https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Glowinglol&feedformat=atomAoPS Wiki - User contributions [en]2024-03-29T14:43:41ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2003_AMC_8_Problems/Problem_18&diff=419362003 AMC 8 Problems/Problem 182011-08-25T14:14:38Z<p>Glowinglol: Created page with "There are three people who are only friends with each other who won't be invited. Also, there are two people who are friends of friends of friends who won't be invited, to give u..."</p>
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<div>There are three people who are only friends with each other who won't be invited. Also, there are two people who are friends of friends of friends who won't be invited, to give us a total of (C) - 5 people.</div>Glowinglolhttps://artofproblemsolving.com/wiki/index.php?title=2003_AMC_8_Problems/Problem_24&diff=419352003 AMC 8 Problems/Problem 242011-08-25T14:13:11Z<p>Glowinglol: Created page with "The distance from X remains constant for the semicircle, so the first part of the graph has to be a straight line. Then, the line gets closer, and then farther away from X. So, (..."</p>
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<div>The distance from X remains constant for the semicircle, so the first part of the graph has to be a straight line. Then, the line gets closer, and then farther away from X. So, (A) would be the best option.</div>Glowinglolhttps://artofproblemsolving.com/wiki/index.php?title=2003_AMC_8_Problems/Problem_25&diff=419342003 AMC 8 Problems/Problem 252011-08-25T14:11:18Z<p>Glowinglol: Created page with "The side lengths of square WXYZ must be 5 cm, since the area is 25 cm ^2. First, you should determine the height of triangle ABC. The distance from O to line WZ must be 2.5 cm, s..."</p>
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<div>The side lengths of square WXYZ must be 5 cm, since the area is 25 cm ^2. First, you should determine the height of triangle ABC. The distance from O to line WZ must be 2.5 cm, since line WX = 5 cm, and the distance from O to Z is half of that. The distance from line WZ to line BC must be 2, since the side lengths of the small squares are 1, and there are two squares from line WZ to line BC. So, the height of ABC must be 4.5, which is 2.5 + 2. The length of BC can be determined by subtracting 2 from 5, since the length of WZ is 5, and the two squares in the corners give us 2 together. This gives us the base for ABC, which is 3. Then, we multiply 4.5 by 3 and divide by 2, to get an answer of (C) - 27/4.</div>Glowinglolhttps://artofproblemsolving.com/wiki/index.php?title=2003_AMC_8_Problems/Problem_14&diff=419332003 AMC 8 Problems/Problem 142011-08-25T13:29:35Z<p>Glowinglol: Created page with "Since both T's are 7, then O has to equal 4, because 7 + 7 = 14. Then, F has to equal 1. To get R, we do 4 + 4 (since O = 4) to get R = 8. The value for W then has to be a number..."</p>
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<div>Since both T's are 7, then O has to equal 4, because 7 + 7 = 14. Then, F has to equal 1. To get R, we do 4 + 4 (since O = 4) to get R = 8. The value for W then has to be a number less than 5, otherwise it will change the value of O, and can't be a number that has already been used, like 4 or 1. The only other possibilities are 2 and 3. 2 doesn't work because it makes U = 4, which is what O already equals. So, the only possible value of W is (D) - 3.</div>Glowinglolhttps://artofproblemsolving.com/wiki/index.php?title=2003_AMC_8_Problems/Problem_12&diff=419322003 AMC 8 Problems/Problem 122011-08-25T13:24:20Z<p>Glowinglol: Created page with "All the possibilities where 6 is on any of the five sides is always divisible by six, and 1*2*3*4*5 is divisible by 6 since 2*3 is equal to six. So, the answer is (E)- 1 because ..."</p>
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<div>All the possibilities where 6 is on any of the five sides is always divisible by six, and 1*2*3*4*5 is divisible by 6 since 2*3 is equal to six. So, the answer is (E)- 1 because the outcome is always divisible by 6.</div>Glowinglolhttps://artofproblemsolving.com/wiki/index.php?title=2003_AMC_8_Problems/Problem_8&diff=419312003 AMC 8 Problems/Problem 82011-08-25T13:01:48Z<p>Glowinglol: Created page with "(A) Art is the right answer because out of all the cookies, Art's had an area of 10 in^2, which was the greatest area out of all the cookies' areas. Roger's cookie had an area of..."</p>
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<div>(A) Art is the right answer because out of all the cookies, Art's had an area of 10 in^2, which was the greatest area out of all the cookies' areas. Roger's cookie had an area of 8 in ^2, and both Paul and Trisha's cookies had an area of 6 in^2. This means Art makes less cookies, since his cookie area is the greatest. The answer is not that there is a tie between Paul and Trisha because they can make the most cookies with a given amount of cookie dough, not the least.</div>Glowinglolhttps://artofproblemsolving.com/wiki/index.php?title=2003_AMC_8_Problems&diff=419302003 AMC 8 Problems2011-08-25T12:55:24Z<p>Glowinglol: /* Problem */</p>
<hr />
<div>==Problem 1==<br />
Jamie counted the number of edges of a cube, Jimmy counted the numbers of corners, and Judy counted the number of faces. They then added the three numbers. What was the resulting sum? <br />
<br />
<math>\mathrm{(A)}\ 12 \qquad\mathrm{(B)}\ 16 \qquad\mathrm{(C)}\ 20 \qquad\mathrm{(D)}\ 22 \qquad\mathrm{(E)}\ 26</math><br />
<br />
[[2003 AMC 8 Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Which of the following numbers has the smallest prime factor?<br />
<br />
<math>\mathrm{(A)}\ 55 \qquad\mathrm{(B)}\ 57 \qquad\mathrm{(C)}\ 58 \qquad\mathrm{(D)}\ 59 \qquad\mathrm{(E)}\ 61</math><br />
<br />
[[2003 AMC 8 Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
A burger at Ricky C's weighs <math>120</math> grams, of which <math>30</math> grams are filler. <br />
What percent of the burger is not filler?<br />
<br />
<math>\mathrm{(A)}\ 60\% \qquad\mathrm{(B)}\ 65\% \qquad\mathrm{(C)}\ 70\% \qquad\mathrm{(D)}\ 75\% \qquad\mathrm{(E)}\ 90\%</math><br />
<br />
[[2003 AMC 8 Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
A group of children riding on bicycles and tricycles rode past Billy Bob's house. Billy Bob counted <math>7</math> children and <math>19</math> wheels. How many tricycles were there?<br />
<br />
<math>\mathrm{(A)}\ 2 \qquad\mathrm{(B)}\ 4 \qquad\mathrm{(C)}\ 5 \qquad\mathrm{(D)}\ 6 \qquad\mathrm{(E)}\ 7</math><br />
<br />
[[2003 AMC 8 Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
If <math> 20\% </math> of a number is <math>12</math>, what is <math> 30\% </math> of the same number?<br />
<br />
<math>\mathrm{(A)}\ 15\qquad\mathrm{(B)}\ 18 \qquad\mathrm{(C)}\ 20 \qquad\mathrm{(D)}\ 24 \qquad\mathrm{(E)}\ 30</math><br />
<br />
[[2005 AMC 8 Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Given the areas of the three squares in the figure, what is the area of the interior triangle? [[File:AMC8 problem 6 2003image.png]]<br />
<br />
<math>\mathrm{(A)}\ 13 \qquad\mathrm{(B)}\ 30 \qquad\mathrm{(C)}\ 60 \qquad\mathrm{(D)}\ 300 \qquad\mathrm{(E)}\ 1800</math><br />
<br />
[[2003 AMC 8 Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Blake and Jenny each took four <math>100</math>-point tests. Blake averaged <math>78</math> on the four tests. Jenny scored <math>10</math> points higher than Blake on the first test, <math>10</math> points lower than him on the second test, and <math>20</math> points higher on both the third and fourth tests. What is the difference between Jenny's average and Blake's average on these four tests?<br />
<br />
<math> \mathrm{(A)}\ 10 \qquad\mathrm{(B)}\ 15 \qquad\mathrm{(C)}\ 20 \qquad\mathrm{(D)}\ 25 \qquad\mathrm{(E)}\ 40 </math><br />
<br />
[[2003 AMC 8 Problems/Problem 7|Solution]]<br />
<br />
==Bake Sale==<br />
Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures<br />
<br />
Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.<br />
<br />
<math>\circ</math> Art's cookies are trapezoids: <br />
<asy>size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8));<br />
draw(origin--(5,0)--(5,3)--(2,3)--cycle);<br />
draw(rightanglemark((5,3), (5,0), origin));<br />
label("5 in", (2.5,0), S);<br />
label("3 in", (5,1.5), E);<br />
label("3 in", (3.5,3), N);</asy><br />
<br />
<math>\circ</math> Roger's cookies are rectangles: <br />
<asy>size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8));<br />
draw(origin--(4,0)--(4,2)--(0,2)--cycle);<br />
draw(rightanglemark((4,2), (4,0), origin));<br />
draw(rightanglemark((0,2), origin, (4,0)));<br />
label("4 in", (2,0), S);<br />
label("2 in", (4,1), E);</asy><br />
<br />
<math>\circ</math> Paul's cookies are parallelograms: <br />
<asy>size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8));<br />
draw(origin--(3,0)--(2.5,2)--(-0.5,2)--cycle);<br />
draw((2.5,2)--(2.5,0), dashed);<br />
draw(rightanglemark((2.5,2),(2.5,0), origin));<br />
label("3 in", (1.5,0), S);<br />
label("2 in", (2.5,1), W);</asy><br />
<br />
<math>\circ</math> Trisha's cookies are triangles: <br />
<asy>size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8));<br />
draw(origin--(3,0)--(3,4)--cycle);<br />
draw(rightanglemark((3,4),(3,0), origin));<br />
label("3 in", (1.5,0), S);<br />
label("4 in", (3,2), E);</asy><br />
<br />
===Problem 8===<br />
Who gets the fewest cookies from one batch of cookie dough?<br />
<br />
<math> \textbf{(A)}\ \text{Art} \qquad\textbf{(B)}\ \text{Paul}\qquad\textbf{(C)}\ \text{Roger}\qquad\textbf{(D)}\ \text{Trisha}\qquad\textbf{(E)}\ \text{There is a tie for fewest}</math><br />
<br />
[[2003 AMC 8 Problems/Problem 8|Solution]]<br />
<br />
===Problem 9===<br />
Each friend uses the same amount of dough, and Art makes exactly <math>12</math> cookies. Art's cookies sell for <math>60</math> cents each. To earn the same amount from a single batch, how much should one of Roger's cookies cost in cents?<br />
<br />
<math> \textbf{(A)}\ 18\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 75\qquad\textbf{(E)}\ 90</math><br />
<br />
[[2003 AMC 8 Problems/Problem 9|Solution]]<br />
<br />
===Problem 10===<br />
How many cookies will be in one batch of Trisha's cookies?<br />
<br />
<math> \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 24</math><br />
<br />
[[2003 AMC 8 Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
Business is a little slow at Lou's Fine Shoes, so Lou decides to have a<br />
sale. On Friday, Lou increases all of Thursday's prices by <math> 10% </math>. Over the<br />
weekend, Lou advertises the sale: Ten percent off the listed price. Sale<br />
starts Monday." How much does a pair of shoes cost on Monday that<br />
cost <math> 40 </math> dollars on Thursday?<br />
<br />
<math> \textbf{(A)}\ 36\qquad\textbf{(B)}\ 39.60\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 40.40\qquad\textbf{(E)}\ 44 </math><br />
<br />
[[2003 AMC 8 Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
When a fair six-sided die is tossed on a table top, the bottom face cannot<br />
be seen. What is the probability that the product of the numbers on the<br />
five faces that can be seen is divisible by 6?<br />
<br />
<math> \textbf{(A)}\ \frac{1}{3}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ \frac{2}{3}\qquad\textbf{(D)}\ \frac{5}{6}\qquad\textbf{(E)}\ 1</math><br />
<br />
[[2003 AMC 8 Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
Fourteen white cubes are put together to form the figure on the right. The complete surface of the figure, including the bottom, is painted red. The figure is then separated into individual cubes. How many of the individual cubes have exactly four red faces?<br />
<br />
<asy><br />
import three;<br />
defaultpen(linewidth(0.8));<br />
real r=0.5;<br />
currentprojection=orthographic(3/4,8/15,7/15);<br />
draw(unitcube, white, thick(), nolight);<br />
draw(shift(1,0,0)*unitcube, white, thick(), nolight);<br />
draw(shift(2,0,0)*unitcube, white, thick(), nolight);<br />
draw(shift(0,0,1)*unitcube, white, thick(), nolight);<br />
draw(shift(2,0,1)*unitcube, white, thick(), nolight);<br />
draw(shift(0,1,0)*unitcube, white, thick(), nolight);<br />
draw(shift(2,1,0)*unitcube, white, thick(), nolight);<br />
draw(shift(0,2,0)*unitcube, white, thick(), nolight);<br />
draw(shift(2,2,0)*unitcube, white, thick(), nolight);<br />
draw(shift(0,3,0)*unitcube, white, thick(), nolight);<br />
draw(shift(0,3,1)*unitcube, white, thick(), nolight);<br />
draw(shift(1,3,0)*unitcube, white, thick(), nolight);<br />
draw(shift(2,3,0)*unitcube, white, thick(), nolight);<br />
draw(shift(2,3,1)*unitcube, white, thick(), nolight);</asy><br />
<br />
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12</math><br />
<br />
[[2003 AMC 8 Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
In this addition problem, each letter stands for a different digit. <br />
<br />
<math> \setlength{\tabcolsep}{0.5mm}\begin{array}{cccc}&T & W & O\\ \plus{} &T & W & O\\ \hline F& O & U & R\end{array} </math><br />
<br />
If T = 7 and the letter O represents an even number, what is the only possible value for W?<br />
<br />
<math>\textbf{(A)}\ 0 \qquad<br />
\textbf{(B)}\ 1 \qquad<br />
\textbf{(C)}\ 2\qquad<br />
\textbf{(D)}\ 3\qquad<br />
\textbf{(E)}\ 4</math><br />
<br />
[[2003 AMC 8 Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
A figure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a figure with the front and side views shown?<br />
<br />
<asy><br />
defaultpen(linewidth(0.8));<br />
path p=unitsquare;<br />
draw(p^^shift(0,1)*p^^shift(1,0)*p);<br />
draw(shift(4,0)*p^^shift(5,0)*p^^shift(5,1)*p);<br />
label("FRONT", (1,0), S);<br />
label("SIDE", (5,0), S);</asy><br />
<br />
<math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7</math><br />
<br />
[[2003 AMC 8 Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
Ali, Bonnie, Carlo, and Dianna are going to drive together to a nearby theme park. The car they are using has <math>4</math> seats: <math>1</math> Driver seat, <math>1</math> front passenger seat, and <math>2</math> back passenger seat. Bonnie and Carlo are the only ones who know how to drive the car. How many possible seating arrangements are there?<br />
<br />
<math>\textbf{(A)}\ 2 \qquad<br />
\textbf{(B)}\ 4 \qquad<br />
\textbf{(C)}\ 6 \qquad<br />
\textbf{(D)}\ 12 \qquad<br />
\textbf{(E)}\ 24</math><br />
<br />
[[2003 AMC 8 Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
The six children listed below are from two families of three siblings each. Each child has blue or brown eyes and black or blond hair. Children from the same family have at least one of these characteristics in common. Which two children are Jim's siblings?<br />
<cmath> \begin{array}{c|c|c}\text{Child}&\text{Eye Color}&\text{Hair Color}\\ \hline\text{Benjamin}&\text{Blue}&\text{Black}\\ \hline\text{Jim}&\text{Brown}&\text{Blonde}\\ \hline\text{Nadeen}&\text{Brown}&\text{Black}\\ \hline\text{Austin}&\text{Blue}&\text{Blonde}\\ \hline\text{Tevyn}&\text{Blue}&\text{Black}\\ \hline\text{Sue}&\text{Blue}&\text{Blonde}\\ \hline\end{array} </cmath><br />
<math> \textbf{(A)}\ \text{Nadeen and Austin}\qquad\textbf{(B)}\ \text{Benjamin and Sue}\qquad\textbf{(C)}\ \text{Benjamin and Austin}\qquad\textbf{(D)}\ \text{Nadeen and Tevyn}\qquad </math><br />
<math> \textbf{(E)}\ \text{Austin and Sue} </math><br />
<br />
[[2003 AMC 8 Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
Each of the twenty dots on the graph below represents one of Sarah's classmates. Classmates who are friends are connected with a line segment. For her birthday party, Sarah is inviting only the following: all of her friends and all of those classmates who are friends with at least one of her friends. How many classmates will not be invited to Sarah's party?<br />
<asy>/* AMC8 2003 #18 Problem */<br />
pair a=(102,256), b=(68,131), c=(162,101), d=(134,150);<br />
pair e=(269,105), f=(359,104), g=(303,12), h=(579,211);<br />
pair i=(534, 342), j=(442,432), k=(374,484), l=(278,501);<br />
pair m=(282,411), n=(147,451), o=(103,437), p=(31,373);<br />
pair q=(419,175), r=(462,209), s=(477,288), t=(443,358);<br />
pair oval=(282,303);<br />
draw(l--m--n--cycle);<br />
draw(p--oval);<br />
draw(o--oval);<br />
draw(b--d--oval);<br />
draw(c--d--e--oval);<br />
draw(e--f--g--h--i--j--oval);<br />
draw(k--oval);<br />
draw(q--oval);<br />
draw(s--oval);<br />
draw(r--s--t--oval);<br />
dot(a); dot(b); dot(c); dot(d); dot(e); dot(f); dot(g); dot(h);<br />
dot(i); dot(j); dot(k); dot(l); dot(m); dot(n); dot(o); dot(p);<br />
dot(q); dot(r); dot(s); dot(t);<br />
filldraw(yscale(.5)*Circle((282,606),80),white,black);<br />
label(scale(0.75)*"Sarah", oval);</asy><br />
<br />
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7</math><br />
<br />
[[2003 AMC 8 Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
How many integers between <math>1000</math> and <math>2000</math> have all three of the numbers <math>15</math>, <math>20</math>, and <math>25</math> as factors?<br />
<br />
<math>\textbf{(A)}\ 1 \qquad<br />
\textbf{(B)}\ 2 \qquad<br />
\textbf{(C)}\ 3 \qquad<br />
\textbf{(D)}\ 4 \qquad<br />
\textbf{(E)}\ 5</math><br />
<br />
[[2003 AMC 8 Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
What is the measure of the acute angle formed by the hands of the clock at <math>4:20</math> PM?<br />
<br />
<math>\textbf{(A)}\ 0 \qquad<br />
\textbf{(B)}\ 5 \qquad<br />
\textbf{(C)}\ 8 \qquad<br />
\textbf{(D)}\ 10 \qquad<br />
\textbf{(E)}\ 12</math><br />
<br />
[[2003 AMC 8 Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
The area of trapezoid <math> ABCD</math> is <math> 164 \text{cm}^2</math>. The altitude is <math> 8 \text{cm}</math>, <math> AB</math> is <math> 10 \text{cm}</math>, and <math> CD</math> is <math> 17 \text{cm}</math>. What is <math> BC</math>, in centimeters?<br />
<asy>/* AMC8 2003 #21 Problem */<br />
size(4inch,2inch);<br />
draw((0,0)--(31,0)--(16,8)--(6,8)--cycle);<br />
draw((11,8)--(11,0), linetype("8 4"));<br />
draw((11,1)--(12,1)--(12,0));<br />
label("$A$", (0,0), SW);<br />
label("$D$", (31,0), SE);<br />
label("$B$", (6,8), NW);<br />
label("$C$", (16,8), NE);<br />
label("10", (3,5), W);<br />
label("8", (11,4), E);<br />
label("17", (22.5,5), E);</asy><br />
<br />
<math> \textbf{(A)}\ 9\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20</math><br />
<br />
[[2003 AMC 8 Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
The following figures are composed of squares and circles. Which figure has a shaded region with largest area?<br />
<asy>/* AMC8 2003 #22 Problem */<br />
size(3inch, 2inch);<br />
unitsize(1cm);<br />
pen outline = black+linewidth(1);<br />
filldraw((0,0)--(2,0)--(2,2)--(0,2)--cycle, mediumgrey, outline);<br />
filldraw(shift(3,0)*((0,0)--(2,0)--(2,2)--(0,2)--cycle), mediumgrey, outline);<br />
filldraw(Circle((7,1), 1), mediumgrey, black+linewidth(1));<br />
filldraw(Circle((1,1), 1), white, outline);<br />
filldraw(Circle((3.5,.5), .5), white, outline);<br />
filldraw(Circle((4.5,.5), .5), white, outline);<br />
filldraw(Circle((3.5,1.5), .5), white, outline);<br />
filldraw(Circle((4.5,1.5), .5), white, outline);<br />
filldraw((6.3,.3)--(7.7,.3)--(7.7,1.7)--(6.3,1.7)--cycle, white, outline);<br />
label("A", (1, 2), N);<br />
label("B", (4, 2), N);<br />
label("C", (7, 2), N);<br />
draw((0,-.5)--(.5,-.5), BeginArrow);<br />
draw((1.5, -.5)--(2, -.5), EndArrow);<br />
label("2 cm", (1, -.5));<br />
<br />
draw((3,-.5)--(3.5,-.5), BeginArrow);<br />
draw((4.5, -.5)--(5, -.5), EndArrow);<br />
label("2 cm", (4, -.5));<br />
<br />
draw((6,-.5)--(6.5,-.5), BeginArrow);<br />
draw((7.5, -.5)--(8, -.5), EndArrow);<br />
label("2 cm", (7, -.5));<br />
<br />
draw((6,1)--(6,-.5), linetype("4 4"));<br />
draw((8,1)--(8,-.5), linetype("4 4"));</asy><br />
<br />
<math> \textbf{(A)}\ \text{A only}\qquad\textbf{(B)}\ \text{B only}\qquad\textbf{(C)}\ \text{C only}\qquad\textbf{(D)}\ \text{both A and B}\qquad\textbf{(E)}\ \text{all are equal}</math><br />
<br />
[[2003 AMC 8 Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
In the pattern below, the cat (denoted as a large circle in the figures below) moves clockwise through the four squares and the mouse (denoted as a dot in the figures below) moves counterclockwise through the eight exterior segments of the four squares.<br />
<br />
<asy>defaultpen(linewidth(0.8));<br />
size(350);<br />
path p=unitsquare;<br />
int i;<br />
for(i=0; i<5; i=i+1) {<br />
draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p));<br />
}<br />
path cat=Circle((0.5,0.5), 0.3);<br />
draw(shift(0,1)*cat^^shift(4,1)*cat^^shift(7,0)*cat^^shift(9,0)*cat^^shift(12,1)*cat);<br />
dot((1.5,0)^^(5,0.5)^^(8,1.5)^^(10.5,2)^^(12.5,2));<br />
<br />
label("1", (1,2), N);<br />
label("2", (4,2), N);<br />
label("3", (7,2), N);<br />
label("4", (10,2), N);<br />
label("5", (13,2), N);<br />
</asy><br />
<br />
If the pattern is continued, where would the cat and mouse be after the 247th move?<br />
<br />
<math>\textbf{(A)}</math><br />
<asy>defaultpen(linewidth(0.8));<br />
size(60);<br />
path p=unitsquare;<br />
int i=0;<br />
draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p));<br />
path cat=Circle((0.5,0.5), 0.3);<br />
draw(shift(1,0)*cat);<br />
dot((0,0.5));<br />
</asy><br />
<br />
<math>\textbf{(B)}</math><br />
<asy>defaultpen(linewidth(0.8));<br />
size(60);<br />
path p=unitsquare;<br />
int i=0;<br />
draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p));<br />
path cat=Circle((0.5,0.5), 0.3);<br />
draw(shift(1,1)*cat);<br />
dot((0,0.5));<br />
</asy><br />
<br />
<math>\textbf{(C)}</math><br />
<asy>defaultpen(linewidth(0.8));<br />
size(60);<br />
path p=unitsquare;<br />
int i=0;<br />
draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p));<br />
path cat=Circle((0.5,0.5), 0.3);<br />
draw(shift(1,0)*cat);<br />
dot((0,1.5));<br />
</asy><br />
<br />
<math>\textbf{(D)}</math><br />
<asy>defaultpen(linewidth(0.8));<br />
size(60);<br />
path p=unitsquare;<br />
int i=0;<br />
draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p));<br />
path cat=Circle((0.5,0.5), 0.3);<br />
draw(shift(0,0)*cat);<br />
dot((0,1.5));<br />
</asy><br />
<br />
<math>\textbf{(E)}</math><br />
<asy>defaultpen(linewidth(0.8));<br />
size(60);<br />
path p=unitsquare;<br />
int i=0;<br />
draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p));<br />
path cat=Circle((0.5,0.5), 0.3);<br />
draw(shift(0,1)*cat);<br />
dot((1.5,0));<br />
</asy><br />
<br />
[[2003 AMC 8 Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
A ship travels from point A to point B along a semicircular path, centered at Island X. Then it travels along a straight path from B to C. Which of these graphs best shows the ship's distance from Island X as it moves along its course?<br />
<br />
<asy>size(150);<br />
pair X=origin, A=(-5,0), B=(5,0), C=(0,5);<br />
draw(Arc(X, 5, 180, 360)^^B--C);<br />
dot(X);<br />
label("$X$", X, NE);<br />
label("$C$", C, N);<br />
label("$B$", B, E);<br />
label("$A$", A, W);</asy><br />
<br />
<math>\textbf{(A)}</math><br />
<asy><br />
defaultpen(fontsize(7));<br />
size(80);<br />
draw((0,16)--origin--(16,0), linewidth(0.9));<br />
label("distance traveled", (8,0), S);<br />
label(rotate(90)*"distance to X", (0,8), W);<br />
draw(Arc((4,10), 4, 0, 180)^^(8,10)--(16,12));<br />
</asy><br />
<br />
<math>\textbf{(B)}</math><br />
<asy><br />
defaultpen(fontsize(7));<br />
size(80);<br />
draw((0,16)--origin--(16,0), linewidth(0.9));<br />
label("distance traveled", (8,0), S);<br />
label(rotate(90)*"distance to X", (0,8), W);<br />
draw(Arc((12,10), 4, 180, 360)^^(0,10)--(8,10));<br />
</asy><br />
<br />
<math>\textbf{(C)}</math><br />
<asy><br />
defaultpen(fontsize(7));<br />
size(80);<br />
draw((0,16)--origin--(16,0), linewidth(0.9));<br />
label("distance traveled", (8,0), S);<br />
label(rotate(90)*"distance to X", (0,8), W);<br />
draw((0,8)--(10,10)--(16,8));<br />
</asy><br />
<br />
<math>\textbf{(D)}</math><br />
<asy><br />
defaultpen(fontsize(7));<br />
size(80);<br />
draw((0,16)--origin--(16,0), linewidth(0.9));<br />
label("distance traveled", (8,0), S);<br />
label(rotate(90)*"distance to X", (0,8), W);<br />
draw(Arc((12,10), 4, 0, 180)^^(0,10)--(8,10));<br />
</asy><br />
<br />
<math>\textbf{(E)}</math><br />
<asy><br />
defaultpen(fontsize(7));<br />
size(80);<br />
draw((0,16)--origin--(16,0), linewidth(0.9));<br />
label("distance traveled", (8,0), S);<br />
label(rotate(90)*"distance to X", (0,8), W);<br />
draw((0,6)--(6,6)--(16,10));<br />
</asy><br />
<br />
[[2003 AMC 8 Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
In the figure, the area of square WXYZ is <math>25 \text{cm}^2</math>. The four smaller squares have sides 1 cm long, either parallel to or coinciding with the sides of the large square. In <math>\Delta ABC</math>, <math>AB = AC</math>, and when <math>\Delta ABC</math> is folded over side BC, point A coincides with O, the center of square WXYZ. What is the area of <math>\Delta ABC</math>, in square centimeters?<br />
<br />
<asy><br />
defaultpen(fontsize(8));<br />
size(225);<br />
pair Z=origin, W=(0,10), X=(10,10), Y=(10,0), O=(5,5), B=(-4,8), C=(-4,2), A=(-13,5);<br />
draw((-4,0)--Y--X--(-4,10)--cycle);<br />
draw((0,-2)--(0,12)--(-2,12)--(-2,8)--B--A--C--(-2,2)--(-2,-2)--cycle);<br />
dot(O);<br />
label("$A$", A, NW);<br />
label("$O$", O, NE);<br />
label("$B$", B, SW);<br />
label("$C$", C, NW);<br />
label("$W$",W , NE);<br />
label("$X$", X, N);<br />
label("$Y$", Y, N);<br />
label("$Z$", Z, SE);<br />
</asy><br />
<br />
<math> \textbf{(A)}\ \frac{15}4\qquad\textbf{(B)}\ \frac{21}4\qquad\textbf{(C)}\ \frac{27}4\qquad\textbf{(D)}\ \frac{21}2\qquad\textbf{(E)}\ \frac{27}2</math><br />
<br />
[[2003 AMC 8 Problems/Problem 25|Solution]]</div>Glowinglol