2002 AMC 10

AMC 10 2002

– A

102000 + 102002
1 The ratio is closest to which of the following numbers?
102001 + 102001
(A) 0.1 (B) 0.2 (C) 1 (D) 5 (E) 10

2 For the nonzero numbers a, b, c, define

a b c
(a, b, c) = + + .
b c a
Find (2, 12, 9).
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

3 According to the standard convention for exponentiation,

2( )
 
22
2
22
2 =2 = 216 = 65,536.

If the order in which the exponentiations are performed is changed, how many
other values are possible?
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

4 For how many positive integers m does there exist at least one positive integer
n such that m · n ≤ m + n?
(A) 4 (B) 6 (C) 9 (D) 12 (E) infinitely many

5 Each of the small circles in the figure has radius one. The innermost circle is
tangent to the six circles that surround it, and each of those circles is tangent
to the large circle and to its small-circle neighbors. Find the area of the shaded
region.

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(A) π (B) 1.5π (C) 2π (D) 3π (E) 3.5π

6 Cindy was asked by her teacher to subtract 3 from a certain number and then
divide the result by 9. Instead, she subtracted 9 and then divided the result by
3, giving an answer of 43. What would her answer have been had she worked
the problem correctly?
(A) 15 (B) 34 (C) 43 (D) 51 (E) 138

7 If an arc of 45◦ on circle A has the same length as an arc of 30◦ on circle B,
then the ratio of the area of circle A to the area of circle B is
(A) 94 (B) 23 (C) 56 (D) 23 (E) 94

8 Betsy designed a flag using blue triangles, small white squares, and a red center
square, as shown. Let B be the total area of the blue triangles, W the total area
of the white squares, and R the area of the red square. Which of the following
is correct?

(A) B = W (B) W = R (C) B = R (D) 3B = 2R (E) 2R =

W

9 Suppose A, B, and C are three numbers for which 1001C − 2002A = 4004 and
1001B + 3003A = 5005.The average of the three numbers A, B, and C is
(A) 1 (B) 3 (C) 6 (D) 9 (E) not uniquely determined

10 Compute the sum of all the roots of (2x + 3)(x − 4) + (2x + 3)(x − 6) = 0.
(A) 7/2 (B) 4 (C) 5 (D) 7 (E) 13

11 Jamal wants to store 30 computer files on floppy disks, each of which has a
capacity of 1.44 megabytes (MB). Three of his files require 0.8 MB of memory
each, 12 more require 0.7 MB each, and the remaining 15 require 0.4 MB each.

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No file can be split between floppy disks. What is the minimal number of floppy
disks that will hold all the files?
(A) 12 (B) 13 (C) 14 (D) 15 (E) 16

12 Mr. Earl E. Bird leaves his house for work at exactly 8:00 A.M. every morning.
When he averages 40 miles per hour, he arrives at his workplace three minutes
late. When he averages 60 miles per hour, he arrives three minutes early. At
what average speed, in miles per hour, should Mr. Bird drive to arrive at his
workplace precisely on time?
(A) 45 (B) 48 (C) 50 (D) 55 (E) 58

13 The sides of a triangle have lengths of 15, 20, and 25. Find the length of the
shortest altitude.
(A) 6 (B) 12 (C) 12.5 (D) 13 (E) 15

14 Both roots of the quadratic equation x2 − 63x + k = 0 are prime numbers. The
number of possible values of k is
(A) 0 (B) 1 (C) 2 (D) 3 (E) more than four

15 The digits 1, 2, 3, 4, 5, 6, 7, and 9 are used to form four two-digit prime

numbers, with each digit used exactly once. What is the sum of these four
primes?
(A) 150 (B) 160 (C) 170 (D) 180 (E) 190

16 If a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5, then a + b + c + d is
(A) − 5 (B) − 10/3 (C) − 7/3 (D) 5/3 (E) 5

17 Sarah pours four ounces of coffee into an eight-ounce cup and four ounces of
cream into a second cup of the same size. She then transfers half the coffee
from the first cup to the second and, after stirring thoroughly, transfers half
the liquid in the second cup back to the first. What fraction of the liquid in
the first cup is now cream?
(A) 1/4 (B) 1/3 (C) 3/8 (D) 2/5 (E) 1/2

18 A 3 × 3 × 3 cube is formed by gluing together 27 standard cubical dice. (On a

standard die, the sum of the numbers on any pair of opposite faces is 7.) The
smallest possible sum of all the numbers showing on the surface of the 3 × 3 × 3
cube is
(A) 60 (B) 72 (C) 84 (D) 90 (E) 96

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19 Spot’s doghouse has a regular hexagonal base that measures one yard on each
side. He is tethered to a vertex with a two-yard rope. What is the area, in
square yards, of the region outside of the doghouse that Spot can reach?
(A) 2π/3 (B) 2π (C) 5π/2 (D) 8π/3 (E) 3π

20 Points A, B, C, D, E and F lie, in that order, on AF , dividing it into five seg-

ments, each of length 1. Point G is not on line AF . Point H lies on GD, and
point J lies on GF . The line segments HC, JE, and AG are parallel. Find
HC/JE.
(A) 5/4 (B) 4/3 (C) 3/2 (D) 5/3 (E) 2

21 The mean, median, unique mode, and range of a collection of eight integers are
all equal to 8. The largest integer that can be an element of this collection is
(A) 11 (B) 12 (C) 13 (D) 14 (E) 15

22 A sit of tiles numbered 1 through 100 is modified repeatedly by the following

operation: remove all tiles numbered with a perfect square, and renumber the
remaining tiles consecutively starting with 1. How many times must the oper-
ation be performed to reduce the number of tiles in the set to one?
(A) 10 (B) 11 (C) 18 (D) 19 (E) 20

23 Points A, B, C and D lie on a line, in that order, with AB = CD and BC = 12.

Point E is not on the line, and BE = CE = 10. The perimeter of △AED is
twice the perimeter of △BEC. Find AB.
(A) 15/2 (B) 8 (C) 17/2 (D) 9 (E) 19/2

24 Tina randomly selects two distinct numbers from the set {1, 2, 3, 4, 5} and Sergio
randomly selects a number from the set {1, 2, ..., 10}. The probability that
Sergio’s number is larger than the sum of the two numbers chosen by Tina is
(A) 2/5 (B) 9/20 (C) 1/2 (D) 11/20 (E) 24/25

25 In trapezoid ABCD with bases AB and CD, we have AB = 52, BC = 12,

CD = 39, and DA = 5. The area of ABCD is

D 39 C

5 12

A 52 B

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(A) 182 (B) 195 (C) 210 (D) 234 (E) 260

– B
2001 2003
1 The ratio 2 62002
·3
is
(A) 16 (B) 13 (C) 1
2 (D) 2
3 (E) 3
2

2 For the nonzero numbers a, b, and c, define

abc
(a, b, c) = .
a+b+c
Find (2, 4, 6).
(A) 1 (B) 2 (C) 4 (D) 6 (E) 24

3 The arithmetic mean of the nine numbers in the set {9, 99, 999, 9999, ..., 999999999}
is a 9-digit number M , all of whose digits are distinct. The number M does
not contain the digit
(A) 0 (B) 2 (C) 4 (D) 6 (E) 8

4 What is the value of

(3x − 2)(4x + 1) − (3x − 2)4x + 1
when x = 4?
(A) 0 (B) 1 (C) 10 (D) 11 (E) 12

5 Circles of radius 2 and 3 are externally tangent and are circumscribed by a

third circle, as shown in the figure. Find the area of the shaded region.

3 2

(A) 3π (B) 4π (C) 6π (D) 9π (E) 12π

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6 For how many positive integers n is n2 − 3n + 2 a prime number?

(A) none (B) one (C) two (D) more than two, but finitely many
(E) infinitely many

1
7 Let n be a positive integer such that 2 + 31 + 17 + 1
n is an integer. Which of the
following statements is not true?
(A) 2 divides n (B) 3 divides n (C) 6 divides n (D) 7 divides n
(E) n > 84

8 Suppose July of year N has five Mondays. Which of the following must occur
five times in August of year N ? (Note: Both months have 31 days.)
(A) Monday (B) Tuesday (C) Wednesday (D) Thursday (E) Friday

9 Using the letters A, M , O, S, and U , we can form 120 five-letter ”words”. If

these ”words” are arranged in alphabetical order, then the ”word” U SAM O
occupies position
(A) 112 (B) 113 (C) 114 (D) 115 (E) 116

10 Suppose that a and b are are nonzero real numbers, and that the equation
x2 + ax + b = 0 has solutions a and b. Then the pair (a, b) is
(A) (−2, 1) (B) (−1, 2) (C) (1, −2) (D) (2, −1) (E) (4, 4)

11 The product of three consecutive positive integers is 8 times their sum. What
is the sum of their squares?
(A) 50 (B) 77 (C) 110 (D) 149 (E) 194

x−1 x−k
12 For which of the following values of k does the equation x−2 = x−6 have no
solution for x?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

13 Find the value(s) of x such that 8xy − 12y + 2x − 3 = 0 is true for all values of
y.
(A) 23 (B) 32 or − 41 (C) − 32 or − 14 (D) 32 (E) − 32 or − 41

14 The number 2564 · 6425 is the square of a positive integer N . In decimal repre-
sentation, the sum of the digits of N is
(A) 7 (B) 14 (C) 21 (D) 28 (E) 35

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15 The positive integers A, B, A − B, and A + B are all prime numbers. The sum
of these four primes is
(A) even (B) divisible by 3 (C) divisible by 5 (D) divisible by 7
(E) prime

n
16 For how many integers n is 20−n the square of an integer?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 10

17 A regular octagon ABCDEF GH has sides of length two. Find the area of
△ADG. √ √ √ √ √
(A) 4 + 2 2 (B) 6 + 2 (C) 4 + 3 2 (D) 3 + 4 2 (E) 8 + 2

18 Four distinct circles are drawn in a plane. What is the maximum number of
points where at least two of the circles intersect?
(A) 8 (B) 9 (C) 10 (D) 12 (E) 16

19 Suppose that {an } is an arithmetic sequence with

a1 + a2 + · · · + a100 = 100 and a101 + a102 + · · · + a200 = 200.

What is the value of a2 − a1 ?

(A) 0.0001 (B) 0.001 (C) 0.01 (D) 0.1 (E) 1

20 Let a, b, and c be real numbers such that a − 7b + 8c = 4 and 8a + 4b − c = 7.

Then a2 − b2 + c2 is
(A) 0 (B) 1 (C) 4 (D) 7 (E) 8

21 Andy’s lawn has twice as much area as Beth’s lawn and three times as much
area as Carlos’ lawn. Carlos’ lawn mower cuts half as fast as Beth’s mower and
one third as fast as Andy’s mower. If they all start to mow their lawns at the
same time, who will finish first?
(A) Andy (B) Beth (C) Carlos (D) Andy and Carlos tie for first.
(E) All three tie.

22 Let △XOY be a right-angled triangle with m∠XOY = 90◦ . Let M and N

be the midpoints of legs OX and OY , respectively. Given that XN = 19 and
Y M = 22, find XY .
(A) 24 (B) 26 (C) 28 (D) 30 (E) 32

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23 Let {ak } be a sequence of integers such that a1 = 1 and am+n = am + an + mn,

for all positive integers m and n. Then a12 is
(A) 45 (B) 56 (C) 67 (D) 78 (E) 89

24 Riders on a Ferris wheel travel in a circle in a vertical plane. A particular

wheel has radius 20 feet and revolves at the constant rate of one revolution per
minute. How many seconds does it take a rider to travel from the bottom of
the wheel to a point 10 vertical feet above the bottom?
(A) 5 (B) 6 (C) 7.5 (D) 10 (E) 15

25 When 15 is appended to a list of integers, the mean is increased by 2. When 1

is appended to the enlarged list, the mean of the enlarged list is decreased by
1. How many integers were in the original list?
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

– P

-1 This test and the matching AMC 12P were developed for the use of a group of
Taiwan schools, in early January of 2002. When Taiwan had taken the contests,
the AMC released the questions here as a set of practice questions for the 2002
AMC 10 and AMC 12 contests.

(24 )8
1 The ratio equals
(48 )2
1 1
(A) (B) (C) 1 (D) 2 (E) 8
4 2

2 The sum of eleven consecutive integers is 2002. What is the smallest of these
integers?
(A) 175 (B) 177 (C) 179 (D) 180 (E) 181

3 Mary typed a six-digit number, but the two 1s she typed didn’t show. What
appeared was 2002. How many different six-digit numbers could she have typed?
(A) 4 (B) 8 (C) 10 (D) 15 (E) 20

4 Which of the following numbers is a perfect square?

(A) 44 55 66 (B) 44 56 65 (C) 45 54 66 (D) 46 54 65 (E) 46 55 64

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5 Let (an )n≥1 be a sequence such that a1 = 1 and 3an+1 − 3an = 1 for all n ≥ 1.
Find a2002 .
(A) 666 (B) 667 (C) 668 (D) 669 (E) 670

6 The perimeter of a rectangle is 100 and its diagonal has length x. What is the
area of this rectangle?
x2 x2
(A) 625−x2 (B) 625− (C) 1250−x2 (D) 1250− (E) 2500−
2 2
x2
2

7 The dimensions of a rectangular box in inches are all positive integers and the
volume of the box is 2002 in3 . Find the minimum possible sum in inches of the
three dimensions.
(A) 36 (B) 38 (C) 42 (D) 44 (E) 92

8 How many ordered triples of positive integers (x, y, z) satisfy (xy )z = 64?
(A) 5 (B) 6 (C) 7 (D) 8 (E) 9

9 The function f is given by the table

x 1 2 3 4 5
f (x) 4 1 3 5 2

If u0 = 4 and un+1 = f (un ) for n ≥ 0, find u2002 .

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

10 Let a and b be distinct real numbers for which

a a + 10b
+ = 2.
b b + 10a
a
Find .
b
(A) 0.6 (B) 0.7 (C) 0.8 (D) 0.9 (E) 1

11 Let P (x) = kx3 + 2k 2 x2 + k 3 . Find the sum of all real numbers k for which
x − 2 is a factor of P (x).
(A) − 8 (B) − 4 (C) 0 (D) 5 (E) 8

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12 For fn (x) = xn and a 6= 1 consider

I. (f11 (a)f13 (a))14
II. f11 (a)f13 (a)f14 (a)
III. (f11 (f13 (a)))14
IV. f11 (f13 (f14 (a)))
Which of these equal f2002 (a)?
(A) I and II only (B) II and III only (C) III and IV only (D) II, III, and IV onl

13 Participation in the local soccer league this year is 10% higher than last year.
The number of males increased by 5% and the number of females increased by
20%. What fraction of the soccer league is now female?
1 4 2 4 1
(A) (B) (C) (D) (E)
3 11 5 9 2

14 The vertex E of a square EF GH is at the center of square ABCD. The length

of a side of ABCD is 1 and the length of a side of EF GH is 2. Side EF
intersects CD at I and EH intersects AD at J. If angle EID = 60◦ , the area
of quadrilateral EIDJ is
√ √ √
1 3 1 2 3
(A) (B) (C) (D) (E)
4 6 3 4 2

15 What is the smallest integer n for which any subset of {1, 2, 3, . . . , 20} of size
n must contain two numbers that differ by 8?
(A) 2 (B) 8 (C) 12 (D) 13 (E) 15

16 Two walls and the ceiling of a room meet at right angles at point P . A fly is in
the air one meter from one wall, eight meters from the other wall, and 9 meters
from point P . How many meters is the fly from the ceiling?
√ √ √ √
(A) 13 (B) 14 (C) 15 (D) 4 (E) 17

17 There are 1001 red marbles and 1001 black marbles in a box. Let Ps be the
probability that two marbles drawn at random from the box are the same color,
and let Pd be the probability that they are different colors. Find |Ps − Pd |.
1 1 2 1
(A) 0 (B) (C) (D) (E)
2002 2001 2001 1000

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18 For how many positive integers n is n3 − 8n2 + 20n − 13 a prime number?

(A) 1 (B) 2 (C) 3 (D) 4 (E) more than 4

19 If a, b, c are real numbers such that a2 +2b = 7, b2 +4c = −7, and c2 +6a = −14,
find a2 + b2 + c2 .
(A) 14 (B) 21 (C) 28 (D) 35 (E) 49

20 How many three-digit numbers have at least one 2 and at least one 3?
(A) 52 (B) 54 (C) 56 (D) 58 (E) 60

21 Let f be a real-valued function such that

 
2002
f (x) + 2f = 3x
x

for all x > 0. Find f (2).

(A) 1000 (B) 2000 (C) 3000 (D) 4000 (E) 6000

2002!
22 In how many zeroes does the number end?
(1001!)2
(A) 0 (B) 1 (C) 2 (D) 200 (E) 400

23 Let
12 22 32 10012
a= + + + ··· +
1 3 5 2001
and
12 22 32 10012
b= + + + ··· + .
3 5 7 2003
Find the integer closest to a − b.
(A) 500 (B) 501 (C) 999 (D) 1000 (E) 1001

24 What is the maximum value of n for which there is a set of distinct positive
integers k1 , k2 , . . . , kn for which

k12 + k22 + . . . + kn2 = 2002?

(A) 14 (B) 15 (C) 16 (D) 17 (E) 18

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25 Under the new AMC 10, 12 scoring method, 6 points are given for each correct
answer, 2.5 points are given for each unanswered question, and no points are
given for an incorrect answer. Some of the possible scores between 0 and 150
can be obtained in only one way, for example, the only way to obtain a score
of 146.5 is to have 24 correct answers and one unanswered question. Some
scores can be obtained in exactly two ways; for example, a score of 104.5 can
be obtained with 17 correct answers, 1 unanswered question, and 7 incorrect,
and also with 12 correct answers and 13 unanswered questions. There are three
scores that can be obtained in exactly three ways. What is their sum?
(A) 175 (B) 179.5 (C) 182 (D) 188.5 (E) 201

–
c Mathematical Association of America (http:
These problems are copyright
//maa.org).

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