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Difference between revisions of "2007 AMC 8 Problems"

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 +
{{AMC8 Problems|year=2007|}}
 
==Problem 1==
 
==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?
+
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?
  
−
(A) 9 (B) 10 (C) 11 (D) 12 (E) 13
+
<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]]
  
 
==Problem 2==
 
==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?
+
<math>650</math> 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?
  
−
[[Image:Graph.jpg]]
+
<center>[[Image:AMC8_2007_2.png]]</center>
 +
 
 +
<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 15: Line 18:
 
==Problem 3==
 
==Problem 3==
  
−
What is the sum of the two smallest prime factors of 250?
+
What is the sum of the two smallest prime factors of <math>250</math>?
−
 
 
−
(A) 2 (B) 5 (C) 7 (D) 10 (E) 12
 
  
 +
<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]]
  
 
==Problem 4==
 
==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?
 +
 +
<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]]
  
 
==Problem 5==
 
==Problem 5==
 +
 +
Chandler wants to buy a <math>\textdollar 500</math> mountain bike. For his birthday, his grandparents
 +
send him <math>\textdollar 50</math>, his aunt sends him <math>\textdollar 35</math> and his cousin gives him <math>\textdollar 15</math>. He earns
 +
<math>\textdollar 16</math> 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?
 +
 +
<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]]
  
 
==Problem 6==
 
==Problem 6==
 +
 +
The average cost of a long-distance call in the USA in <math>1985</math> was
 +
<math>41</math> cents per minute, and the average cost of a long-distance
 +
call in the USA in <math>2005</math> was <math>7</math> cents per minute. Find the
 +
approximate percent decrease in the cost per minute of a long-
 +
distance call.
 +
 +
<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]]
  
 
==Problem 7==
 
==Problem 7==
 +
 +
The average age of <math>5</math> people in a room is <math>30</math> years. An <math>18</math>-year-old person leaves
 +
the room. What is the average age of the four remaining people?
 +
 +
<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]]
  
 
==Problem 8==
 
==Problem 8==
 +
 +
In trapezoid <math>ABCD</math>, <math>\overline{AD}</math> is perpendicular to <math>\overline{DC}</math>,
 +
<math>AD = AB = 3</math>, and <math>DC = 6</math>. In addition, <math>E</math> is on <math>\overline{DC}</math>, and <math>\overline{BE}</math> is parallel to <math>\overline{AD}</math>. Find the area of <math>\triangle BEC</math>.
 +
<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>
 +
 +
<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]]
  
 
==Problem 9==
 
==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?
 +
 +
<center>[[Image:AMC8_2007_9.png]]</center>
 +
 +
<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]]
  
 
==Problem 10==
 
==Problem 10==
 +
 +
For any positive integer <math>n</math>, define <math>\boxed{n}</math> to be the sum of the positive factors of <math>n</math>.
 +
For example, <math>\boxed{6} = 1 + 2 + 3 + 6 = 12</math>. Find <math>\boxed{\boxed{11}}</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]]
  
 
==Problem 11==
 
==Problem 11==
 +
 +
Tiles <math>I, II, III</math> and <math>IV</math> are translated so one tile coincides with each of the rectangles <math>A, B, C</math> and <math>D</math>. 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 <math>C</math>?
 +
 +
<center>[[Image:AMC8_2007_11.png]]</center>
 +
 +
<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]]
  
 
==Problem 12==
 
==Problem 12==
 +
 +
A unit hexagram is composed of a regular hexagon of side length <math>1</math> and its <math>6</math>
 +
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?
 +
 +
<center>[[Image:AMC8_2007_12.png]]</center>
 +
 +
<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]]
  
 
==Problem 13==
 
==Problem 13==
 +
 +
Sets <math>A</math> and <math>B</math>, shown in the Venn diagram, have the same number of elements.
 +
Their union has <math>2007</math> elements and their intersection has <math>1001</math> elements. Find
 +
the number of elements in <math>A</math>.
 +
 +
<center>[[Image:AMC8_2007_13.png]]</center>
 +
 +
<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]]
  
 
==Problem 14==
 
==Problem 14==
 +
 +
The base of isosceles <math>\triangle ABC</math> is <math>24</math> and its area is <math>60</math>. What is the length of one
 +
of the congruent sides?
 +
 +
<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]]
  
 
==Problem 15==
 
==Problem 15==
 +
 +
Let <math>a, b</math> and <math>c</math> be numbers with <math>0 < a < b < c</math>. Which of the following is
 +
impossible?
 +
 +
<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]]
  
 
==Problem 16==
 
==Problem 16==
 +
 +
Amanda draws five circles with radii <math>1, 2, 3,
 +
4</math> and <math>5</math>. Then for each circle she plots the point <math>(C,A)</math>,
 +
where <math>C</math> is its circumference and <math>A</math> is its area. Which of the
 +
following could be her graph?
 +
 +
<center>[[Image:AMC8_2007_16.png]]</center>
  
 
[[2007 AMC 8 Problems/Problem 16|Solution]]
 
[[2007 AMC 8 Problems/Problem 16|Solution]]
  
 
==Problem 17==
 
==Problem 17==
 +
 +
A mixture of <math>30</math> liters of paint is <math>25\%</math> red tint, <math>30\%</math> yellow
 +
tint and <math>45\%</math> water. Five liters of yellow tint are added to
 +
the original mixture. What is the percent of yellow tint
 +
in the new mixture?
 +
 +
<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]]
  
 
==Problem 18==
 
==Problem 18==
 +
 +
The product of the two <math>99</math>-digit numbers
 +
 +
<math>303,\!030,\!303,\!...,\!030,\!303</math> and <math>505,\!050,\!505,\!...,\!050,\!505</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>\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]]
  
 
==Problem 19==
 
==Problem 19==
 +
 +
Pick two consecutive positive integers whose sum is less than <math>100</math>. Square both
 +
of those integers and then find the difference of the squares. Which of the
 +
following could be the difference?
 +
 +
<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]]
  
 
==Problem 20==
 
==Problem 20==
 +
 +
Before district play, the Unicorns had won <math>45\%</math> 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?
 +
 +
<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]]
  
 
==Problem 21==
 
==Problem 21==
 +
 +
Two cards are dealt from a deck of four red cards labeled <math>A, B, C, D</math> and four
 +
green cards labeled <math>A, B, C, D</math>. A winning pair is two of the same color or two
 +
of the same letter. What is the probability of drawing a winning pair?
 +
 +
<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]]
  
 
==Problem 22==
 
==Problem 22==
 +
 +
A lemming sits at a corner of a square with side length <math>10</math> meters. The lemming
 +
runs <math>6.2</math> meters along a diagonal toward the opposite corner. It stops, makes
 +
a <math>90</math> degree right turn and runs <math>2</math> 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?
 +
 +
<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]]
  
 
==Problem 23==
 
==Problem 23==
 +
 +
What is the area of the shaded part shown in the <math>5</math> x <math>5</math> grid?
 +
 +
<center>[[Image:AMC8_2007_23.png]]</center>
 +
 +
<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]]
  
 
==Problem 24==
 
==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?
 +
 +
<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]]
  
 
==Problem 25==
 
==Problem 25==
 +
 +
On the dart board shown in the Figure, the outer circle has radius <math>6</math> 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>
 +
 +
<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]]
  
−
==See also==
+
==See Also==
 +
{{AMC8 box|year=2007|before=[[2006 AMC 8 Problems|2006 AMC 8]]|after=[[2008 AMC 8 Problems|2008 AMC 8]]}}
 
* [[AMC 8]]
 
* [[AMC 8]]
 
* [[AMC 8 Problems and Solutions]]
 
* [[AMC 8 Problems and Solutions]]
 +
* [[Mathematics competition resources]]
 +
 +
 +
{{MAA Notice}}

Latest revision as of 19:01, 8 May 2023

2007 AMC 8 (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 1 point for each correct answer. There is no penalty for wrong answers.
  3. No aids are permitted other than plain scratch paper, writing utensils, ruler, and erasers. In particular, graph paper, compass, protractor, calculators, computers, smartwatches, and smartphones are not permitted. Rules
  4. Figures are not necessarily drawn to scale.
  5. You will have 40 minutes working time to complete the test.
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

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 $ABCD$, $\overline{AD}$ is perpendicular to $\overline{DC}$, $AD = AB = 3$, and $DC = 6$. In addition, $E$ is on $\overline{DC}$, and $\overline{BE}$ is parallel to $\overline{AD}$. 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


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