) x2

4(E) x

3. Joe uses a ridiculous phone provider which charges an amount x cents per minute for whichhe talks in a month. The phone company also has a flat rate for which it charges all customersregardless of how much they talk. In January, the phone company charges Joe $14.20. Thefollowing month, Joe does a lot of business conferences and talks on the phone for four timesas long as he did in January. His monthly bill is $37.30. What is the fixed rate that thephone company charges to the nearest cent?

(A) $3.55 (B) $6.50 (C) $9.33 (D) $5.78 (E) $23.10

4. What is the maximum possible number of intersection points between two circles and twolines?

(A) 8 (B) 9 (C) 11 (D) 12 (E) 15

5. In a standard cube, an ant is set at a corner. What is the maximum possible lengths an antcan walk if it cannot visit any point twice or overlap any of the lengths, if the ant must endup in the point it started at?

(A) 5 (B) 6 (C) 7 (D) 8 (E) 9

6. In the diagram below, we have AB = 3 and right triangles ABD and ADE. The point Dbisects BC. IfAC =

73, what is the value ofAE, in simplest radical form? Note: the

diagram is not necessarily drawn to scale.


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Mock AMC 10 2008 3

(A)

31/2 (B)

39

73(C)

25

73

41(D)

55 (E)

70

7. We have the letters and numbers A,B,C,0 in the set. What is the probability that if werandomly select three of these with replacement, we will have a string in the format of LLN,where L symbolizes a letter and N a number?

(A)1

64(B)

20

21(C)

5

6(D)

3

4(E)

3

64

8. There are five women and five men at a party. Each person shakes hands with their spouseand with the other people of their gender at a party. How many handshakes are there?

(A) 10 (B) 15 (C) 20 (D) 25 (E) 30

9. Bob finds a weird calculator which only has two keys - [+1] and [*2]. The number 1 ison the screen. Bob wants to make this number 71. What is the minimal number of clicksrequired to reach the desired number?

(A) 9 (B) 10 (C) 11 (D) 12 (E) 13

10. If the sides of a triangle are 8, 15, and 17, what is the length of the shortest altitude to thenearest tenth?

(A) 6.2 (B) 6.9 (C) 7.1 (D) 7.7 (E) 8.0

11. A monster buys a camera and a case worth 1/2 of the cameras cost, an ice cream worth 1/2of the cases value, and a cookie for 1/2 of the ice creams value, all for $105. How much didthe cookie cost?

(A) $7 (B) $14 (C) $56 (D) $57.25 (E) $105


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Mock AMC 10 2008 4

12. How many distinct solutions exist for the equation 2x3 5x2 + 4x 1 = 0?(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

13. What is the area of a polygon with vertices at: (0,0), (3,4), (6,0), (3,-4), and (0,3)?

(A) 4.5 (B) 19.5 (C) 21.5 (D) 24.0 (E) 28.5

14. What is the average of all two-digit integers that satisfy the condition that the number isequal to the sum of the tens digit and the units digit and the product of the tens digit andthe units digit?

(A) 56 (B) 57 (C) 58 (D) 59 (E) 60

15. What is the units digit of the following sum:

(1!)2 + (2!)2 + + (1337!)2?

(A) 5 (B) 6 (C) 7 (D) 9 (E) 0

16. In the expansion of (a + b)15, what is the last number of the coefficient of the a7b9 term?

(A) 0 (B) 3 (C) 4 (D) 5 (E) 9

17. If two of the sides of a rectangular prism have measures 3 and 4, and the surface area is 48,then what is the volume?

(A) 72/7 (B) 100/7 (C) 144/7 (D) 180/7 (E) 242/7

18. How many ways can 4 people sit at a round table if there are seven chairs?

(A) 24 (B) 120 (C) 240 (D) 840 (E) 1200

19. The area of a kite is 24 square units. What is the area of a kite whose diagonals are threetimes the length of the original kite?

(A) 24 (B) 72 (C) 108 (D) 216 (E) 240


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Mock AMC 10 2008 5

20. A cross-section of a river is a isosceles trapezoid with bases 10 and 16 and slanted sides oflength 5. The water flows at mph. In a dam, the water flows through 4 identical circular

holes at 16 mph. What is the radius of these holes?

(A) 20

3 (B)

15

4(C) 5 (D) 36

2 (E)

13

4

21. How many ways can one cover a 3 x 8 rectangle with 12 dominoes?

(A) 220 (B) 153 (C) 300 (D) 452 (E) 479

22. If a beast randomly throws a dart at a square dartboard with side length 2, what is theprobability that the dart hits closer to the center of the dartboard than any corner, but

within a distance of 1 from a corner?(A) (B)

24

(C) 4

(D)1

4(E)

1

8

23. We can define a cool point to be a point on a coordinate plane which has integral coordi-nates. For example, (7,8) and (5,9) are cool points. How many cool points exist within oron the graph of the circle (x 5)2 + (y 2)2 = 100?(A) 317 (B) 420 (C) 100 (D) 25 (E) 10, 000

24. Bill has a 2 2 2, a 3 3 3, a 4 4 4, and a 5 5 5 Rubiks cube. Sticky Fingerssue comes along and steals one all with equal probability, then Pick-pocket Pearl comes andtakes either the largest one or the smallest one still there, again with equal probability. Billcomes back, but he only wants to play with his 3 3 3. What is the probability that hecan?

(A) 1/8 (B) 1/4 (C) 3/8 (D) 1/2 (E) 5/8

25. Two friends get home at 2:30 P.M. They both go to bed 10 P.M. If both friends will go to apark twice for 15 minutes, what is the probability that they will be able to see each other atany random time between the two time intervals? Assume that the travel to and from the

park takes no time.

(A)25

225(B)

8

15(C)

27

225(D)

29

225(E)

3

17

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