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AMC 10 Preparation Practice Test 1

American Mathematics Competitions

Practice 1
AMC 10
(American Mathematics Contest 10)

INSTRUCTIONS

1. This is a twenty-five question multiple choice test. Each question is followed

by answers marked A, B, C, D and E. Only one of these is correct.

2. You will have 75 minutes to complete the test.

3. No aids are permitted other than scratch paper, graph paper, rulers, and erasers.
No problems on the test will require the use of a calculator.

4. Figures are not necessarily drawn to scale.

5. SCORING: You will receive 6 points for each correct answer, 1.5 points for
each problem left unanswered, and 0 points for each incorrect answer.

6. Do not guess the answer.

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AMC 10 Preparation Practice Test 1

1. Let M, I, and T be distinct positive integers such that the product M × I × T =

2015. What is the largest possible value of the sum M + I + T ?

(A) 8 (B) 44 (C) 155 (D) 403 (E) 409

(2  1)(2 2  1)(2 4  1)(28  1)(216  1)(232  1)

2. =
264  1

(A) 233 (B) 2 64 (C) 1024 (D) 2 (E) 1

3. Each day, Jenny ate 30% of the jellybeans that were in her jar at the beginning
of that day. At the end of second day, 98 remained. How many jellybeans were in
the jar originally?

(A) 100 (B) 150 (C) 200 (D) 250 (E) 3005

4. Catherine pays an on-line service provider a fixed monthly fee plus an hourly
charge for connect time. Her March bill was $64.48, but in April her bill was
$51.54 because she used half as much connect time as in March. What is the fixed
monthly fee?

(A) $47.54 (B) $46.06 (C) $43.24 (D) $40.42 (E) $38.60

5. The lengths of three sides of a triangle are all integers. The perimeter of the
triangle is an odd number and the difference of the lengths of two sides is 11. The
possible length of the third side is:

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

2

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AMC 10 Preparation Practice Test 1

6. The Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21,… starts with two 1’s, and each
term afterwards is the sum of its two predecessors. How many odd numbers are
there in the first 2015 terms in the Fibonacci sequence?

(A) 1344 (B) 1342 (C) 1024 (D) 1007 (E) 1006

7. In rectangle ABCD, BE : EC = 5 : 2, DF : CF = 2 : 1. Find the area of triangle

AEF if the area of the rectangle ABCD is 1764.

(A) 134 (B) 342 (C) 462 (D) 707 (E) 764

8. At Hope High School, 3/7 of the freshmen and 6/7 of the sophomores took the
AMC 10. Given that the number of freshmen contestants is twice as many as the
number of sophomore contestants, which of the following must be true?

(A) There are four times as many sophomores as freshmen.

(B) There are twice as many sophomores as freshmen.
(C) There are as many freshmen as sophomores.
(D) There are twice as many freshmen as sophomores.
(E) There are four times as many freshmen as sophomores.

9. Suppose that x    2q , where x < . What is the value of x + 3q ?

(A)  (B) 3 (C)   2q (D)  + q (E)  + 5q

3

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AMC 10 Preparation Practice Test 1

10. The length and width of a rectangle are positive integers. The perimeter and
the area have the same numerical values. What is the sum of all possible values of
the area of the rectangle?

(A) 16 (B) 18 (C) 24 (D) 34 (E) 46

11. Two different prime numbers between 12 and 32 are chosen. When their sum
is subtracted from their product, which of the following numbers could be
obtained?

(A) 159 (B) 210 (C) 359 (D) 802 (E) 1079

12. At each stage, a new square is drawn on each side of the perimeter of the
figure in the previous stage. Figures show four stages of 1, 5, 13, and 25
nonoverlapping unit squares, respectively. If the pattern were continued, how
many unit squares will be in Stage 200?

A) 79601 (B) 78804 (C) 49402 (D) 40199 (E) 39402

Stage 1 Stage 2 Stage 3 Stage 4

13. How many ways can five people line up behind four registers?

(A) 6720 (B) 120 (C) 720 (D) 394 (E) 6240

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AMC 10 Preparation Practice Test 1

14. Mr. Wang gave an exam in a mathematics class of five students. He entered
the scores in random order into a spreadsheet, which recalculated the class
average after each score was entered. Mr. Wang noticed that after each score was
entered, the average was always an integer. The scores (listed in ascending order)
were 74, 77, 88, 94, and 97. What was the last score Mr. Wang entered?

(A) 77 (B) 88 (C) 74 (D) 94 (E) 97

a b
15. What is the value of if a 2  b2  4ab , where a  b  0 .
ab
3 3
(A) 3 (B)  (C) (D)  3 (E) 1
3 3

16. The horizontally and vertically adjacent points in this square grid are 1 cm
apart. Segment AB meets segment CD at E. Find the length of segment BE.

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

17. Betsy has an incredible coin machine. When she puts in a quarter, it returns
five nickels; when he puts in a nickel, it returns five pennies; and when she puts in
a penny, it returns nine quarters. Betsy starts with just one quarter. Which of the
following amounts could she have after using the machine many times?

(A) $7.73 (B) $7.27 (C) $7.16 (D) $7.05 (E) $6.97

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AMC 10 Preparation Practice Test 1

18. Charles walks completely around the boundary of a square. From any point on
his path he can see exactly 2 km horizontally in all directions. The area of the
region consisting of all points Charles can see is 144 + 4 during his walk. What
is the length of the side of the square?

(A) 4 (B) 6 (C) 8 (D) 10 (E) 12

19. In square units, what is the largest possible area a rectangle inscribed in the
triangle shown here can have? BC = 100 cm. AH = 80 cm.

(A) 2000 (B) 6000 (C) 5000 (D) 4000 (E) 1200

20. Let a, b, and c be distinct real numbers. Find a : b : c if a, b, and c form an

arithmetic sequence and b, a, and c form a geometric sequence.

(A) 2 : 1 : 4 (B) ( 2) : 1 : 4 (C) 2 : ( 2) : 4 (D) 2 : 1 : ( 2) (E) 3 : 1 : 4

21. Alex, Bob, and Charlie are students who participate in three different clubs
(Math, Science, and Writing) with each person in only one club. They are from
three schools: Hope School, Cox School, and Ashley School, not necessarily in
that order. Alex is not from Hope School, Bob is not from Cox School. The
student from Hope School is not in the Science club, the student from Cox School
is in the Math club. Bob is not in the Writing club. Which school is Charlie from
and what club is he in?

(A) Hope School and Writing club. (B) Cox School and Math club.
(C) Ashley and Science club. (D) Hope School and Science club.
(E) Cox School and Science club.
6

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AMC 10 Preparation Practice Test 1

22. One morning Alex and Bob took a math contest. Alex got a ninth of the total
number of problems on the contest wrong and Bob got seven problems correct.
The number of the problems answered correctly by both people is one sixth of the
total number of problems on the contest. How many problem were answered
correctly by Alex?

(A) 32 (B) 22 (C) 12 (D) 8 (E) 7

23. When the mean, median, and mode of the list 11, 3, 6, 3, 5, 3, x are arranged
in increasing order, they form a non-constant arithmetic progression. What is the
sum of all possible real value of x ?

(A) 4 (B) 25 (C) 39 (D) 43 (E) 54

24. What is the sum of all values of z for which f (5z) = 21 and f is a function
x
defined by f ( )  x 2  x  1.
5
1 1 5
(A)  (B)  (C) 5 (D) (E) 5
13 25 7

25. In year N , the 200th day of the year is a Sunday. In year N + 1, the 100th day
is also a Sunday. On what day of the week did the 300th day of year N − 1 occur?

(A) Thursday (B) Friday (C) Saturday (D) Sunday (E) Monday

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AMC 10 Preparation Practice Test 1

ANSWER KEYS

1. E.
2. E.
3. C.
4. E.
5. B.
6. A.
7. C.
8. E.
9. D.
10. D.
11. C.
12. A.
13. A.
14. C.
15. C.
16. B.
17. E.
18. D.
19. A.
20. B.
21. A.
22. A.
23. E.
24. B.
25. E.

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