View source for 2004 AMC 8 Problems

==Problem 1==

Ona map, a <math>12</math>-centimeter length represents <math>72</math> kilometers. How many kilometers does a <math>17</math>-centimeter length represent?

<math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 102\qquad\textbf{(C)}\ 204\qquad\textbf{(D)}\ 864\qquad\textbf{(E)}\ 1224 </math>

[[2004 AMC 8 Problems/Problem 1|Solution]]

==Problem 2==

How many different four-digit numbers can be formed be rearranging the four digits in <math>2004</math>?

<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 81 </math>

[[2004 AMC 8 Problems/Problem 2|Solution]]

==Problem 3==

Twelve friends met for dinner at Oscar's Overstuffed Oyster House, and each ordered one meal. The portions were so large, there was enough food for <math>18</math> people. If they shared, how many meals should they have ordered to have just enough food for the <math>12</math> of them?

<math> \textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 18 </math>

[[2004 AMC 8 Problems/Problem 3|Solution]]

==Problem 4==

'''The following information is needed to solve problems 4, 5 and 6.'''

Ms. Hamilton’s eighth-grade class wants to participate in the annual three-person-team basketball tournament.

Lance, Sally, Joy, and Fred are chosen for the team. In how many ways can the three starters be chosen?

<math> \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 10 </math>

[[2004 AMC 8 Problems/Problem 4|Solution]]

==Problem 5==

The losing team of each game is eliminated from the tournament. If sixteen teams compete, how many games will be played to determine the winner?

<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 16 </math>

[[2004 AMC 8 Problems/Problem 5|Solution]]

==Problem 6==

After Sally takes <math>20</math> shots, she has made <math>55\%</math> of her shots. After she takes <math>5</math> more shots, she raises her percentage to <math>56\%</math>. How many of the last <math>5</math> shots did she make?

<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 </math>

[[2004 AMC 8 Problems/Problem 6|Solution]]

==Problem 7==

An athlete's target heart rate, in beats per minute, is <math>80\%</math> of the theoretical maximum heart rate. The maximum heart rate is found by subtracting the athlete's age, in years, from <math>220</math>. To the nearest whole number, what is the target heart rate of an athlete who is <math>26</math> years old?

<math> \textbf{(A)}\ 134\qquad\textbf{(B)}\ 155\qquad\textbf{(C)}\ 176\qquad\textbf{(D)}\ 194\qquad\textbf{(E)}\ 243 </math>

[[2004 AMC 8 Problems/Problem 7|Solution]]

==Problem 8==

Find the number of two-digit positive integers whose digits total <math>7</math>.

<math> \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ 10  </math>

[[2004 AMC 8 Problems/Problem 8|Solution]]

==Problem 9==

The average of the five numbers in a list is <math>54</math>. The average of the first two
numbers is <math>48</math>. What is the average of the last three numbers?

<math> \textbf{(A)}\ 55\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 57\qquad\textbf{(D)}\ 58\qquad\textbf{(E)}\ 59</math>

[[2004 AMC 8 Problems/Problem 9|Solution]]

==Problem 10==

Handy Aaron helped a neighbor <math>1 \frac14</math> hours on Monday, <math>50</math> minutes on Tuesday, from 8:20 to 10:45 on Wednesday morning, and a half-hour on Friday. He is paid <math>\textdollar 3</math> per hour. How much did he earn for the week?

<math>\textbf{(A)}\ \textdollar 8 \qquad \textbf{(B)}\ \textdollar 9 \qquad \textbf{(C)}\ \textdollar 10 \qquad \textbf{(D)}\ \textdollar 12 \qquad \textbf{(E)}\ \textdollar 15</math>

[[2004 AMC 8 Problems/Problem 10|Solution]]

==Problem 11==

The numbers <math>-2, 4, 6, 9</math> and <math>12</math> are rearranged according to these rules:
        
         1. The largest isn’t first, but it is in one of the first three places. 
         2. The smallest isn’t last, but it is in one of the last three places. 
         3. The median isn’t first or last.

What is the average of the first and last numbers?

<math>\textbf{(A)}\ 3.5 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6.5 \qquad \textbf{(D)}\ 7.5 \qquad \textbf{(E)}\ 8</math>

[[2004 AMC 8 Problems/Problem 11|Solution]]

==Problem 12==

Niki usually leaves her cell phone on. If her cell phone is on but
she is not actually using it, the battery will last for <math>24</math> hours. If
she is using it constantly, the battery will last for only <math>3</math> hours.
Since the last recharge, her phone has been on <math>9</math> hours, and during
that time she has used it for <math>60</math> minutes. If she doesn’t talk any
more but leaves the phone on, how many more hours will the battery last?

<math>\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 11 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15</math>

[[2004 AMC 8 Problems/Problem 12|Solution]]

==Problem 13==
Amy, Bill and Celine are friends with different ages. Exactly one of the following statements is true.
 I. Bill is the oldest.
 II. Amy is not the oldest.
 III. Celine is not the youngest.
Rank the friends from the oldest to the youngest.

<math>\textbf{(A)}\ \text{Bill, Amy, Celine}\qquad \textbf{(B)}\ \text{Amy, Bill, Celine}\qquad \textbf{(C)}\ \text{Celine, Amy, Bill}\\
\textbf{(D)}\ \text{Celine, Bill, Amy} \qquad \textbf{(E)}\ \text{Amy, Celine, Bill}</math>

[[2004 AMC 8 Problems/Problem 13|Solution]]

==Problem 14==
What is the area enclosed by the geoboard quadrilateral below?

<asy>
unitsize(3mm);
defaultpen(linewidth(.8pt));
dotfactor=2;

for(int a=0; a<=10; ++a)
for(int b=0; b<=10; ++b)
 {
  dot((a,b));
 };

draw((4,0)--(0,5)--(3,4)--(10,10)--cycle);
</asy>

<math>\textbf{(A)}\ 15\qquad \textbf{(B)}\ 18\frac12 \qquad \textbf{(C)}\ 22\frac12 \qquad \textbf{(D)}\27 \qquad \textbf{(E)}\ 41</math>

[[2004 AMC 8 Problems/Problem 14|Solution]]

==Problem 15==
Thirteen black and six white hexagonal tiles were used to create the figure below. If a new figures is created by attaching a border of white tiles with the same size and shape as the others, what will be the difference between the total number of white tiles and the total number of black tiles in the new figure?

<center>
[[Image:AMC8200415.gif]]
</center>

<math>\textbf{(A)}\ 5\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 11\qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 18</math>

[[2004 AMC 8 Problems/Problem 15|Solution]]

==Problem 16==
Two <math>600</math> mL pitchers contain orange juice. One pitcher is <math>1/3</math> full and the other pitcher is <math>2/5</math> full. Water is added to fill each pitcher completely, then both pitchers are poured into one large container. What fraction of the mixture in the large container is orange juice?

<math>\textbf{(A)}\ \frac18 \qquad \textbf{(B)}\ \frac{3}{16} \qquad \textbf{(C)}\ \frac{11}{30} \qquad \textbf{(D)}\ \frac{11}{19}\qquad \textbf{(E)}\ \frac{11}{15}</math>

[[2004 AMC 8 Problems/Problem 16|Solution]]

==Problem 17==
Three friends have a total of <math>6</math> identical pencils, and each one has at least one pencil. In how many ways can this happen?

<math>\textbf{(A)}\ 1\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12</math>

[[2004 AMC 8 Problems/Problem 17|Solution]]

==Problem 18==
Five friends compete in a dart-throwing contest. Each one has two darts to throw at the same circular target, and each individual's score is the sum of the scores in the target regions that are hit. The scores for the target regions are the whole numbers <math>1</math> through <math>10</math>. Each throw hits the target in a region with a different value. The scores are: Alice <math>16</math> points, Ben <math>4</math> points, Cindy <math>7</math> points, Dave <math>11</math> points, and Ellen <math>17</math> points. Who hits the region worth <math>6</math> points?

<math>\textbf{(A)}\ \text{Alice}\qquad \textbf{(B)}\ \text{Ben}\qquad \textbf{(C)}\ \text{Cindy}\qquad \textbf{(D)}\ \text{Dave} \qquad \textbf{(E)}\ \text{Ellen}</math>

[[2004 AMC 8 Problems/Problem 18|Solution]]

==Problem 19==
A whole number larger than <math>2</math> leaves a remainder of <math>2</math> when divided by each of the numbers <math>3, 4, 5,</math> and <math>6</math>. The smallest such number lies between which two numbers?

<math>\textbf{(A)}\ 40\ \text{and}\ 49 \qquad \textbf{(B)}\ 60 \text{ and } 79 \qquad \textbf{(C)}\ 100\ \text{and}\ 129 \qquad \textbf{(D)}\ 210\ \text{and}\ 249\qquad \textbf{(E)}\ 320\ \text{and}\ 369</math>

[[2004 AMC 8 Problems/Problem 19|Solution]]

==Problem 20==
Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If there are <math>6</math> empty chairs, how many people are in the room?

<math>\textbf{(A)}\ 12\qquad \textbf{(B)}\ 18\qquad \textbf{(C)}\ 24\qquad \textbf{(D)}\ 27\qquad \textbf{(E)}\ 36</math>

[[2004 AMC 8 Problems/Problem 20|Solution]]

==Problem 21==
Spinners <math>A</math> and <math>B</math> are spun. On each spinner, the arrow is equally likely to land on each number. What is the probability that the product of the two spinners' numbers is even?

<asy>
pair A=(0,0); pair B=(3,0);
draw(Circle(A,1)); draw(Circle(B,1));

draw((-1,0)--(1,0)); draw((0,1)--(0,-1));
draw((3,0)--(3,1)); draw((3+sqrt(3)/2,-.5)--(3,0)); draw((3,0)--(3-sqrt(3)/2,-.5)); 

label("$A$",(-1,1));
label("$B$",(2,1));

label("$1$",(-.4,.4)); label("$2$",(.4,.4)); label("$3$",(.4,-.4)); label("$4$",(-.4,-.4));
label("$1$",(2.6,.4)); label("$2$",(3.4,.4)); label("$3$",(3,-.5));

</asy>

<math>\textbf{(A)}\ \frac14\qquad \textbf{(B)}\ \frac13\qquad \textbf{(C)}\ \frac12\qquad \textbf{(D)}\ \frac23\qquad \textbf{(E)}\ \frac34</math>

[[2004 AMC 8 Problems/Problem 21|Solution]]

==Problem 22==
At a party there are only single women and married men with their wives. The probability that a randomly selected woman is single is <math>\frac25</math>. What fraction of the people in the room are married men.

<math>\textbf{(A)}\ \frac13\qquad \textbf{(B)}\ \frac38\qquad \textbf{(C)}\ \frac25\qquad \textbf{(D)}\ \frac{5}{12}\qquad \textbf{(E)}\ \frac35</math>

[[2004 AMC 8 Problems/Problem 22|Solution]]

==Problem 23==
Tess runs counterclockwise around rectangular block <math>JKLM</math>. She lives at corner <math>J</math>. Which graph could represent her straight-line distance from home?

<asy>
unitsize(5mm);
pair J=(-3,2); pair K=(-3,-2); pair L=(3,-2); pair M=(3,2); 
draw(J--K--L--M--cycle);
label("$J$",J,NW);
label("$K$",K,SW);
label("$L$",L,SE);
label("$M$",M,NE);
</asy>

[[Image:AMC8200423.gif]]

[[2004 AMC 8 Problems/Problem 23|Solution]]

==Problem 24==
In the figure, <math>ABCD</math> is a rectangle and <math>EFGH</math> is a parallelogram. Using the measurements given in the figure, what is the length <math>d</math> of the segment that is perpendicular to <math>\overline{HE}</math> and <math>\overline{FG}</math>?

<asy>
unitsize(3mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));

pair D=(0,0), C=(10,0), B=(10,8), A=(0,8);
pair E=(4,8), F=(10,3), G=(6,0), H=(0,5);

draw(A--B--C--D--cycle);
draw(E--F--G--H--cycle);

label("$A$",A,NW);
label("$B$",B,NE);
label("$C$",C,SE);
label("$D$",D,SW);

label("$E$",E,N);
label("$F$",(10.8,3));
label("$G$",G,S);
label("$H$",H,W);

label("$4$",A--E,N);
label("$6$",B--E,N);
label("$5$",(10.8,5.5));
label("$3$",(10.8,1.5));
label("$4$",G--C,S);
label("$6$",G--D,S);
label("$5$",D--H,W);
label("$3$",A--H,W);

draw((3,7.25)--(7.56,1.17));
label("$d$",(3,7.25)--(7.56,1.17), NE);

</asy>

<math>\textbf{(A)}\ 6.8\qquad \textbf{(B)}\ 7.1\qquad \textbf{(C)}\ 7.6\qquad \textbf{(D)}\ 7.8\qquad \textbf{(E)}\ 8.1</math>

[[2004 AMC 8 Problems/Problem 24|Solution]]

==Problem 25==
Two <math>4 \times 4</math> squares intersect at right angles, bisecting their intersecting sides, as shown. The circle's diameter is the segment between the two points of intersection. What is the area of the shaded region created by removing the circle from the squares?

<asy>
unitsize(6mm);
draw(unitcircle);
filldraw((0,1)--(1,2)--(3,0)--(1,-2)--(0,-1)--(-1,-2)--(-3,0)--(-1,2)--cycle,lightgray,black);
filldraw(unitcircle,white,black);
</asy>

<math>\textbf{(A)}\ 16-4\pi\qquad \textbf{(B)}\ 16-2\pi \qquad \textbf{(C)}\ 28-4\pi \qquad \textbf{(D)}\ 28-2\pi \qquad \textbf{(E)}\ 32-2\pi</math>

[[2004 AMC 8 Problems/Problem 25|Solution]]

==See Also==
{{AMC8 box|year=2004|before=[[2003 AMC 8 Problems|2003 AMC 8]]|after=[[2005 AMC 8 Problems|2005 AMC 8]]}}
* [[AMC 8]]
* [[AMC 8 Problems and Solutions]]
* [[Mathematics competition resources]]


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