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 <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>\ | + | <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]] | ||
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<center>[[Image:AMC8_2007_2.png]]</center> | <center>[[Image:AMC8_2007_2.png]]</center> | ||
− | <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]] | ||
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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>\ | + | <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]] | ||
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by a different window? | by a different window? | ||
− | <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]] | ||
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to buy the mountain bike? | to buy the mountain bike? | ||
− | <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]] | ||
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distance call. | distance call. | ||
− | <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]] | ||
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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>\ | + | <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]] | ||
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==Problem 8== | ==Problem 8== | ||
− | In trapezoid <math>ABCD</math>, <math>AD</math> is perpendicular to <math>DC</math>, | + | In trapezoid <math>ABCD</math>, <math>\overline{AD}</math> is perpendicular to <math>\overline{DC}</math>, |
− | <math>AD | + | <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>. |
− | <math>DC</math>, and <math>BE</math> is parallel to <math>AD</math>. Find the area of | + | <asy> |
− | <math>\triangle BEC</math>. | + | 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>\ | + | <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]] | ||
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<center>[[Image:AMC8_2007_9.png]]</center> | <center>[[Image:AMC8_2007_9.png]]</center> | ||
− | <math>\ | + | <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]] | ||
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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>\ | + | <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]] | ||
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<center>[[Image:AMC8_2007_11.png]]</center> | <center>[[Image:AMC8_2007_11.png]]</center> | ||
− | <math>\ | + | <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]] | ||
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<center>[[Image:AMC8_2007_12.png]]</center> | <center>[[Image:AMC8_2007_12.png]]</center> | ||
− | <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]] | ||
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<center>[[Image:AMC8_2007_13.png]]</center> | <center>[[Image:AMC8_2007_13.png]]</center> | ||
− | <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]] | ||
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of the congruent sides? | of the congruent sides? | ||
− | <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]] | ||
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impossible? | impossible? | ||
− | <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]] | ||
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in the new mixture? | in the new mixture? | ||
− | <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]] | ||
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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>\ | + | <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]] | ||
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following could be the difference? | following could be the difference? | ||
− | <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]] | ||
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all? | all? | ||
− | <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]] | ||
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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>\ | + | <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]] | ||
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of these four distances in meters? | of these four distances in meters? | ||
− | <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]] | ||
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<center>[[Image:AMC8_2007_23.png]]</center> | <center>[[Image:AMC8_2007_23.png]]</center> | ||
− | <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]] | ||
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the three-digit number is a multiple of 3? | the three-digit number is a multiple of 3? | ||
− | <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 274: | Line 285: | ||
</asy> | </asy> | ||
− | <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]] |
Latest revision as of 20:01, 8 May 2023
2007 AMC 8 (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
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 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See Also
Problem 1
Theresa's parents have agreed to buy her tickets to see her favorite band if she spends an average of hours per week helping around the house for
weeks. For the first
weeks she helps around the house for
,
,
,
and
hours. How many hours must she work for the final week to earn the tickets?
Problem 2
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](https://wiki-images.artofproblemsolving.com//a/a3/AMC8_2007_2.png)
Problem 3
What is the sum of the two smallest prime factors of ?
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?
Problem 5
Chandler wants to buy a mountain bike. For his birthday, his grandparents
send him
, his aunt sends him
and his cousin gives him
. He earns
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?
Problem 6
The average cost of a long-distance call in the USA in was
cents per minute, and the average cost of a long-distance
call in the USA in
was
cents per minute. Find the
approximate percent decrease in the cost per minute of a long-
distance call.
Problem 7
The average age of people in a room is
years. An
-year-old person leaves
the room. What is the average age of the four remaining people?
Problem 8
In trapezoid ,
is perpendicular to
,
, and
. In addition,
is on
, and
is parallel to
. Find the area of
.
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](https://wiki-images.artofproblemsolving.com//e/e5/AMC8_2007_9.png)
cannot be determined
Problem 10
For any positive integer , define
to be the sum of the positive factors of
.
For example,
. Find
.
Problem 11
Tiles and
are translated so one tile coincides with each of the rectangles
and
. 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
?
![AMC8 2007 11.png](https://wiki-images.artofproblemsolving.com//5/5c/AMC8_2007_11.png)
cannot be determined
Problem 12
A unit hexagram is composed of a regular hexagon of side length and its
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](https://wiki-images.artofproblemsolving.com//7/7a/AMC8_2007_12.png)
Problem 13
Sets and
, shown in the Venn diagram, have the same number of elements.
Their union has
elements and their intersection has
elements. Find
the number of elements in
.
![AMC8 2007 13.png](https://wiki-images.artofproblemsolving.com//0/0f/AMC8_2007_13.png)
Problem 14
The base of isosceles is
and its area is
. What is the length of one
of the congruent sides?
Problem 15
Let and
be numbers with
. Which of the following is
impossible?
Problem 16
Amanda draws five circles with radii and
. Then for each circle she plots the point
,
where
is its circumference and
is its area. Which of the
following could be her graph?
![AMC8 2007 16.png](https://wiki-images.artofproblemsolving.com//f/ff/AMC8_2007_16.png)
Problem 17
A mixture of liters of paint is
red tint,
yellow
tint and
water. Five liters of yellow tint are added to
the original mixture. What is the percent of yellow tint
in the new mixture?
Problem 18
The product of the two -digit numbers
and
has thousands digit and units digit
. What is the sum of
and
?
Problem 19
Pick two consecutive positive integers whose sum is less than . Square both
of those integers and then find the difference of the squares. Which of the
following could be the difference?
Problem 20
Before district play, the Unicorns had won 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?
Problem 21
Two cards are dealt from a deck of four red cards labeled and four
green cards labeled
. A winning pair is two of the same color or two
of the same letter. What is the probability of drawing a winning pair?
Problem 22
A lemming sits at a corner of a square with side length meters. The lemming
runs
meters along a diagonal toward the opposite corner. It stops, makes
a
degree right turn and runs
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?
Problem 23
What is the area of the shaded part shown in the x
grid?
![AMC8 2007 23.png](https://wiki-images.artofproblemsolving.com//e/e5/AMC8_2007_23.png)
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?
Problem 25
On the dart board shown in the Figure, the outer circle has radius 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?
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.