2003 AMC 10

Part of AMC 10

Subcontests

(25)

2

Toothpick Triangles

A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the ﬁgure we have

3

rows of small congruent equilateral triangles, with

5

small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of

2003

small equilateral triangles? [asy]unitsize(15mm); defaultpen(linewidth(.8pt)+fontsize(8pt));
pair Ap=(0,0), Bp=(1,0), Cp=(2,0), Dp=(3,0), Gp=dir(60); pair Fp=shift(Gp)*Bp, Ep=shift(Gp)*Cp; pair Hp=shift(Gp)*Gp, Ip=shift(Gp)*Fp; pair Jp=shift(Gp)*Hp; pair[] points={Ap,Bp,Cp,Dp,Ep,Fp,Gp,Hp,Ip,Jp};
draw(Ap--Dp--Jp--cycle); draw(Gp--Bp--Ip--Hp--Cp--Ep--cycle);
for(pair p : points) { fill(circle(p, 0.07),white); }
pair[] Cn=new pair[5]; Cn[0]=centroid(Ap,Bp,Gp); Cn[1]=centroid(Gp,Bp,Fp); Cn[2]=centroid(Bp,Fp,Cp); Cn[3]=centroid(Cp,Fp,Ep); Cn[4]=centroid(Cp,Ep,Dp); label("

1

",Cn[0]); label("

2

",Cn[1]); label("

3

",Cn[2]); label("

4

",Cn[3]); label("

5

",Cn[4]); for (pair p : Cn) { draw(circle(p,0.1)); }[/asy]

<span class='latex-bold'>(A)</span>\ 1,\!004,\!004 \qquad <span class='latex-bold'>(B)</span>\ 1,\!005,\!006 \qquad <span class='latex-bold'>(C)</span>\ 1,\!507,\!509 \qquad <span class='latex-bold'>(D)</span>\ 3,\!015,\!018 \qquad <span class='latex-bold'>(E)</span>\ 6,\!021,\!018

Octagon/Rectangle Area

A regular octagon

ABCDEFGH

has an area of one square unit. What is the area of the rectangle

ABEF

? [asy]unitsize(8mm); defaultpen(linewidth(.8pt)+fontsize(6pt));
pair C=dir(22.5), B=dir(67.5), A=dir(112.5), H=dir(157.5), G=dir(202.5), F=dir(247.5), E=dir(292.5), D=dir(337.5);
draw(A--B--C--D--E--F--G--H--cycle); label("

A

",A,NNW); label("

B

",B,NNE); label("

C

",C,ENE); label("

D

",D,ESE); label("

E

",E,SSE); label("

F

",F,SSW); label("

G

",G,WSW); label("

H

",H,WNW);[/asy] (A)\ 1\minus{}\frac{\sqrt2}{2} \qquad (B)\ \frac{\sqrt2}{4} \qquad (C)\ \sqrt2\minus{}1 \qquad (D)\ \frac12 \qquad (E)\ \frac{1\plus{}\sqrt2}{4}

2

Rectangle and Lines

ABCD

, we have AB\equal{}8, BC\equal{}9,

H

\overline{BC}

with BH\equal{}6,

E

\overline{AD}

with DE\equal{}4, line

EC

intersects line

AH

G

F

AD

\overline{GF}\perp\overline{AF}

. Find the length

GF

. [asy]unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(8pt));
pair D=(0,0), Ep=(4,0), A=(9,0), B=(9,8), H=(3,8), C=(0,8), G=(-6,20), F=(-6,0);
draw(D--A--B--C--D--F--G--Ep); draw(A--G); label("

F

",F,W); label("

G

",G,W); label("

C

",C,WSW); label("

H

",H,NNE); label("

6

",(6,8),N); label("

B

",B,NE); label("

A

",A,SW); label("

E

",Ep,S); label("

4

",(2,0),S); label("

D

<span class='latex-bold'>(A)</span>\ 16 \qquad <span class='latex-bold'>(B)</span>\ 20 \qquad <span class='latex-bold'>(C)</span>\ 24 \qquad <span class='latex-bold'>(D)</span>\ 28 \qquad <span class='latex-bold'>(E)</span>\ 30

Clock's Chimes

A clock chimes once at

30

minutes past each hour and chimes on the hour according to the hour. For example, at 1 PM there is one chime and at noon and midnight there are twelve chimes. Starting at 11:15 AM on February

26

2003

, on what date will the

2003^{\text{rd}}

<span class='latex-bold'>(A)</span>\ \text{March 8} \qquad <span class='latex-bold'>(B)</span>\ \text{March 9} \qquad <span class='latex-bold'>(C)</span>\ \text{March 10} \qquad <span class='latex-bold'>(D)</span>\ \text{March 20} \qquad <span class='latex-bold'>(E)</span>\ \text{March 21}

2

Pat's Cookies

Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How many different assortments of six cookies can be selected?

<span class='latex-bold'>(A)</span>\ 22 \qquad <span class='latex-bold'>(B)</span>\ 25 \qquad <span class='latex-bold'>(C)</span>\ 27 \qquad <span class='latex-bold'>(D)</span>\ 28 \qquad <span class='latex-bold'>(E)</span>\ 29

Bead Probability

A bag contains two red beads and two green beads. You reach into the bag and pull out a bead, replacing it with a red bead regardless of the color you pulled out. What is the probability that all beads in the bag are red after three such replacements?

<span class='latex-bold'>(A)</span>\ \frac{1}{8} \qquad <span class='latex-bold'>(B)</span>\ \frac{5}{32} \qquad <span class='latex-bold'>(C)</span>\ \frac{9}{32} \qquad <span class='latex-bold'>(D)</span>\ \frac{3}{8} \qquad <span class='latex-bold'>(E)</span>\ \frac{7}{16}

2

Random Three-Digit Number

10

three-digit number

n

is selected at random. Which of the following is closest to the probability that the base-

9

representation and the base-

11

representation of

n

are both three-digit numerals?

<span class='latex-bold'>(A)</span>\ 0.3 \qquad <span class='latex-bold'>(B)</span>\ 0.4 \qquad <span class='latex-bold'>(C)</span>\ 0.5 \qquad <span class='latex-bold'>(D)</span>\ 0.6 \qquad <span class='latex-bold'>(E)</span>\ 0.7

Area of Triangle

ABCD

, AB\equal{}5 and BC\equal{}3. Points

F

G

\overline{CD}

so that DF\equal{}1 and GC\equal{}2. Lines

AF

BG

E

. Find the area of

\triangle{AEB}

. [asy]unitsize(6mm); defaultpen(linewidth(.8pt)+fontsize(8pt));
pair A=(0,0), B=(5,0), C=(5,3), D=(0,3), F=(1,3), G=(3,3); pair E=extension(A,F,B,G); draw(A--B--C--D--A--E--B);
label("

A

",A,SW); label("

B

",B,SE); label("

C

",C,NE); label("

D

",D,NW); label("

E

",E,N); label("

F

",F,SE); label("

G

",G,SW); label("

B

",B,SE);
label("1",midpoint(D--F),N); label("2",midpoint(G--C),N); label("3",midpoint(B--C),E); label("3",midpoint(A--D),W); label("5",midpoint(A--B),S);[/asy]

<span class='latex-bold'>(A)</span>\ 10 \qquad <span class='latex-bold'>(B)</span>\ \frac{21}{2} \qquad <span class='latex-bold'>(C)</span>\ 12 \qquad <span class='latex-bold'>(D)</span>\ \frac{25}{2} \qquad <span class='latex-bold'>(E)</span>\ 15

2

Sum of Root Reciprocals

What is the sum of the reciprocals of the roots of the equation \frac {2003}{2004}x \plus{} 1 \plus{} \frac {1}{x} \equal{} 0? (A)\ \minus{}\! \frac {2004}{2003} \qquad (B)\ \minus{} \!1 \qquad (C)\ \frac {2003}{2004} \qquad (D)\ 1 \qquad (E)\ \frac {2004}{2003}

Polynomial Divisor

What is the largest integer that is a divisor of (n\plus{}1)(n\plus{}3)(n\plus{}5)(n\plus{}7)(n\plus{}9) for all positive even integers

n

<span class='latex-bold'>(A)</span>\ 3 \qquad <span class='latex-bold'>(B)</span>\ 5 \qquad <span class='latex-bold'>(C)</span>\ 11 \qquad <span class='latex-bold'>(D)</span>\ 15 \qquad <span class='latex-bold'>(E)</span>\ 165

2

Circling the Triangle

The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle?

<span class='latex-bold'>(A)</span>\ \frac{3\sqrt2}{\pi} \qquad <span class='latex-bold'>(B)</span>\ \frac{3\sqrt3}{\pi} \qquad <span class='latex-bold'>(C)</span>\ \sqrt3 \qquad <span class='latex-bold'>(D)</span>\ \frac{6}{\pi} \qquad <span class='latex-bold'>(E)</span>\ \sqrt3\pi

Ice Cream Cone

An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly ﬁll the cone. Assume that the melted ice cream occupies

75\%

of the volume of the frozen ice cream. What is the ratio of the cone’s height to its radius?

<span class='latex-bold'>(A)</span>\ 2: 1 \qquad <span class='latex-bold'>(B)</span>\ 3: 1 \qquad <span class='latex-bold'>(C)</span>\ 4: 1 \qquad <span class='latex-bold'>(D)</span>\ 16: 3 \qquad <span class='latex-bold'>(E)</span>\ 6: 1

2

Units Digit

What is the units digit of

13^{2003}

<span class='latex-bold'>(A)</span>\ 1 \qquad <span class='latex-bold'>(B)</span>\ 3 \qquad <span class='latex-bold'>(C)</span>\ 7 \qquad <span class='latex-bold'>(D)</span>\ 8 \qquad <span class='latex-bold'>(E)</span>\ 9

Restaurant Dinner

A restaurant offers three desserts, and exactly twice as many appetizers as main courses. A dinner consists of an appetizer, a main course, and a dessert. What is the least number of main courses that the restaurant should offer so that a customer could have a different dinner each night in the year

2003

<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

2

Divisibility Probability

What is the probability that an integer in the set

\{1,2,3,\ldots,100\}

is divisible by

2

and not divisible by

3

<span class='latex-bold'>(A)</span>\ \frac{1}{6} \qquad <span class='latex-bold'>(B)</span>\ \frac{33}{100} \qquad <span class='latex-bold'>(C)</span>\ \frac{17}{50} \qquad <span class='latex-bold'>(D)</span>\ \frac{1}{2} \qquad <span class='latex-bold'>(E)</span>\ \frac{18}{25}

Singles Tennis

100

players in a singles tennis tournament. The tournament is single elimination, meaning that a player who loses a match is eliminated. In the first round, the strongest

28

players are given a bye, and the remaining

72

players are paired off to play. After each round, the remaining players play in the next round. The match continues until only one player remains unbeaten. The total number of matches played is

<span class='latex-bold'>(A)</span>\ \text{a prime number} \qquad <span class='latex-bold'>(B)</span>\ \text{divisible by 2} \qquad <span class='latex-bold'>(C)</span>\ \text{divisible by 5}

<span class='latex-bold'>(D)</span>\ \text{divisible by 7} \qquad <span class='latex-bold'>(E)</span>\ \text{divisible by 11}

2

Product of Three Primes

n

be the largest integer that is the product of exactly

3

distinct prime numbers,

d

e

, and 10d\plus{}e, where

d

e

are single digits. What is the sum of the digits of

n

<span class='latex-bold'>(A)</span>\ 12 \qquad <span class='latex-bold'>(B)</span>\ 15 \qquad <span class='latex-bold'>(C)</span>\ 18 \qquad <span class='latex-bold'>(D)</span>\ 21 \qquad <span class='latex-bold'>(E)</span>\ 24

Power Minimization

Given that 3^8\cdot5^2 \equal{} a^b, where both

a

b

are positive integers, find the smallest possible value for a \plus{} b.

<span class='latex-bold'>(A)</span>\ 25 \qquad <span class='latex-bold'>(B)</span>\ 34 \qquad <span class='latex-bold'>(C)</span>\ 351 \qquad <span class='latex-bold'>(D)</span>\ 407 \qquad <span class='latex-bold'>(E)</span>\ 900

2

Three Numbers

The sum of three numbers is

20

. The ﬁrst is

4

times the sum of the other two. The second is seven times the third. What is the product of all three?

<span class='latex-bold'>(A)</span>\ 28 \qquad <span class='latex-bold'>(B)</span>\ 40 \qquad <span class='latex-bold'>(C)</span>\ 100 \qquad <span class='latex-bold'>(D)</span>\ 400 \qquad <span class='latex-bold'>(E)</span>\ 800

Digital Sum

\clubsuit(x)

denote the sum of the digits of the positive integer

x

. For example, \clubsuit(8)\equal{}8 and \clubsuit(123)\equal{}1\plus{}2\plus{}3\equal{}6. For how many two-digit values of

x

is \clubsuit(\clubsuit(x))\equal{}3?

<span class='latex-bold'>(A)</span>\ 3 \qquad <span class='latex-bold'>(B)</span>\ 4 \qquad <span class='latex-bold'>(C)</span>\ 6 \qquad <span class='latex-bold'>(D)</span>\ 9 \qquad <span class='latex-bold'>(E)</span>\ 10

2

Random Point

(x,y)

is randomly picked from inside the rectangle with vertices

(0,0)

(4,0)

(4,1)

(0,1)

. What is the probability that

x<y

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

Friends' Investments

Al, Betty, and Clare split

\$1000

among them to be invested in different ways. Each begins with a different amount. At the end of one year they have a total of

\$1500

. Betty and Clare have both doubled their money, whereas Al has managed to lose

\$100

. What was Al’s original portion?

<span class='latex-bold'>(A)</span>\ \$ 250 \qquad <span class='latex-bold'>(B)</span>\ \$ 350 \qquad <span class='latex-bold'>(C)</span>\ \$ 400 \qquad <span class='latex-bold'>(D)</span>\ \$ 450 \qquad <span class='latex-bold'>(E)</span>\ \$ 500

2

Nested Roots

\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}}

<span class='latex-bold'>(A)</span>\ \sqrt{x} \qquad <span class='latex-bold'>(B)</span>\ \sqrt[3]{x^2} \qquad <span class='latex-bold'>(C)</span>\ \sqrt[27]{x^2} \qquad <span class='latex-bold'>(D)</span>\ \sqrt[54]{x} \qquad <span class='latex-bold'>(E)</span>\ \sqrt[81]{x^{80}}

Exponent Solving

Find the value of

x

that satisfies the equation 25^{\minus{}2}\equal{}\frac{5^{48/x}}{5^{26/x}\cdot25^{17/x}}.

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

2

Root Product

d

e

denote the solutions of 2x^2\plus{}3x\minus{}5\equal{}0. What is the value of (d\minus{}1)(e\minus{}1)? (A)\ \minus{}\frac{5}{2} \qquad (B)\ 0 \qquad (C)\ 3 \qquad (D)\ 5 \qquad (E)\ 6

Moeing the Lawn

Moe uses a mower to cut his rectangular

90

150

-foot lawn. The swath he cuts is

28

inches wide, but he overlaps each cut by

4

inches to make sure that no grass is missed. He walks at the rate of

5000

feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow his lawn?

<span class='latex-bold'>(A)</span>\ 0.75 \qquad <span class='latex-bold'>(B)</span>\ 0.8 \qquad <span class='latex-bold'>(C)</span>\ 1.35 \qquad <span class='latex-bold'>(D)</span>\ 1.5 \qquad <span class='latex-bold'>(E)</span>\ 3

2

5-Digit Number Divided by 100

n

5

-digit number, and let

q

r

be the quotient and remainder, respectively, when

n

100

. For how many values of

n

is q \plus{} r divisible by

11

<span class='latex-bold'>(A)</span>\ 8180 \qquad <span class='latex-bold'>(B)</span>\ 8181 \qquad <span class='latex-bold'>(C)</span>\ 8182 \qquad <span class='latex-bold'>(D)</span>\ 9000 \qquad <span class='latex-bold'>(E)</span>\ 9090

Counting Four-Digit Numbers

How many distinct four-digit numbers are divisible by

3

23

as their last two digits?

<span class='latex-bold'>(A)</span>\ 27 \qquad <span class='latex-bold'>(B)</span>\ 30 \qquad <span class='latex-bold'>(C)</span>\ 33 \qquad <span class='latex-bold'>(D)</span>\ 81 \qquad <span class='latex-bold'>(E)</span>\ 90

2

Lune

A semicircle of diameter

1

sits at the top of a semicircle of diameter

2

, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune. [asy]unitsize(2.5cm); defaultpen(fontsize(10pt)+linewidth(.8pt));
filldraw(Circle((0,.866),.5),grey,black); label("1",(0,.866),S); filldraw(Circle((0,0),1),white,black); draw((-.5,.866)--(.5,.866),linetype("4 4")); clip((-1,0)--(1,0)--(1,2)--(-1,2)--cycle); draw((-1,0)--(1,0)); label("2",(0,0),S);[/asy] (A)\ \frac {1}{6}\pi \minus{} \frac {\sqrt {3}}{4} \qquad (B)\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{12}\pi \qquad (C)\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{24}\pi\qquad(D)\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{24}\pi (E)\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{12}\pi

Four Semicircles

Three semicircles of radius

1

are constructed on diameter

AB

of a semicircle of radius

2

. The centers of the small semicircles divide

\overline{AB}

into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles? [asy]import graph; unitsize(14mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dashed=linetype("4 4"); dotfactor=3;
pair A=(-2,0), B=(2,0);
fill(Arc((0,0),2,0,180)--cycle,mediumgray); fill(Arc((-1,0),1,0,180)--cycle,white); fill(Arc((0,0),1,0,180)--cycle,white); fill(Arc((1,0),1,0,180)--cycle,white); draw(Arc((-1,0),1,60,180)); draw(Arc((0,0),1,0,60),dashed); draw(Arc((0,0),1,60,120)); draw(Arc((0,0),1,120,180),dashed); draw(Arc((1,0),1,0,120)); draw(Arc((0,0),2,0,180)--cycle);
dot((0,0)); dot((-1,0)); dot((1,0));
draw((-2,-0.1)--(-2,-0.3),gray); draw((-1,-0.1)--(-1,-0.3),gray); draw((1,-0.1)--(1,-0.3),gray); draw((2,-0.1)--(2,-0.3),gray);
label("

A

",A,W); label("

B

",B,E); label("1",(-1.5,-0.1),S); label("2",(0,-0.1),S); label("1",(1.5,-0.1),S);[/asy] (A)\ \pi\minus{}\sqrt3 \qquad (B)\ \pi\minus{}\sqrt2 \qquad (C)\ \frac{\pi\plus{}\sqrt2}{2} \qquad (D)\ \frac{\pi\plus{}\sqrt3}{2} (E)\ \frac{7}{6}\pi\minus{}\frac{\sqrt3}{2}

2

Pentomino to Cube?

The polygon enclosed by the solid lines in the ﬁgure consists of

4

congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing? [asy]unitsize(10mm); defaultpen(fontsize(10pt)); pen finedashed=linetype("4 4");
filldraw((1,1)--(2,1)--(2,2)--(4,2)--(4,3)--(1,3)--cycle,grey,black+linewidth(.8pt)); draw((0,1)--(0,3)--(1,3)--(1,4)--(4,4)--(4,3)-- (5,3)--(5,2)--(4,2)--(4,1)--(2,1)--(2,0)--(1,0)--(1,1)--cycle,finedashed); draw((0,2)--(2,2)--(2,4),finedashed); draw((3,1)--(3,4),finedashed); label("

1

",(1.5,0.5)); draw(circle((1.5,0.5),.17)); label("

2

",(2.5,1.5)); draw(circle((2.5,1.5),.17)); label("

3

",(3.5,1.5)); draw(circle((3.5,1.5),.17)); label("

4

",(4.5,2.5)); draw(circle((4.5,2.5),.17)); label("

5

",(3.5,3.5)); draw(circle((3.5,3.5),.17)); label("

6

",(2.5,3.5)); draw(circle((2.5,3.5),.17)); label("

7

",(1.5,3.5)); draw(circle((1.5,3.5),.17)); label("

8

",(0.5,2.5)); draw(circle((0.5,2.5),.17)); label("

9

",(0.5,1.5)); draw(circle((0.5,1.5),.17));[/asy]

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

Nebraskan License Plates

Nebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four digits. Each new license plate consists of three letters followed by three digits. By how many times is the number of possible license plates increased?

<span class='latex-bold'>(A)</span>\ \frac{26}{10} \qquad <span class='latex-bold'>(B)</span>\ \frac{26^2}{10^2} \qquad <span class='latex-bold'>(C)</span>\ \frac{26^2}{10} \qquad <span class='latex-bold'>(D)</span>\ \frac{26^3}{10^3} \qquad <span class='latex-bold'>(E)</span>\ \frac{26^3}{10^2}

2

Sally's Weird Stack of Cards

Sally has ﬁve red cards numbered

1

5

and four blue cards numbered

3

6

. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?

<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>\ 11 \qquad <span class='latex-bold'>(E)</span>\ 12

Arithmetic Sequence

The ﬁrst four terms in an arithmetic sequence are x \plus{} y, x \minus{} y,

xy

x/y

, in that order. What is the ﬁfth term? (A)\ \minus{}\frac{15}{8} \qquad (B)\ \minus{}\frac{6}{5} \qquad (C)\ 0 \qquad (D)\ \frac{27}{20} \qquad (E)\ \frac{123}{40}

2

Factors of 60

What is the probability that a randomly drawn positive factor of

60

7

?

<span class='latex-bold'>(A)</span>\ \frac{1}{10} \qquad <span class='latex-bold'>(B)</span>\ \frac{1}{6} \qquad <span class='latex-bold'>(C)</span>\ \frac{1}{4} \qquad <span class='latex-bold'>(D)</span>\ \frac{1}{3} \qquad <span class='latex-bold'>(E)</span>\ \frac{1}{2}

Geometric Sequence

The second and fourth terms of a geometric sequence are

2

6

. Which of the following is a possible first term? (A)\ \minus{}\!\sqrt3 \qquad (B)\ \minus{}\!\frac{2\sqrt3}{3} \qquad (C)\ \minus{}\!\frac{\sqrt3}{3} \qquad (D)\ \sqrt3 \qquad (E)\ 3

2

Perimeter 7 Triangles

How many non-congruent triangles with perimeter

7

have integer side lengths?

<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

Floor Sum

\lfloor x\rfloor

denotes the largest integer not exceeding

x

. For example. \lfloor3\rfloor\equal{}3, and \lfloor 9/2\rfloor\equal{}4. Compute \lfloor\sqrt1\rfloor\plus{}\lfloor\sqrt2\rfloor\plus{}\lfloor\sqrt3\rfloor\plus{}\cdots\plus{}\lfloor\sqrt{16}\rfloor.

<span class='latex-bold'>(A)</span>\ 35 \qquad <span class='latex-bold'>(B)</span>\ 38 \qquad <span class='latex-bold'>(C)</span>\ 40 \qquad <span class='latex-bold'>(D)</span>\ 42 \qquad <span class='latex-bold'>(E)</span>\ 136

2

Heart Function

x \heartsuit y

to be |x\minus{}y| for all real numbers

x

y

. Which of the following statements is not true?
(A)\ x \heartsuit y \equal{} y \heartsuit x \text{ for all } x \text{ and } y (B)\ 2(x \heartsuit y) \equal{} (2x) \heartsuit (2y) \text{ for all } x \text{ and } y (C)\ x \heartsuit 0 \equal{} x \text{ for all } x (D)\ x \heartsuit x \equal{} 0 \text{ for all } x

<span class='latex-bold'>(E)</span>\ x \heartsuit y > 0 \text{ if } x \ne y

Big TV Screen

Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is

4 : 3

. The horizontal length of a “

27

-inch” television screen is closest, in inches, to which of the following? [asy]import math; unitsize(7mm); defaultpen(linewidth(.8pt)+fontsize(8pt));
draw((0,0)--(4,0)--(4,3)--(0,3)--(0,0)--(4,3)); fill((0,0)--(4,0)--(4,3)--cycle,mediumgray); label(rotate(aTan(3.0/4.0))*"Diagonal",(2,1.5),NW); label(rotate(90)*"Height",(4,1.5),E); label("Length",(2,0),S);[/asy]

<span class='latex-bold'>(A)</span>\ 20 \qquad <span class='latex-bold'>(B)</span>\ 20.5 \qquad <span class='latex-bold'>(C)</span>\ 21 \qquad <span class='latex-bold'>(D)</span>\ 21.5 \qquad <span class='latex-bold'>(E)</span>\ 22

2

Numbers AMC10+AMC12

The sum of the two

5

AMC10

AMC12

123422

. What is A\plus{}M\plus{}C?

<span class='latex-bold'>(A)</span>\ 10 \qquad <span class='latex-bold'>(B)</span>\ 11 \qquad <span class='latex-bold'>(C)</span>\ 12 \qquad <span class='latex-bold'>(D)</span>\ 13 \qquad <span class='latex-bold'>(E)</span>\ 14

x-intercepts

A line with slope

3

intersects a line with slope

5

(10, 15)

. What is the distance between the

x

-intercepts of these two lines?

<span class='latex-bold'>(A)</span>\ 2 \qquad <span class='latex-bold'>(B)</span>\ 5 \qquad <span class='latex-bold'>(C)</span>\ 7 \qquad <span class='latex-bold'>(D)</span>\ 12 \qquad <span class='latex-bold'>(E)</span>\ 20

2

Mary's Uphill Journey

30

minutes to walk uphill

1

km from her home to school, but it takes her only

10

minutes to walk from school to home along the same route. What is her average speed, in km/hr, for the round trip?

<span class='latex-bold'>(A)</span>\ 3 \qquad <span class='latex-bold'>(B)</span>\ 3.125 \qquad <span class='latex-bold'>(C)</span>\ 3.5 \qquad <span class='latex-bold'>(D)</span>\ 4 \qquad <span class='latex-bold'>(E)</span>\ 4.5

Rose's Rectangular Flower Bed

Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the ﬁgure. She plants one flower per square foot in each region. Asters cost

\$

1 each, begonias

\$

1.50 each, cannas

\$

2 each, dahlias

\$

2.50 each, and Easter lilies

\$

3 each. What is the least possible cost, in dollars, for her garden? [asy]unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(8pt));
draw((6,0)--(0,0)--(0,1)--(6,1)); draw((0,1)--(0,6)--(4,6)--(4,1)); draw((4,6)--(11,6)--(11,3)--(4,3)); draw((11,3)--(11,0)--(6,0)--(6,3));
label("1",(0,0.5),W); label("5",(0,3.5),W); label("3",(11,1.5),E); label("3",(11,4.5),E); label("4",(2,6),N); label("7",(7.5,6),N); label("6",(3,0),S); label("5",(8.5,0),S);[/asy]

<span class='latex-bold'>(A)</span>\ 108 \qquad <span class='latex-bold'>(B)</span>\ 115 \qquad <span class='latex-bold'>(C)</span>\ 132 \qquad <span class='latex-bold'>(D)</span>\ 144 \qquad <span class='latex-bold'>(E)</span>\ 156

2

Removing Box Corners

15

10

8

cm. A new solid is formed by removing a cube

3

cm on a side from each corner of this box. What percent of the original volume is removed?

<span class='latex-bold'>(A)</span>\ 4.5 \qquad <span class='latex-bold'>(B)</span>\ 9 \qquad <span class='latex-bold'>(C)</span>\ 12 \qquad <span class='latex-bold'>(D)</span>\ 18 \qquad <span class='latex-bold'>(E)</span>\ 24

Consecutive Integers

The sum of 5 consecutive even integers is

4

less than the sum of the ﬁrst

8

consecutive odd counting numbers. What is the smallest of the even integers?

<span class='latex-bold'>(A)</span>\ 6 \qquad <span class='latex-bold'>(B)</span>\ 8 \qquad <span class='latex-bold'>(C)</span>\ 10 \qquad <span class='latex-bold'>(D)</span>\ 12 \qquad <span class='latex-bold'>(E)</span>\ 14

2

Rockham Soccer League

Members of the Rockham Soccer League buy socks and T-shirts. Socks cost

\$4

per pair and each T-shirt costs

\$5

more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is

\$2366

, how many members are in the League?

<span class='latex-bold'>(A)</span>\ 77 \qquad <span class='latex-bold'>(B)</span>\ 91 \qquad <span class='latex-bold'>(C)</span>\ 143 \qquad <span class='latex-bold'>(D)</span>\ 182 \qquad <span class='latex-bold'>(E)</span>\ 286

Algebritis

Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs

\$1

more than a pink pill, and Al’s pills cost a total of

\$546

for the two weeks. How much does one green pill cost?

<span class='latex-bold'>(A)</span>\ \$7 \qquad <span class='latex-bold'>(B)</span>\ \$14 \qquad <span class='latex-bold'>(C)</span>\ \$19 \qquad <span class='latex-bold'>(D)</span>\ \$20 \qquad <span class='latex-bold'>(E)</span>\ \$39

2

Even + Odd Counting Numbers

What is the difference between the sum of the first

2003

even counting numbers and the sum of the first

2003

odd counting numbers?

<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>\ 2003 \qquad <span class='latex-bold'>(E)</span>\ 4006

Fractions?

Which of the following is the same as \frac{2\minus{}4\plus{}6\minus{}8\plus{}10\minus{}12\plus{}14}{3\minus{}6\plus{}9\minus{}12\plus{}15\minus{}18\plus{}21}? (A)\ \minus{}1 \qquad (B)\ \minus{}\frac23 \qquad (C)\ \frac23 \qquad (D)\ 1 \qquad (E)\ \frac{14}{3}