Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus
ONLINE AMC 8 PREP WITH AOPS
Top scorers around the country use AoPS. Join training courses for beginners and advanced students.
VIEW CATALOG

Difference between revisions of "2007 AMC 8 Problems"

(→‎Problem 8)
(Changed all options to textbf.)
Line 2: Line 2:
 
Theresa's parents have agreed to buy her tickets to see her favorite band if she spends an average of <math>10</math> hours per week helping around the house for <math>6</math> weeks. For the first <math>5</math> weeks she helps around the house for <math>8</math>, <math>11</math>, <math>7</math>, <math>12</math> and <math>10</math> hours. How many hours must she work for the final week to earn the tickets?
 
Theresa's parents have agreed to buy her tickets to see her favorite band if she spends an average of <math>10</math> hours per week helping around the house for <math>6</math> weeks. For the first <math>5</math> weeks she helps around the house for <math>8</math>, <math>11</math>, <math>7</math>, <math>12</math> and <math>10</math> hours. How many hours must she work for the final week to earn the tickets?
  
−
<math>\mathrm{(A)}\ 9 \qquad\mathrm{(B)}\ 10 \qquad\mathrm{(C)}\ 11 \qquad\mathrm{(D)}\ 12 \qquad\mathrm{(E)}\ 13</math>
+
<math>\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13</math>
  
 
[[2007 AMC 8 Problems/Problem 1|Solution]]
 
[[2007 AMC 8 Problems/Problem 1|Solution]]
Line 11: Line 11:
 
<center>[[Image:AMC8_2007_2.png]]</center>
 
<center>[[Image:AMC8_2007_2.png]]</center>
  
−
<math>\mathrm{(A)} \frac{2}{5} \qquad \mathrm{(B)} \frac{1}{2} \qquad \mathrm{(C)} \frac{5}{4} \qquad \mathrm{(D)} \frac{5}{3} \qquad \mathrm{(E)} \frac{5}{2}</math>
+
<math>\textbf{(A)} \frac{2}{5} \qquad \textbf{(B)} \frac{1}{2} \qquad \textbf{(C)} \frac{5}{4} \qquad \textbf{(D)} \frac{5}{3} \qquad \textbf{(E)} \frac{5}{2}</math>
  
 
[[2007 AMC 8 Problems/Problem 2|Solution]]
 
[[2007 AMC 8 Problems/Problem 2|Solution]]
Line 19: Line 19:
 
What is the sum of the two smallest prime factors of <math>250</math>?
 
What is the sum of the two smallest prime factors of <math>250</math>?
  
−
<math>\mathrm{(A)}\ 2 \qquad\mathrm{(B)}\ 5 \qquad\mathrm{(C)}\ 7 \qquad\mathrm{(D)}\ 10 \qquad\mathrm{(E)}\ 12</math>
+
<math>\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 12</math>
  
 
[[2007 AMC 8 Problems/Problem 3|Solution]]
 
[[2007 AMC 8 Problems/Problem 3|Solution]]
Line 29: Line 29:
 
by a different window?
 
by a different window?
  
−
<math>\mathrm{(A)}\ 12 \qquad\mathrm{(B)}\ 15 \qquad\mathrm{(C)}\ 18 \qquad\mathrm{(D)}\ 30 \qquad\mathrm{(E)}\ 36</math>
+
<math>\textbf{(A)}\ 12 \qquad\textbf{(B)}\ 15 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 30 \qquad\textbf{(E)}\ 36</math>
  
 
[[2007 AMC 8 Problems/Problem 4|Solution]]
 
[[2007 AMC 8 Problems/Problem 4|Solution]]
Line 41: Line 41:
 
to buy the mountain bike?
 
to buy the mountain bike?
  
−
<math>\mathrm{(A)}\ 24 \qquad\mathrm{(B)}\ 25 \qquad\mathrm{(C)}\ 26 \qquad\mathrm{(D)}\ 27 \qquad\mathrm{(E)}\ 28</math>
+
<math>\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 25 \qquad\textbf{(C)}\ 26 \qquad\textbf{(D)}\ 27 \qquad\textbf{(E)}\ 28</math>
  
 
[[2007 AMC 8 Problems/Problem 5|Solution]]
 
[[2007 AMC 8 Problems/Problem 5|Solution]]
Line 53: Line 53:
 
distance call.
 
distance call.
  
−
<math>\mathrm{(A)}\ 7 \qquad\mathrm{(B)}\ 17 \qquad\mathrm{(C)}\ 34 \qquad\mathrm{(D)}\ 41 \qquad\mathrm{(E)}\ 80</math>
+
<math>\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 17 \qquad\textbf{(C)}\ 34 \qquad\textbf{(D)}\ 41 \qquad\textbf{(E)}\ 80</math>
  
 
[[2007 AMC 8 Problems/Problem 6|Solution]]
 
[[2007 AMC 8 Problems/Problem 6|Solution]]
Line 62: Line 62:
 
the room. What is the average age of the four remaining people?
 
the room. What is the average age of the four remaining people?
  
−
<math>\mathrm{(A)}\ 25 \qquad\mathrm{(B)}\ 26 \qquad\mathrm{(C)}\ 29 \qquad\mathrm{(D)}\ 33 \qquad\mathrm{(E)}\ 36</math>
+
<math>\textbf{(A)}\ 25 \qquad\textbf{(B)}\ 26 \qquad\textbf{(C)}\ 29 \qquad\textbf{(D)}\ 33 \qquad\textbf{(E)}\ 36</math>
  
 
[[2007 AMC 8 Problems/Problem 7|Solution]]
 
[[2007 AMC 8 Problems/Problem 7|Solution]]
Line 90: Line 90:
 
</center>
 
</center>
  
−
<math>\text{(A)}\ 3 \qquad \text{(B)}\ 4.5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 18</math>
+
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4.5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 18</math>
  
 
[[2007 AMC 8 Problems/Problem 8|Solution]]
 
[[2007 AMC 8 Problems/Problem 8|Solution]]
Line 102: Line 102:
 
<center>[[Image:AMC8_2007_9.png]]</center>
 
<center>[[Image:AMC8_2007_9.png]]</center>
  
−
<math>\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 2 \qquad \mathrm{(C)}\ 3 \qquad \mathrm{(D)}\ 4 \qquad \mathrm{(E)}</math> cannot be determined
+
<math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}</math> cannot be determined
  
 
[[2007 AMC 8 Problems/Problem 9|Solution]]
 
[[2007 AMC 8 Problems/Problem 9|Solution]]
Line 111: Line 111:
 
For example, <math>\boxed{6} = 1 + 2 + 3 + 6 = 12</math>. Find <math>\boxed{\boxed{11}}</math> .
 
For example, <math>\boxed{6} = 1 + 2 + 3 + 6 = 12</math>. Find <math>\boxed{\boxed{11}}</math> .
  
−
<math>\mathrm{(A)}\ 13 \qquad \mathrm{(B)}\ 20 \qquad \mathrm{(C)}\ 24 \qquad \mathrm{(D)}\ 28 \qquad \mathrm{(E)}\ 30</math>
+
<math>\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 30</math>
  
 
[[2007 AMC 8 Problems/Problem 10|Solution]]
 
[[2007 AMC 8 Problems/Problem 10|Solution]]
Line 121: Line 121:
 
<center>[[Image:AMC8_2007_11.png]]</center>
 
<center>[[Image:AMC8_2007_11.png]]</center>
  
−
<math>\mathrm{(A)}\ I \qquad \mathrm{(B)}\ II \qquad \mathrm{(C)}\ III \qquad \mathrm{(D)}\ IV \qquad \mathrm{(E)}</math> cannot be determined
+
<math>\textbf{(A)}\ I \qquad \textbf{(B)}\ II \qquad \textbf{(C)}\ III \qquad \textbf{(D)}\ IV \qquad \textbf{(E)}</math> cannot be determined
  
 
[[2007 AMC 8 Problems/Problem 11|Solution]]
 
[[2007 AMC 8 Problems/Problem 11|Solution]]
Line 133: Line 133:
 
<center>[[Image:AMC8_2007_12.png]]</center>
 
<center>[[Image:AMC8_2007_12.png]]</center>
  
−
<math>\mathrm{(A)}\ 1:1 \qquad \mathrm{(B)}\ 6:5  \qquad \mathrm{(C)}\ 3:2 \qquad \mathrm{(D)}\ 2:1 \qquad \mathrm{(E)}\ 3:1</math>  
+
<math>\textbf{(A)}\ 1:1 \qquad \textbf{(B)}\ 6:5  \qquad \textbf{(C)}\ 3:2 \qquad \textbf{(D)}\ 2:1 \qquad \textbf{(E)}\ 3:1</math>  
  
 
[[2007 AMC 8 Problems/Problem 12|Solution]]
 
[[2007 AMC 8 Problems/Problem 12|Solution]]
Line 145: Line 145:
 
<center>[[Image:AMC8_2007_13.png]]</center>
 
<center>[[Image:AMC8_2007_13.png]]</center>
  
−
<math>\mathrm{(A)}\ 503 \qquad \mathrm{(B)}\ 1006 \qquad \mathrm{(C)}\ 1504 \qquad \mathrm{(D)}\ 1507 \qquad \mathrm{(E)}\ 1510</math>
+
<math>\textbf{(A)}\ 503 \qquad \textbf{(B)}\ 1006 \qquad \textbf{(C)}\ 1504 \qquad \textbf{(D)}\ 1507 \qquad \textbf{(E)}\ 1510</math>
  
 
[[2007 AMC 8 Problems/Problem 13|Solution]]
 
[[2007 AMC 8 Problems/Problem 13|Solution]]
Line 154: Line 154:
 
of the congruent sides?
 
of the congruent sides?
  
−
<math>\mathrm{(A)}\ 5 \qquad \mathrm{(B)}\ 8 \qquad \mathrm{(C)}\ 13 \qquad \mathrm{(D)}\ 14 \qquad \mathrm{(E)}\ 18</math>
+
<math>\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 18</math>
  
 
[[2007 AMC 8 Problems/Problem 14|Solution]]
 
[[2007 AMC 8 Problems/Problem 14|Solution]]
Line 163: Line 163:
 
impossible?
 
impossible?
  
−
<math>\mathrm{(A)} \ a + c < b  \qquad \mathrm{(B)} \ a \cdot b < c \qquad \mathrm{(C)} \ a + b < c \qquad \mathrm{(D)} \ a \cdot c < b \qquad \mathrm{(E)}\frac{b}{c} = a</math>
+
<math>\textbf{(A)} \ a + c < b  \qquad \textbf{(B)} \ a \cdot b < c \qquad \textbf{(C)} \ a + b < c \qquad \textbf{(D)} \ a \cdot c < b \qquad \textbf{(E)}\frac{b}{c} = a</math>
  
 
[[2007 AMC 8 Problems/Problem 15|Solution]]
 
[[2007 AMC 8 Problems/Problem 15|Solution]]
Line 185: Line 185:
 
in the new mixture?
 
in the new mixture?
  
−
<math>\mathrm{(A)}\ 25 \qquad \mathrm{(B)}\ 35 \qquad \mathrm{(C)}\ 40 \qquad \mathrm{(D)}\ 45 \qquad \mathrm{(E)}\ 50</math>
+
<math>\textbf{(A)}\ 25 \qquad \textbf{(B)}\ 35 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 45 \qquad \textbf{(E)}\ 50</math>
  
 
[[2007 AMC 8 Problems/Problem 17|Solution]]
 
[[2007 AMC 8 Problems/Problem 17|Solution]]
Line 197: Line 197:
 
has thousands digit <math>A</math> and units digit <math>B</math>. What is the sum of <math>A</math> and <math>B</math>?
 
has thousands digit <math>A</math> and units digit <math>B</math>. What is the sum of <math>A</math> and <math>B</math>?
  
−
<math>\mathrm{(A)}\ 3 \qquad \mathrm{(B)}\ 5 \qquad \mathrm{(C)}\ 6 \qquad \mathrm{(D)}\ 8 \qquad \mathrm{(E)}\ 10</math>
+
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10</math>
  
 
[[2007 AMC 8 Problems/Problem 18|Solution]]
 
[[2007 AMC 8 Problems/Problem 18|Solution]]
Line 207: Line 207:
 
following could be the difference?
 
following could be the difference?
  
−
<math>\mathrm{(A)}\ 2 \qquad \mathrm{(B)}\ 64 \qquad \mathrm{(C)}\ 79 \qquad \mathrm{(D)}\ 96 \qquad \mathrm{(E)}\ 131</math>
+
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 64 \qquad \textbf{(C)}\ 79 \qquad \textbf{(D)}\ 96 \qquad \textbf{(E)}\ 131</math>
  
 
[[2007 AMC 8 Problems/Problem 19|Solution]]
 
[[2007 AMC 8 Problems/Problem 19|Solution]]
Line 219: Line 219:
 
all?
 
all?
  
−
<math>\mathrm{(A)}\ 48 \qquad \mathrm{(B)}\ 50 \qquad \mathrm{(C)}\ 52 \qquad \mathrm{(D)}\ 54 \qquad \mathrm{(E)}\ 60</math>
+
<math>\textbf{(A)}\ 48 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 52 \qquad \textbf{(D)}\ 54 \qquad \textbf{(E)}\ 60</math>
  
 
[[2007 AMC 8 Problems/Problem 20|Solution]]
 
[[2007 AMC 8 Problems/Problem 20|Solution]]
Line 229: Line 229:
 
of the same letter. What is the probability of drawing a winning pair?
 
of the same letter. What is the probability of drawing a winning pair?
  
−
<math>\mathrm{(A)} \frac{2}{7} \qquad \mathrm{(B)} \frac{3}{8} \qquad \mathrm{(C)} \frac{1}{2} \qquad \mathrm{(D)} \frac{4}{7} \qquad \mathrm{(E)} \frac{5}{8}</math>
+
<math>\textbf{(A)} \frac{2}{7} \qquad \textbf{(B)} \frac{3}{8} \qquad \textbf{(C)} \frac{1}{2} \qquad \textbf{(D)} \frac{4}{7} \qquad \textbf{(E)} \frac{5}{8}</math>
  
 
[[2007 AMC 8 Problems/Problem 21|Solution]]
 
[[2007 AMC 8 Problems/Problem 21|Solution]]
Line 241: Line 241:
 
of these four distances in meters?
 
of these four distances in meters?
  
−
<math>\mathrm{(A)}\ 2 \qquad \mathrm{(B)}\ 4.5 \qquad \mathrm{(C)}\ 5 \qquad \mathrm{(D)}\ 6.2 \qquad \mathrm{(E)}\ 7</math>
+
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4.5 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6.2 \qquad \textbf{(E)}\ 7</math>
  
 
[[2007 AMC 8 Problems/Problem 22|Solution]]
 
[[2007 AMC 8 Problems/Problem 22|Solution]]
Line 251: Line 251:
 
<center>[[Image:AMC8_2007_23.png]]</center>
 
<center>[[Image:AMC8_2007_23.png]]</center>
  
−
<math>\mathrm{(A)}\ 4 \qquad\mathrm{(B)}\ 6 \qquad\mathrm{(C)}\ 8 \qquad\mathrm{(D)}\ 10 \qquad\mathrm{(E)}\ 12</math>
+
<math>\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 12</math>
  
 
[[2007 AMC 8 Problems/Problem 23|Solution]]
 
[[2007 AMC 8 Problems/Problem 23|Solution]]
Line 262: Line 262:
 
the three-digit number is a multiple of 3?
 
the three-digit number is a multiple of 3?
  
−
<math>\mathrm{(A)} \frac{1}{4} \qquad \mathrm{(B)} \frac{1}{3} \qquad \mathrm{(C)} \frac{1}{2} \qquad \mathrm{(D)} \frac{2}{3} \qquad \mathrm{(E)} \frac{3}{4}</math>
+
<math>\textbf{(A)} \frac{1}{4} \qquad \textbf{(B)} \frac{1}{3} \qquad \textbf{(C)} \frac{1}{2} \qquad \textbf{(D)} \frac{2}{3} \qquad \textbf{(E)} \frac{3}{4}</math>
  
 
[[2007 AMC 8 Problems/Problem 24|Solution]]
 
[[2007 AMC 8 Problems/Problem 24|Solution]]
Line 289: Line 289:
 
</asy>
 
</asy>
  
−
<math>\mathrm{(A)} \frac{17}{36} \qquad \mathrm{(B)} \frac{35}{72} \qquad \mathrm{(C)} \frac{1}{2} \qquad \mathrm{(D)} \frac{37}{72} \qquad \mathrm{(E)} \frac{19}{36}</math>
+
<math>\textbf{(A)} \frac{17}{36} \qquad \textbf{(B)} \frac{35}{72} \qquad \textbf{(C)} \frac{1}{2} \qquad \textbf{(D)} \frac{37}{72} \qquad \textbf{(E)} \frac{19}{36}</math>
  
 
[[2007 AMC 8 Problems/Problem 25|Solution]]
 
[[2007 AMC 8 Problems/Problem 25|Solution]]

Revision as of 13:57, 15 July 2021

Problem 1

Theresa's parents have agreed to buy her tickets to see her favorite band if she spends an average of $10$ hours per week helping around the house for $6$ weeks. For the first $5$ weeks she helps around the house for $8$, $11$, $7$, $12$ and $10$ hours. How many hours must she work for the final week to earn the tickets?

$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13$

Solution

Problem 2

$650$ students were surveyed about their pasta preferences. The choices were lasagna, manicotti, ravioli and spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti?

AMC8 2007 2.png

$\textbf{(A)} \frac{2}{5} \qquad \textbf{(B)} \frac{1}{2} \qquad \textbf{(C)} \frac{5}{4} \qquad \textbf{(D)} \frac{5}{3} \qquad \textbf{(E)} \frac{5}{2}$

Solution

Problem 3

What is the sum of the two smallest prime factors of $250$?

$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 12$

Solution

Problem 4

A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window?

$\textbf{(A)}\ 12 \qquad\textbf{(B)}\ 15 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 30 \qquad\textbf{(E)}\ 36$

Solution

Problem 5

Chandler wants to buy a $\textdollar 500$ mountain bike. For his birthday, his grandparents send him $\textdollar 50$, his aunt sends him $\textdollar 35$ and his cousin gives him $\textdollar 15$. He earns $\textdollar 16$ per week for his paper route. He will use all of his birthday money and all of the money he earns from his paper route. In how many weeks will he be able to buy the mountain bike?

$\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 25 \qquad\textbf{(C)}\ 26 \qquad\textbf{(D)}\ 27 \qquad\textbf{(E)}\ 28$

Solution

Problem 6

The average cost of a long-distance call in the USA in $1985$ was $41$ cents per minute, and the average cost of a long-distance call in the USA in $2005$ was $7$ cents per minute. Find the approximate percent decrease in the cost per minute of a long- distance call.

$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 17 \qquad\textbf{(C)}\ 34 \qquad\textbf{(D)}\ 41 \qquad\textbf{(E)}\ 80$

Solution

Problem 7

The average age of $5$ people in a room is $30$ years. An $18$-year-old person leaves the room. What is the average age of the four remaining people?

$\textbf{(A)}\ 25 \qquad\textbf{(B)}\ 26 \qquad\textbf{(C)}\ 29 \qquad\textbf{(D)}\ 33 \qquad\textbf{(E)}\ 36$

Solution

Problem 8

In trapezoid $ABCD$, $AD$ is perpendicular to $DC$, $AD$ = $AB$ = $3$, and $DC$ = $6$. In addition, $E$ is on $DC$, and $BE$ is parallel to $AD$. Find the area of $\triangle BEC$.

[asy] defaultpen(linewidth(0.7)); pair A=(0,3), B=(3,3), C=(6,0), D=origin, E=(3,0); draw(E--B--C--D--A--B); draw(rightanglemark(A, D, C)); label("$A$", A, NW); label("$B$", B, NW); label("$C$", C, SE); label("$D$", D, SW); label("$E$", E, NW); label("$3$", A--D, W); label("$3$", A--B, N); label("$6$", E, S); [/asy]

$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4.5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 18$

Solution

Problem 9

To complete the grid below, each of the digits 1 through 4 must occur once in each row and once in each column. What number will occupy the lower right-hand square?

AMC8 2007 9.png

$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}$ cannot be determined

Solution

Problem 10

For any positive integer $n$, define $\boxed{n}$ to be the sum of the positive factors of $n$. For example, $\boxed{6} = 1 + 2 + 3 + 6 = 12$. Find $\boxed{\boxed{11}}$ .

$\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 30$

Solution

Problem 11

Tiles $I, II, III$ and $IV$ are translated so one tile coincides with each of the rectangles $A, B, C$ and $D$. In the final arrangement, the two numbers on any side common to two adjacent tiles must be the same. Which of the tiles is translated to Rectangle $C$?

AMC8 2007 11.png

$\textbf{(A)}\ I \qquad \textbf{(B)}\ II \qquad \textbf{(C)}\ III \qquad \textbf{(D)}\ IV \qquad \textbf{(E)}$ cannot be determined

Solution

Problem 12

A unit hexagram is composed of a regular hexagon of side length $1$ and its $6$ equilateral triangular extensions, as shown in the diagram. What is the ratio of the area of the extensions to the area of the original hexagon?

AMC8 2007 12.png

$\textbf{(A)}\ 1:1 \qquad \textbf{(B)}\ 6:5 \qquad \textbf{(C)}\ 3:2 \qquad \textbf{(D)}\ 2:1 \qquad \textbf{(E)}\ 3:1$

Solution

Problem 13

Sets $A$ and $B$, shown in the Venn diagram, have the same number of elements. Their union has $2007$ elements and their intersection has $1001$ elements. Find the number of elements in $A$.

AMC8 2007 13.png

$\textbf{(A)}\ 503 \qquad \textbf{(B)}\ 1006 \qquad \textbf{(C)}\ 1504 \qquad \textbf{(D)}\ 1507 \qquad \textbf{(E)}\ 1510$

Solution

Problem 14

The base of isosceles $\triangle ABC$ is $24$ and its area is $60$. What is the length of one of the congruent sides?

$\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 18$

Solution

Problem 15

Let $a, b$ and $c$ be numbers with $0 < a < b < c$. Which of the following is impossible?

$\textbf{(A)} \ a + c < b \qquad \textbf{(B)} \ a \cdot b < c \qquad \textbf{(C)} \ a + b < c \qquad \textbf{(D)} \ a \cdot c < b \qquad \textbf{(E)}\frac{b}{c} = a$

Solution

Problem 16

Amanda draws five circles with radii $1, 2, 3, 4$ and $5$. Then for each circle she plots the point $(C,A)$, where $C$ is its circumference and $A$ is its area. Which of the following could be her graph?

AMC8 2007 16.png

Solution

Problem 17

A mixture of $30$ liters of paint is $25\%$ red tint, $30\%$ yellow tint and $45\%$ water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture?

$\textbf{(A)}\ 25 \qquad \textbf{(B)}\ 35 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 45 \qquad \textbf{(E)}\ 50$

Solution

Problem 18

The product of the two $99$-digit numbers

$303,\!030,\!303,\!...,\!030,\!303$ and $505,\!050,\!505,\!...,\!050,\!505$

has thousands digit $A$ and units digit $B$. What is the sum of $A$ and $B$?

$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10$

Solution

Problem 19

Pick two consecutive positive integers whose sum is less than $100$. Square both of those integers and then find the difference of the squares. Which of the following could be the difference?

$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 64 \qquad \textbf{(C)}\ 79 \qquad \textbf{(D)}\ 96 \qquad \textbf{(E)}\ 131$

Solution

Problem 20

Before district play, the Unicorns had won $45\%$ of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all?

$\textbf{(A)}\ 48 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 52 \qquad \textbf{(D)}\ 54 \qquad \textbf{(E)}\ 60$

Solution

Problem 21

Two cards are dealt from a deck of four red cards labeled $A, B, C, D$ and four green cards labeled $A, B, C, D$. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair?

$\textbf{(A)} \frac{2}{7} \qquad \textbf{(B)} \frac{3}{8} \qquad \textbf{(C)} \frac{1}{2} \qquad \textbf{(D)} \frac{4}{7} \qquad \textbf{(E)} \frac{5}{8}$

Solution

Problem 22

A lemming sits at a corner of a square with side length $10$ meters. The lemming runs $6.2$ meters along a diagonal toward the opposite corner. It stops, makes a $90$ degree right turn and runs $2$ more meters. A scientist measures the shortest distance between the lemming and each side of the square. What is the average of these four distances in meters?

$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4.5 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6.2 \qquad \textbf{(E)}\ 7$

Solution

Problem 23

What is the area of the shaded part shown in the $5$ x $5$ grid?

AMC8 2007 23.png

$\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 12$

Solution

Problem 24

A bag contains four pieces of paper, each labeled with one of the digits "1, 2, 3" or "4", with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. What is the probability that the three-digit number is a multiple of 3?

$\textbf{(A)} \frac{1}{4} \qquad \textbf{(B)} \frac{1}{3} \qquad \textbf{(C)} \frac{1}{2} \qquad \textbf{(D)} \frac{2}{3} \qquad \textbf{(E)} \frac{3}{4}$

Solution

Problem 25

On the dart board shown in the Figure, the outer circle has radius $6$ and the inner circle has a radius of 3. Three radii divide each circle into three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to the area of the region. When two darts hit this board, the score is the sum of the point values in the regions. What is the probability that the score is odd?

[asy] draw(Circle(origin, 2)); draw(Circle(origin, 1)); draw(origin--2*dir(90)); draw(origin--2*dir(210)); draw(origin--2*dir(330)); label("$1$", 0.35*dir(150), dir(150)); label("$1$", 1.3*dir(30), dir(30)); label("$1$", (0,-1.3), dir(270)); label("$2$", 1.3*dir(150), dir(150)); label("$2$", 0.35*dir(30), dir(30)); label("$2$", (0,-0.35), dir(270)); [/asy]

$\textbf{(A)} \frac{17}{36} \qquad \textbf{(B)} \frac{35}{72} \qquad \textbf{(C)} \frac{1}{2} \qquad \textbf{(D)} \frac{37}{72} \qquad \textbf{(E)} \frac{19}{36}$

Solution

See Also

2007 AMC 8 (Problems • Answer Key • Resources)
Preceded by
2006 AMC 8 		Followed by
2008 AMC 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions


The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2007_AMC_8_Problems&oldid=158561"