2002 AMC 10

Part of AMC 10

Subcontests

(26)

1

2002 AMC 10P Introduction

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.

3

Symmetric System's Cyclic Sum

If a\plus{}1\equal{}b\plus{}2\equal{}c\plus{}3\equal{}d\plus{}4\equal{}a\plus{}b\plus{}c\plus{}d\plus{}5, then a\plus{}b\plus{}c\plus{}d is \text{(A)}\ \minus{}5 \qquad \text{(B)}\ \minus{}10/3 \qquad \text{(C)}\ \minus{}7/3 \qquad \text{(D)}\ 5/3 \qquad \text{(E)}\ 5

Fraction Square

For how many integers

n

is \frac{n}{20\minus{}n} the square of an integer?

<span class='latex-bold'>(A)</span>\ 1 \qquad <span class='latex-bold'>(B)</span>\ 2 \qquad <span class='latex-bold'>(C)</span>\ 3 \qquad <span class='latex-bold'>(D)</span>\ 4 \qquad <span class='latex-bold'>(E)</span>\ 10

A Fly in a Box

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?

<span class='latex-bold'>(A) </span>\sqrt{13}\qquad<span class='latex-bold'>(B) </span>\sqrt{14}\qquad<span class='latex-bold'>(C) </span>\sqrt{15}\qquad<span class='latex-bold'>(D) </span>4\qquad<span class='latex-bold'>(E) </span>\sqrt{17}

3

Making Primes from Digits

1

2

3

4

5

6

7

9

are used to form four two-digit prime numbers, with each digit used exactly once. What is the sum of these four primes?

\text{(A)}\ 150 \qquad \text{(B)}\ 160 \qquad \text{(C)}\ 170 \qquad \text{(D)}\ 180 \qquad \text{(E)}\ 190

Sum of Four Primes

The positive integers

A

B

, A \minus{} B, and A \plus{} B are all prime numbers. The sum of these four primes is

<span class='latex-bold'>(A)</span>\ \text{even} \qquad <span class='latex-bold'>(B)</span>\ \text{divisible by }3 \qquad <span class='latex-bold'>(C)</span>\ \text{divisible by }5 \qquad <span class='latex-bold'>(D)</span>\ \text{divisible by }7 \\ <span class='latex-bold'>(E)</span>\ \text{prime}

Two Numbers which Differ by 8

What is the smallest integer

n

for which any subset of

\{1,2,3,\ldots,20\}

n

must contain two numbers that differ by

8

?

<span class='latex-bold'>(A) </span>2\qquad<span class='latex-bold'>(B) </span>8\qquad<span class='latex-bold'>(C) </span>12\qquad<span class='latex-bold'>(D) </span>13\qquad<span class='latex-bold'>(E) </span>15

3

Sarah's weird coffee ritual

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?

<span class='latex-bold'>(A)</span>\ 1/4 \qquad <span class='latex-bold'>(B)</span>\ 1/3 \qquad <span class='latex-bold'>(C)</span>\ 3/8 \qquad <span class='latex-bold'>(D)</span>\ 2/5 \qquad <span class='latex-bold'>(E)</span>\ 1/2

Regular Octagon

A regular octagon

ABCDEFGH

has sides of length two. Find the area of

\triangle{ADG}

. (A)\ 4 \plus{} 2 \sqrt{2} \qquad (B)\ 6 \plus{} \sqrt{2} \qquad (C)\ 4 \plus{} 3 \sqrt{2} \qquad (D)\ 3 \plus{} 4 \sqrt{2} \qquad (E)\ 8 \plus{} \sqrt{2}

Probability of Same vs. Different Color

1001

red marbles and

1001

black marbles in a box. Let

P_s

be the probability that two marbles drawn at random from the box are the same color, and let

P_d

be the probability that they are different colors. Find

|P_s-P_d|

.

<span class='latex-bold'>(A) </span>0\qquad<span class='latex-bold'>(B) </span>\dfrac1{2002}\qquad<span class='latex-bold'>(C) </span>\dfrac1{2001}\qquad<span class='latex-bold'>(D) </span>\dfrac2{2001}\qquad<span class='latex-bold'>(E) </span>\dfrac1{1000}

3

Prime Quadratic Roots

Both roots of the quadratic equation x^2 \minus{} 63x \plus{} k \equal{} 0 are prime numbers. The number of possible values of

k

<span class='latex-bold'>(A)</span>\ 0 \qquad <span class='latex-bold'>(B)</span>\ 1 \qquad <span class='latex-bold'>(C)</span>\ 2 \qquad <span class='latex-bold'>(D)</span>\ 3 \qquad <span class='latex-bold'>(E)</span>\ <span class='latex-bold'>more than four</span>

Sum of Digits of Square

25^{64}\cdot64^{25}

is the square of a positive integer

N

. In decimal representation, the sum of the digits of

N

<span class='latex-bold'>(A)</span>\ 7 \qquad <span class='latex-bold'>(B)</span>\ 14 \qquad <span class='latex-bold'>(C)</span>\ 21 \qquad <span class='latex-bold'>(D)</span>\ 28 \qquad <span class='latex-bold'>(E)</span>\ 35

Intersecting Squares

E

EFGH

is at the center of square

ABCD

. The length of a side of

ABCD

1

and the length of a side of

EFGH

2

EF

CD

I

EH

AD

J

EID=60^\circ

, the area of quadrilateral

EIDJ

<span class='latex-bold'>(A) </span>\dfrac14\qquad<span class='latex-bold'>(B) </span>\dfrac{\sqrt3}6\qquad<span class='latex-bold'>(C) </span>\dfrac13\qquad<span class='latex-bold'>(D) </span>\dfrac{\sqrt2}4\qquad<span class='latex-bold'>(E) </span>\dfrac{\sqrt3}2

3

27 dice make a 3x3x3 cube

3 \times 3 \times 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 \times 3 \times 3

\text{(A)}\ 60 \qquad \text{(B)}\ 72 \qquad \text{(C)}\ 84 \qquad \text{(D)}\ 90 \qquad \text{(E)}\ 96

Four Circles

Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?

<span class='latex-bold'>(A)</span>\ 8 \qquad <span class='latex-bold'>(B)</span>\ 9 \qquad <span class='latex-bold'>(C)</span>\ 10 \qquad <span class='latex-bold'>(D)</span>\ 12 \qquad <span class='latex-bold'>(E)</span>\ 16

Certain Prime Polynomial

For how many positive integers

n

n^3-8n^2+20n-13

a prime number?

<span class='latex-bold'>(A) </span>1\qquad<span class='latex-bold'>(B) </span>2\qquad<span class='latex-bold'>(C) </span>3\qquad<span class='latex-bold'>(D) </span>4\qquad<span class='latex-bold'>(E) </span>\text{more than 4}

3

Shortest Altitude

The sides of a triangle have lengths of

15

20

25

. Find the length of the shortest altitude.

\text{(A)}\ 6 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 12.5 \qquad \text{(D)}\ 13 \qquad \text{(E)}\ 15

True for All Values

Find the value(s) of

x

such that 8xy\minus{}12y\plus{}2x\minus{}3\equal{}0 is true for all values of

y

. (A)\ \frac{2}{3} \qquad (B)\ \frac{3}{2}\text{ or }\minus{}\frac{1}{4} \qquad (C)\ \minus{}\frac{2}{3}\text{ or }\minus{}\frac{1}{4} \qquad (D)\ \frac{3}{2} \qquad (E)\ \minus{}\frac{3}{2}\text{ or }\minus{}\frac{1}{4}

Soccer League Participation Increase

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?

<span class='latex-bold'>(A) </span>\dfrac13\qquad<span class='latex-bold'>(B) </span>\dfrac4{11}\qquad<span class='latex-bold'>(C) </span>\dfrac25\qquad<span class='latex-bold'>(D) </span>\dfrac49\qquad<span class='latex-bold'>(E) </span>\dfrac12

3

Spot rebelling against his leash

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?

\text{(A)}\ 2\pi/3 \qquad \text{(B)}\ 2\pi \qquad \text{(C)}\ 5\pi/2 \qquad \text{(D)}\ 8\pi/3 \qquad \text{(E)}\ 3\pi

Arithmetic Sequence

\{a_n\}

is an arithmetic sequence with a_1 \plus{} a_2 \plus{} \cdots \plus{} a_{100} \equal{} 100 \text{and} a_{101} \plus{} a_{102} \plus{} \cdots \plus{} a_{200} \equal{} 200. What is the value of a_2 \minus{} a_1?

<span class='latex-bold'>(A)</span>\ 0.0001 \qquad <span class='latex-bold'>(B)</span>\ 0.001 \qquad <span class='latex-bold'>(C)</span>\ 0.01 \qquad <span class='latex-bold'>(D)</span>\ 0.1 \qquad <span class='latex-bold'>(E)</span>\ 1

System of Equations

a,b,c

are real numbers such that

a^2+2b=7

b^2+4c=-7

c^2+6a=-14

a^2+b^2+c^2

.

<span class='latex-bold'>(A) </span>14\qquad<span class='latex-bold'>(B) </span>21\qquad<span class='latex-bold'>(C) </span>28\qquad<span class='latex-bold'>(D) </span>35\qquad<span class='latex-bold'>(E) </span>49

3

Earl E. Bird

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?

<span class='latex-bold'>(A)</span>\ 45 \qquad <span class='latex-bold'>(B)</span>\ 48 \qquad <span class='latex-bold'>(C)</span>\ 50 \qquad <span class='latex-bold'>(D)</span>\ 55 \qquad <span class='latex-bold'>(E)</span>\ 58

No Solution

For which of the following values of

k

does the equation \frac{x\minus{}1}{x\minus{}2}\equal{}\frac{x\minus{}k}{x\minus{}6} have no solution for

x

<span class='latex-bold'>(A)</span>\ 1 \qquad <span class='latex-bold'>(B)</span>\ 2 \qquad <span class='latex-bold'>(C)</span>\ 3 \qquad <span class='latex-bold'>(D)</span>\ 4 \qquad <span class='latex-bold'>(E)</span>\ 5

Which Equals our Result?

f_n(x)=x^n

a\neq 1

(f_{11}(a)f_{13}(a))^{14}

f_{11}(a)f_{13}(a)f_{14}(a)

(f_{11}(f_{13}(a)))^{14}

f_{11}(f_{13}(f_{14}(a)))

Which of these equal

f_{2002}(a)

?

<span class='latex-bold'>(A) </span>\text{I and II only}\qquad<span class='latex-bold'>(B) </span>\text{II and III only}\qquad<span class='latex-bold'>(C) </span>\text{III and IV only}\qquad<span class='latex-bold'>(D) </span>\text{II, III, and IV only}\qquad<span class='latex-bold'>(E) </span>\text{all of them}

3

Equal segments and parallel lines

A,B,C,D,E

F

lie, in that order, on

\overline{AF}

, dividing it into five segments, each of length 1. Point

G

AF

H

\overline{GD}

J

\overline{GF}

. The line segments

\overline{HC}, \overline{JE},

\overline{AG}

are parallel. Find

HC/JE

\text{(A)}\ 5/4 \qquad \text{(B)}\ 4/3 \qquad \text{(C)}\ 3/2 \qquad \text{(D)}\ 5/3 \qquad \text{(E)}\ 2

System of Equations

a

b

c

be real numbers such that a \minus{} 7b \plus{} 8c \equal{} 4 and 8a \plus{} 4b \minus{} c \equal{} 7. Then a^2 \minus{} b^2 \plus{} c^2 is

<span class='latex-bold'>(A)</span>\ 0 \qquad <span class='latex-bold'>(B)</span>\ 1 \qquad <span class='latex-bold'>(C)</span>\ 4 \qquad <span class='latex-bold'>(D)</span>\ 7 \qquad <span class='latex-bold'>(E)</span>\ 8

At least one 2, at least one 3

How many three-digit numbers have at least one

2

and at least one

3

?

<span class='latex-bold'>(A) </span>52\qquad<span class='latex-bold'>(B) </span>54\qquad<span class='latex-bold'>(C) </span>56\qquad<span class='latex-bold'>(D) </span>58\qquad<span class='latex-bold'>(E) </span>60

3

Minimizing Disk Quantity

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

0.7

MB each, and the remaining

15

0.4

MB each. No file can be split between floppy disks. What is the minimal number of floppy disks that will hold all the files?

<span class='latex-bold'>(A)</span>\ 12 \qquad <span class='latex-bold'>(B)</span>\ 13 \qquad <span class='latex-bold'>(C)</span>\ 14 \qquad <span class='latex-bold'>(D)</span>\ 15 \qquad <span class='latex-bold'>(E)</span>\ 16

Consecutive Positive Integers

The product of three consecutive positive integers is

8

times their sum. What is the sum of their squares?

<span class='latex-bold'>(A)</span>\ 50 \qquad <span class='latex-bold'>(B)</span>\ 77 \qquad <span class='latex-bold'>(C)</span>\ 110 \qquad <span class='latex-bold'>(D)</span>\ 149 \qquad <span class='latex-bold'>(E)</span>\ 194

P(x) divisible by x-2

P(x)=kx^3+2k^2x^2+k^3

. Find the sum of all real numbers

k

x-2

P(x)

.

<span class='latex-bold'>(A) </span>-8\qquad<span class='latex-bold'>(B) </span>-4\qquad<span class='latex-bold'>(C) </span>0\qquad<span class='latex-bold'>(D) </span>5\qquad<span class='latex-bold'>(E) </span>8

3

Sum of Roots of Sum of Products

Compute the sum of all the roots of (2x \plus{} 3)(x \minus{} 4) \plus{} (2x \plus{} 3)(x \minus{} 6) \equal{} 0.

<span class='latex-bold'>(A)</span>\ 7/2 \qquad <span class='latex-bold'>(B)</span>\ 4 \qquad <span class='latex-bold'>(C)</span>\ 5 \qquad <span class='latex-bold'>(D)</span>\ 7 \qquad <span class='latex-bold'>(E)</span>\ 13

Quadratic Roots and Coefficients

a

b

are are nonzero real numbers, and that the equation x^2\plus{}ax\plus{}b\equal{}0 has solutions

a

b

. Then the pair

(a,b)

is
(A)\ (\minus{}2,1) \qquad (B)\ (\minus{}1,2) \qquad (C)\ (1,\minus{}2) \qquad (D)\ (2,\minus{}1) \qquad (E)\ (4,4)

Ratio of Distinct Real Numbers

a

b

be distinct real numbers for which

\dfrac ab+\dfrac{a+10b}{b+10a}=2.

\dfrac ab

.

<span class='latex-bold'>(A) </span>0.6\qquad<span class='latex-bold'>(B) </span>0.7\qquad<span class='latex-bold'>(C) </span>0.8\qquad<span class='latex-bold'>(D) </span>0.9\qquad<span class='latex-bold'>(E) </span>1

3

System to Average

A

B

C

are three numbers for which 1001C\minus{}2002A\equal{}4004 and 1001B\plus{}3003A\equal{}5005.The average of the three numbers

A

B

C

\text{(A)}\ 1 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ \text{not uniquely determined}

"USAMO" Permutations

Using the letters

A

M

O

S

U

120

five-letter "words". If these "words" are arranged in alphabetical order, then the "word"

USAMO

occupies position

<span class='latex-bold'>(A)</span>\ 112 \qquad <span class='latex-bold'>(B)</span>\ 113 \qquad <span class='latex-bold'>(C)</span>\ 114 \qquad <span class='latex-bold'>(D)</span>\ 115 \qquad <span class='latex-bold'>(E)</span>\ 116

Function Recurrence

f

is given by the table

\begin{array}{|c||c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 4 & 1 & 3 & 5 & 2 \\ \hline \end{array}

u_0=4

u_{n+1}=f(u_n)

n\geq 0

u_{2002}

.

<span class='latex-bold'>(A) </span>1\qquad<span class='latex-bold'>(B) </span>2\qquad<span class='latex-bold'>(C) </span>3\qquad<span class='latex-bold'>(D) </span>4\qquad<span class='latex-bold'>(E) </span>5

3

Betsy's Flag

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? [asy]unitsize(3mm); fill((-4,-4)--(-4,4)--(4,4)--(4,-4)--cycle,blue); fill((-2,-2)--(-2,2)--(2,2)--(2,-2)--cycle,red); path onewhite=(-3,3)--(-2,4)--(-1,3)--(-2,2)--(-3,3)--(-1,3)--(0,4)--(1,3)--(0,2)--(-1,3)--(1,3)--(2,4)--(3,3)--(2,2)--(1,3)--cycle; path divider=(-2,2)--(-3,3)--cycle; fill(onewhite,white); fill(rotate(90)*onewhite,white); fill(rotate(180)*onewhite,white); fill(rotate(270)*onewhite,white);[/asy] (A)\ B \equal{} W \qquad (B)\ W \equal{} R \qquad (C)\ B \equal{} R \qquad (D)\ 3B \equal{} 2R \qquad (E)\ 2R \equal{} W

July and August

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

<span class='latex-bold'>(A)</span>\ \text{Monday} \qquad <span class='latex-bold'>(B)</span>\ \text{Tuesday} \qquad <span class='latex-bold'>(C)</span>\ \text{Wednesday} \qquad <span class='latex-bold'>(D)</span>\ \text{Thursday} \qquad <span class='latex-bold'>(E)</span>\ \text{Friday}

Ordered Triples of Integers

How many ordered triples of positive integers

(x,y,z)

(x^y)^z=64

?

<span class='latex-bold'>(A) </span>5\qquad<span class='latex-bold'>(B) </span>6\qquad<span class='latex-bold'>(C) </span>7\qquad<span class='latex-bold'>(D) </span>8\qquad<span class='latex-bold'>(E) </span>9

3

Arc Ratio

45^\circ

A

has the same length as an arc of

30^\circ

B

, then the ratio of the area of circle

A

to the area of circle

B

<span class='latex-bold'>(A)</span>\ \frac {4}{9} \qquad <span class='latex-bold'>(B)</span>\ \frac {2}{3} \qquad <span class='latex-bold'>(C)</span>\ \frac {5}{6} \qquad <span class='latex-bold'>(D)</span>\ \frac {3}{2} \qquad <span class='latex-bold'>(E)</span>\ \frac {9}{4}

Sum of Fractions

n

be a positive integer such that \tfrac{1}{2}\plus{}\tfrac{1}{3}\plus{}\tfrac{1}{7}\plus{}\tfrac{1}{n} is an integer. Which of the following statements is not true?

<span class='latex-bold'>(A)</span>\ 2\text{ divides }n \qquad <span class='latex-bold'>(B)</span>\ 3\text{ divides }n \qquad <span class='latex-bold'>(C)</span>\ 6\text{ divides }n \qquad <span class='latex-bold'>(D)</span>\ 7\text{ divides }n \\ <span class='latex-bold'>(E)</span>\ n>84

Dimensions of Rectangular Box

The dimensions of a rectangular box in inches are all positive integers and the volume of the box is

2002\text{ in}^3

. Find the minimum possible sum in inches of the three dimensions.

<span class='latex-bold'>(A) </span>36\qquad<span class='latex-bold'>(B) </span>38\qquad<span class='latex-bold'>(C) </span>42\qquad<span class='latex-bold'>(D) </span>44\qquad<span class='latex-bold'>(E) </span>92

3

Cindy's Teacher's Request

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?

<span class='latex-bold'>(A)</span>\ 15 \qquad <span class='latex-bold'>(B)</span>\ 34 \qquad <span class='latex-bold'>(C)</span>\ 43 \qquad <span class='latex-bold'>(D)</span>\ 51 \qquad <span class='latex-bold'>(E)</span>\ 138

Quadratic Prime?

For how many positive integers

n

is n^2\minus{}3n\plus{}2 a prime number?

<span class='latex-bold'>(A)</span>\ \text{none} \qquad <span class='latex-bold'>(B)</span>\ \text{one} \qquad <span class='latex-bold'>(C)</span>\ \text{two} \qquad <span class='latex-bold'>(D)</span>\ \text{more than two, but finitely many}\\ <span class='latex-bold'>(E)</span>\ \text{infinitely many}

Area given Perimeter

The perimeter of a rectangle is

100

and its diagonal has length

x

. What is the area of this rectangle?

<span class='latex-bold'>(A) </span>625-x^2\qquad<span class='latex-bold'>(B) </span>625-\dfrac{x^2}2\qquad<span class='latex-bold'>(C) </span>1250-x^2\qquad<span class='latex-bold'>(D) </span>1250-\dfrac{x^2}2\qquad<span class='latex-bold'>(E) </span>2500-\dfrac{x^2}2

3

8 Circles

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. [asy]unitsize(.3cm); defaultpen(linewidth(.8pt)); path c=Circle((0,2),1); filldraw(Circle((0,0),3),grey,black); filldraw(Circle((0,0),1),white,black); filldraw(c,white,black); filldraw(rotate(60)*c,white,black); filldraw(rotate(120)*c,white,black); filldraw(rotate(180)*c,white,black); filldraw(rotate(240)*c,white,black); filldraw(rotate(300)*c,white,black);[/asy]

<span class='latex-bold'>(A)</span>\ \pi \qquad <span class='latex-bold'>(B)</span>\ 1.5\pi \qquad <span class='latex-bold'>(C)</span>\ 2\pi \qquad <span class='latex-bold'>(D)</span>\ 3\pi \qquad <span class='latex-bold'>(E)</span>\ 3.5\pi

Shaded Region in Circles

Circles of radius

2

3

are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region. [asy]unitsize(3mm); defaultpen(linewidth(0.7)+fontsize(8));
filldraw(Circle((0,0),5),grey,black); filldraw(Circle((-2,0),3),white,black); filldraw(Circle((3,0),2),white,black); dot((-2,0)); dot((3,0)); draw((-2,0)--(1,0)); draw((3,0)--(5,0)); label("

3

",(-0.5,0),N); label("

2

",(4,0),N);[/asy]

<span class='latex-bold'>(A)</span>\ 3\pi \qquad <span class='latex-bold'>(B)</span>\ 4\pi \qquad <span class='latex-bold'>(C)</span>\ 6\pi \qquad <span class='latex-bold'>(D)</span>\ 9\pi \qquad <span class='latex-bold'>(E)</span>\ 12\pi

Recursive Sequence

(a_n)_{n\geq 1}

be a sequence such that

a_1=1

3a_{n+1}-3a_n=1

n\geq 1

a_{2002}

.

<span class='latex-bold'>(A) </span>666\qquad<span class='latex-bold'>(B) </span>667\qquad<span class='latex-bold'>(C) </span>668\qquad<span class='latex-bold'>(D) </span>669\qquad<span class='latex-bold'>(E) </span>670

3

Mean, median, mode

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

<span class='latex-bold'>(A)</span>\ 11 \qquad <span class='latex-bold'>(B)</span>\ 12 \qquad <span class='latex-bold'>(C)</span>\ 13 \qquad <span class='latex-bold'>(D)</span>\ 14 \qquad <span class='latex-bold'>(E)</span>\ 15

Lawnmowers

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?

<span class='latex-bold'>(A)</span>\ \text{Andy} \qquad <span class='latex-bold'>(B)</span>\ \text{Beth} \qquad <span class='latex-bold'>(C)</span>\ \text{Carlos} \qquad <span class='latex-bold'>(D)</span>\ \text{Andy and Carlos tie for first.}

<span class='latex-bold'>(E)</span>\ \text{All three tie.}

Real Valued Function

f

be a real-valued function such that

f(x)+2f\left(\dfrac{2002}x\right)=3x

x>0

f(2)

.

<span class='latex-bold'>(A) </span>1000\qquad<span class='latex-bold'>(B) </span>2000\qquad<span class='latex-bold'>(C) </span>3000\qquad<span class='latex-bold'>(D) </span>4000\qquad<span class='latex-bold'>(E) </span>6000

3

Removing and labelling tiles

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 operation be performed to reduce the number of tiles in the set to one?

\text{(A)}\ 10 \qquad \text{(B)}\ 11 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 19 \qquad \text{(E)}\ 20

Right Triangle Medians

\triangle{XOY}

be a right-angled triangle with m\angle{XOY}\equal{}90^\circ. Let

M

N

be the midpoints of legs

OX

OY

, respectively. Given that XN\equal{}19 and YM\equal{}22, find

XY

<span class='latex-bold'>(A)</span>\ 24 \qquad <span class='latex-bold'>(B)</span>\ 26 \qquad <span class='latex-bold'>(C)</span>\ 28 \qquad <span class='latex-bold'>(D)</span>\ 30 \qquad <span class='latex-bold'>(E)</span>\ 32

Zeroes Ending a Number

In how many zeroes does the number

\dfrac{2002!}{(1001!)^2}

<span class='latex-bold'>(A) </span>0\qquad<span class='latex-bold'>(B) </span>1\qquad<span class='latex-bold'>(C) </span>2\qquad<span class='latex-bold'>(D) </span>200\qquad<span class='latex-bold'>(E) </span>400

3

Triangle

A,B,C

D

lie on a line, in that order, with AB\equal{}CD and BC\equal{}12. Point

E

is not on the line, and BE\equal{}CE\equal{}10. The perimeter of

\triangle AED

is twice the perimeter of

\triangle BEC

AB

\text{(A)}\ 15/2 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 17/2 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 19/2

Sequence

\{a_k\}

be a sequence of integers such that a_1 \equal{} 1 and a_{m \plus{} n} \equal{} a_m \plus{} a_n \plus{} mn, for all positive integers

m

n

a_{12}

<span class='latex-bold'>(A)</span>\ 45 \qquad <span class='latex-bold'>(B)</span>\ 56 \qquad <span class='latex-bold'>(C)</span>\ 67 \qquad <span class='latex-bold'>(D)</span>\ 78 \qquad <span class='latex-bold'>(E)</span>\ 89

Differences in Values

a=\dfrac{1^2}1+\dfrac{2^2}3+\dfrac{3^2}5+\cdots+\dfrac{1001^2}{2001}

b=\dfrac{1^2}3+\dfrac{2^2}5+\dfrac{3^2}7+\cdots+\dfrac{1001^2}{2003}.

Find the integer closest to

a-b

.

<span class='latex-bold'>(A) </span>500\qquad<span class='latex-bold'>(B) </span>501\qquad<span class='latex-bold'>(C) </span>999\qquad<span class='latex-bold'>(D) </span>1000\qquad<span class='latex-bold'>(E) </span>1001

3

Tina and Sergio pick numbers

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

<span class='latex-bold'>(A)</span>\ 2/5 \qquad <span class='latex-bold'>(B)</span>\ 9/20 \qquad <span class='latex-bold'>(C)</span>\ 1/2\qquad <span class='latex-bold'>(D)</span>\ 11/20 \qquad <span class='latex-bold'>(E)</span>\ 24/25

Ferris Wheel

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?

<span class='latex-bold'>(A)</span>\ 5 \qquad <span class='latex-bold'>(B)</span>\ 6 \qquad <span class='latex-bold'>(C)</span>\ 7.5 \qquad <span class='latex-bold'>(D)</span>\ 10 \qquad <span class='latex-bold'>(E)</span>\ 15

Sum of Squares Equals 2002

What is the maximum value of

n

for which there is a set of distinct positive integers

k_1,k_2,\ldots,k_n

k_1^2+k_2^2+\ldots+k_n^2=2002?

<span class='latex-bold'>(A) </span>14\qquad<span class='latex-bold'>(B) </span>15\qquad<span class='latex-bold'>(C) </span>16\qquad<span class='latex-bold'>(D) </span>17\qquad<span class='latex-bold'>(E) </span>18

3

Find area of trapezoid

ABCD

AB

CD

, we have AB\equal{}52, BC\equal{}12, CD\equal{}39, and DA\equal{}5. The area of

ABCD

is
[asy] pair A,B,C,D; A=(0,0); B=(52,0); C=(38,20); D=(5,20); dot(A); dot(B); dot(C); dot(D); draw(A--B--C--D--cycle); label("

A

",A,S); label("

B

",B,S); label("

C

",C,N); label("

D

",D,N); label("52",(A+B)/2,S); label("39",(C+D)/2,N); label("12",(B+C)/2,E); label("5",(D+A)/2,W);[/asy]

\text{(A)}\ 182 \qquad \text{(B)}\ 195 \qquad \text{(C)}\ 210 \qquad \text{(D)}\ 234 \qquad \text{(E)}\ 260

List of Integers

15

is appended to a list of integers, the mean is increased by

2

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?

<span class='latex-bold'>(A)</span>\ 4 \qquad <span class='latex-bold'>(B)</span>\ 5 \qquad <span class='latex-bold'>(C)</span>\ 6 \qquad <span class='latex-bold'>(D)</span>\ 7 \qquad <span class='latex-bold'>(E)</span>\ 8

Exactly Three Ways to Score

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

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?

<span class='latex-bold'>(A) </span>175\qquad<span class='latex-bold'>(B) </span>179.5\qquad<span class='latex-bold'>(C) </span>182\qquad<span class='latex-bold'>(D) </span>188.5\qquad<span class='latex-bold'>(E) </span>201

3

Product-Sum Inequality

For how many positive integers

m

does there exist at least one positive integer

n

such that m\cdot n \le m \plus{} n?

<span class='latex-bold'>(A)</span>\ 4 \qquad <span class='latex-bold'>(B)</span>\ 6 \qquad <span class='latex-bold'>(C)</span>\ 9 \qquad <span class='latex-bold'>(D)</span>\ 12 \qquad <span class='latex-bold'>(E)</span>

infinitely many

Evaluating

What is the value of (3x \minus{} 2)(4x \plus{} 1) \minus{} (3x \minus{} 2)4x \plus{} 1when x \equal{} 4?

<span class='latex-bold'>(A)</span>\ 0 \qquad <span class='latex-bold'>(B)</span>\ 1 \qquad <span class='latex-bold'>(C)</span>\ 10 \qquad <span class='latex-bold'>(D)</span>\ 11 \qquad <span class='latex-bold'>(E)</span>\ 12

Perfect Square

Which of the following numbers is a perfect square?

<span class='latex-bold'>(A) </span>4^45^56^6\qquad<span class='latex-bold'>(B) </span>4^45^66^5\qquad<span class='latex-bold'>(C) </span>4^55^46^6\qquad<span class='latex-bold'>(D) </span>4^65^46^5\qquad<span class='latex-bold'>(E) </span>4^65^56^4

3

Exponentiation Order

According to the standard convention for exponentiation, 2^{2^{2^2}} \equal{} 2^{\left(2^{\left(2^2\right)}\right)} \equal{} 2^{16} \equal{} 65,\!536. If the order in which the exponentiations are performed is changed, how many other values are possible?

<span class='latex-bold'>(A)</span>\ 0 \qquad <span class='latex-bold'>(B)</span>\ 1 \qquad <span class='latex-bold'>(C)</span>\ 2 \qquad <span class='latex-bold'>(D)</span>\ 3 \qquad <span class='latex-bold'>(E)</span>\ 4

Arithmetic Mean

The arithmetic mean of the nine numbers in the set

\{9,99,999,9999,...,999999999\}

9

M

, all of whose digits are distinct. The number

M

does not contain the digit

<span class='latex-bold'>(A)</span>\ 0 \qquad <span class='latex-bold'>(B)</span>\ 2 \qquad <span class='latex-bold'>(C)</span>\ 4 \qquad <span class='latex-bold'>(D)</span>\ 6 \qquad <span class='latex-bold'>(E)</span>\ 8

Typo-ing a Six Digit Number

Mary typed a six-digit number, but the two

1

s she typed didn't show. What appeared was

2002

. How many different six-digit numbers could she have typed?

<span class='latex-bold'>(A) </span>4\qquad<span class='latex-bold'>(B) </span>8\qquad<span class='latex-bold'>(C) </span>10\qquad<span class='latex-bold'>(D) </span>15\qquad<span class='latex-bold'>(E) </span>20

3

Symmetric Quotient Function

For the nonzero numbers

a

b

c

, define (a,b,c)\equal{}\frac{a}{b}\plus{}\frac{b}{c}\plus{}\frac{c}{a}. Find

(2,12,9)

<span class='latex-bold'>(A)</span>\ 4 \qquad <span class='latex-bold'>(B)</span>\ 5 \qquad <span class='latex-bold'>(C)</span>\ 6 \qquad <span class='latex-bold'>(D)</span>\ 7 \qquad <span class='latex-bold'>(E)</span>\ 8

Function on Three Variables

For the nonzero numbers

a

b

c

, define (a,b,c)\equal{}\frac{abc}{a\plus{}b\plus{}c}. Find

(2,4,6)

.

<span class='latex-bold'>(A)</span>\ 1 \qquad <span class='latex-bold'>(B)</span>\ 2 \qquad <span class='latex-bold'>(C)</span>\ 4 \qquad <span class='latex-bold'>(D)</span>\ 6 \qquad <span class='latex-bold'>(E)</span>\ 24

Sum of Eleven Consecutive Integers

The sum of eleven consecutive integers is

2002

. What is the smallest of these integers?

<span class='latex-bold'>(A) </span>175\qquad<span class='latex-bold'>(B) </span>177\qquad<span class='latex-bold'>(C) </span>179\qquad<span class='latex-bold'>(D) </span>180\qquad<span class='latex-bold'>(E) </span>181

3

Powers of 10 Estimation

The ratio \dfrac{10^{2000}\plus{}10^{2002}}{10^{2001}\plus{}10^{2001}} is closest to which of the following numbers?

\text{(A)}\ 0.1\qquad \text{(B)}\ 0.2\qquad \text{(C)}\ 1\qquad \text{(D)}\ 5\qquad \text{(E)}\ 10

Exponent Ratio

\frac{2^{2001}\cdot3^{2003}}{6^{2002}}

<span class='latex-bold'>(A)</span>\ \frac{1}{6} \qquad <span class='latex-bold'>(B)</span>\ \frac{1}{3} \qquad <span class='latex-bold'>(C)</span>\ \frac{1}{2} \qquad <span class='latex-bold'>(D)</span>\ \frac{2}{3} \qquad <span class='latex-bold'>(E)</span>\ \frac{3}{2}

Equaling a Ratio

\dfrac{(2^4)^8}{(4^8)^2}

<span class='latex-bold'>(A) </span>\dfrac14\qquad<span class='latex-bold'>(B) </span>\dfrac12\qquad<span class='latex-bold'>(C) </span>1\qquad<span class='latex-bold'>(D) </span>2\qquad<span class='latex-bold'>(E) </span>8