Difference between revisions of "2008 AMC 10B Problems"
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+ | {{AMC10 Problems|year=2008|ab=B}} | ||
==Problem 1== | ==Problem 1== | ||
− | A basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers could represent the total points scored by the player? | + | A basketball player made <math>5</math> baskets during a game. Each basket was worth either <math>2</math> or <math>3</math> points. How many different numbers could represent the total points scored by the player? |
<math>\mathrm{(A)}\ 2\qquad\mathrm{(B)}\ 3\qquad\mathrm{(C)}\ 4\qquad\mathrm{(D)}\ 5\qquad\mathrm{(E)}\ 6</math> | <math>\mathrm{(A)}\ 2\qquad\mathrm{(B)}\ 3\qquad\mathrm{(C)}\ 4\qquad\mathrm{(D)}\ 5\qquad\mathrm{(E)}\ 6</math> | ||
Line 7: | Line 8: | ||
==Problem 2== | ==Problem 2== | ||
− | A <math>4\times 4</math> block of calendar dates | + | A <math>4\times 4</math> block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums? |
− | <math>\ | + | <math>\begin{tabular}[t]{|c|c|c|c|} |
+ | \multicolumn{4}{c}{}\\\hline | ||
+ | 1&2&3&4\\\hline | ||
+ | 8&9&10&11\\\hline | ||
+ | 15&16&17&18\\\hline | ||
+ | 22&23&24&25\\\hline | ||
+ | \end{tabular}</math> | ||
+ | |||
+ | <math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10</math> | ||
[[2008 AMC 10B Problems/Problem 2|Solution]] | [[2008 AMC 10B Problems/Problem 2|Solution]] | ||
Line 21: | Line 30: | ||
==Problem 4== | ==Problem 4== | ||
− | A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least < | + | A semipro baseball league has teams with <math>21</math> players each. League rules state that a player must be paid at least <math>\textdollar 15,000</math> and that the total of all players' salaries for each team cannot exceed <math>\textdollar 700,000.</math> What is the maximum possible salary, in dollars, for a single player? |
<math>\mathrm{(A)}\ 270,000\qquad\mathrm{(B)}\ 385,000\qquad\mathrm{(C)}\ 400,000\qquad\mathrm{(D)}\ 430,000\qquad\mathrm{(E)}\ 700,000</math> | <math>\mathrm{(A)}\ 270,000\qquad\mathrm{(B)}\ 385,000\qquad\mathrm{(C)}\ 400,000\qquad\mathrm{(D)}\ 430,000\qquad\mathrm{(E)}\ 700,000</math> | ||
Line 28: | Line 37: | ||
==Problem 5== | ==Problem 5== | ||
− | For [[real number]]s <math>a</math> and <math>b</math>, define <math>a\</math> | + | For [[real number]]s <math>a</math> and <math>b</math>, define <math>a \textdollar b</math> <math>=(a-b)^2</math>. What is <math>(x-y)^2\textdollar(y-x)^2</math>? |
<math>\mathrm{(A)}\ 0\qquad\mathrm{(B)}\ x^2+y^2\qquad\mathrm{(C)}\ 2x^2\qquad\mathrm{(D)}\ 2y^2\qquad\mathrm{(E)}\ 4xy</math> | <math>\mathrm{(A)}\ 0\qquad\mathrm{(B)}\ x^2+y^2\qquad\mathrm{(C)}\ 2x^2\qquad\mathrm{(D)}\ 2y^2\qquad\mathrm{(E)}\ 4xy</math> | ||
Line 35: | Line 44: | ||
==Problem 6== | ==Problem 6== | ||
− | {{ | + | Points <math>B</math> and <math>C</math> lie on <math>AD</math>. The length of <math>AB</math> is <math>4</math> times the length of <math>BD</math>, and the length of <math>AC</math> is <math>9</math> times the length of <math>CD</math>. The length of <math>BC</math> is what fraction of the length of <math>AD</math>? |
+ | |||
+ | <math>\mathrm{(A)}\ 1/36\qquad\mathrm{(B)}\ 1/13\qquad\mathrm{(C)}\ 1/10\qquad\mathrm{(D)}\ 5/36\qquad\mathrm{(E)}\ 1/5</math> | ||
[[2008 AMC 10B Problems/Problem 6|Solution]] | [[2008 AMC 10B Problems/Problem 6|Solution]] | ||
==Problem 7== | ==Problem 7== | ||
− | {{ | + | An equilateral triangle of side length <math>10</math> is completely filled in by non-overlapping equilateral triangles of side length <math>1</math>. How many small triangles are required? |
+ | |||
+ | <math>\mathrm{(A)}\ 10\qquad\mathrm{(B)}\ 25\qquad\mathrm{(C)}\ 100\qquad\mathrm{(D)}\ 250\qquad\mathrm{(E)}\ 1000</math> | ||
[[2008 AMC 10B Problems/Problem 7|Solution]] | [[2008 AMC 10B Problems/Problem 7|Solution]] | ||
==Problem 8== | ==Problem 8== | ||
− | {{ | + | A class collects <math>\textdollar50</math> to buy flowers for a classmate who is in the hospital. Roses cost <math>\textdollar3</math> each, and carnations cost <math>\textdollar2</math> each. No other flowers are to be used. How many different bouquets could be purchased for exactly <math>\textdollar50</math>? |
+ | |||
+ | <math> | ||
+ | \mathrm{(A)}\ 1 | ||
+ | \qquad | ||
+ | \mathrm{(B)}\ 7 | ||
+ | \qquad | ||
+ | \mathrm{(C)}\ 9 | ||
+ | \qquad | ||
+ | \mathrm{(D)}\ 16 | ||
+ | \qquad | ||
+ | \mathrm{(E)}\ 17 | ||
+ | </math> | ||
[[2008 AMC 10B Problems/Problem 8|Solution]] | [[2008 AMC 10B Problems/Problem 8|Solution]] | ||
==Problem 9== | ==Problem 9== | ||
− | {{ | + | A quadratic equation <math>ax^2 - 2ax + b = 0</math> has two real solutions. What is the average of these two solutions? |
+ | |||
+ | <math>\mathrm{(A)}\ 1\qquad\mathrm{(B)}\ 2\qquad\mathrm{(C)}\ \frac ba\qquad\mathrm{(D)}\ \frac{2b}a\qquad\mathrm{(E)}\ \sqrt{2b-a}</math> | ||
[[2008 AMC 10B Problems/Problem 9|Solution]] | [[2008 AMC 10B Problems/Problem 9|Solution]] | ||
==Problem 10== | ==Problem 10== | ||
− | {{ | + | Points <math>A</math> and <math>B</math> are on a circle of radius <math>5</math> and <math>AB=6</math>. Point <math>C</math> is the [[midpoint]] of the minor arc <math>AB</math>. What is the length of the line segment <math>AC</math>? |
+ | |||
+ | <math>\mathrm{(A)}\ \sqrt{10}\qquad\mathrm{(B)}\ \frac{7}{2}\qquad\mathrm{(C)}\ \sqrt{14}\qquad\mathrm{(D)}\ \sqrt{15}\qquad\mathrm{(E)}\ 4</math> | ||
[[2008 AMC 10B Problems/Problem 10|Solution]] | [[2008 AMC 10B Problems/Problem 10|Solution]] | ||
==Problem 11== | ==Problem 11== | ||
− | {{ | + | Suppose that <math>(u_n)</math> is a [[sequence]] of real numbers satisfying <math>u_{n+2}=2u_{n+1}+u_n</math>, |
+ | |||
+ | and that <math>u_3=9</math> and <math>u_6=128</math>. What is <math>u_5</math>? | ||
+ | |||
+ | <math>\mathrm{(A)}\ 40\qquad\mathrm{(B)}\ 53\qquad\mathrm{(C)}\ 68\qquad\mathrm{(D)}\ 88\qquad\mathrm{(E)}\ 104</math> | ||
[[2008 AMC 10B Problems/Problem 11|Solution]] | [[2008 AMC 10B Problems/Problem 11|Solution]] | ||
==Problem 12== | ==Problem 12== | ||
− | {{ | + | Postman Pete has a pedometer to count his steps. The pedometer records up to 99999 steps, then flips over to 00000 on the next step. Pete plans to determine his mileage for a year. On January 1 Pete sets the pedometer to 00000. During the year, the pedometer flips from 99999 to 00000 forty-four |
+ | times. On December 31 the pedometer reads 50000. Pete takes 1800 steps per mile. Which of the following is closest to the number of miles Pete walked during the year? | ||
+ | |||
+ | <math>\mathrm{(A)}\ 2500\qquad\mathrm{(B)}\ 3000\qquad\mathrm{(C)}\ 3500\qquad\mathrm{(D)}\ 4000\qquad\mathrm{(E)}\ 4500</math> | ||
[[2008 AMC 10B Problems/Problem 12|Solution]] | [[2008 AMC 10B Problems/Problem 12|Solution]] | ||
==Problem 13== | ==Problem 13== | ||
− | {{ | + | For each positive integer <math>n</math>, the mean of the first <math>n</math> terms of a sequence is <math>n</math>. What is the 2008th term of the sequence? |
+ | |||
+ | <math>\mathrm{(A)}\ {{{2008}}} \qquad \mathrm{(B)}\ {{{4015}}} \qquad \mathrm{(C)}\ {{{4016}}} \qquad \mathrm{(D)}\ {{{4,030,056}}} \qquad \mathrm{(E)}\ {{{4,032,064}}}</math> | ||
[[2008 AMC 10B Problems/Problem 13|Solution]] | [[2008 AMC 10B Problems/Problem 13|Solution]] | ||
==Problem 14== | ==Problem 14== | ||
− | {{ | + | Triangle <math>OAB</math> has <math>O=(0,0)</math>, <math>B=(5,0)</math>, and <math>A</math> in the first quadrant. In addition, <math>\angle ABO=90^\circ</math> and <math>\angle AOB=30^\circ</math>. Suppose that <math>OA</math> is rotated <math>90^\circ</math> counterclockwise about <math>O</math>. What are the coordinates of the image of <math>A</math>? |
+ | |||
+ | <math> | ||
+ | \mathrm{(A)}\ \left( - \frac {10}{3}\sqrt {3},5\right) | ||
+ | \qquad | ||
+ | \mathrm{(B)}\ \left( - \frac {5}{3}\sqrt {3},5\right) | ||
+ | \qquad | ||
+ | \mathrm{(C)}\ \left(\sqrt {3},5\right) | ||
+ | \qquad | ||
+ | \mathrm{(D)}\ \left(\frac {5}{3}\sqrt {3},5\right) | ||
+ | \qquad | ||
+ | \mathrm{(E)}\ \left(\frac {10}{3}\sqrt {3},5\right) | ||
+ | </math> | ||
[[2008 AMC 10B Problems/Problem 14|Solution]] | [[2008 AMC 10B Problems/Problem 14|Solution]] | ||
==Problem 15== | ==Problem 15== | ||
+ | How many right triangles have integer leg lengths <math>a</math> and <math>b</math> and a hypotenuse of length <math>b+1</math>, where <math>b<100</math>? | ||
+ | |||
+ | <math>\mathrm{(A)}\ 6\qquad\mathrm{(B)}\ 7\qquad\mathrm{(C)}\ 8\qquad\mathrm{(D)}\ 9\qquad\mathrm{(E)}\ 10</math> | ||
[[2008 AMC 10B Problems/Problem 15|Solution]] | [[2008 AMC 10B Problems/Problem 15|Solution]] | ||
==Problem 16== | ==Problem 16== | ||
− | {{ | + | Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, the sum is <math>0</math>.) |
+ | |||
+ | <math>\mathrm{(A)}\ {{{\frac{3} {8}}}} \qquad \mathrm{(B)}\ {{{\frac{1} {2}}}} \qquad \mathrm{(C)}\ {{{\frac{43} {72}}}} \qquad \mathrm{(D)}\ {{{\frac{5} {8}}}} \qquad \mathrm{(E)}\ {{{\frac{2} {3}}}}</math> | ||
[[2008 AMC 10B Problems/Problem 16|Solution]] | [[2008 AMC 10B Problems/Problem 16|Solution]] | ||
==Problem 17== | ==Problem 17== | ||
− | {{ | + | A poll shows that <math>70\%</math> of all voters approve of the mayor's work. On three separate occasions a pollster selects a voter at random. What is the probability that on exactly one of these three occasions the voter approves of the mayor's work? |
+ | |||
+ | <math>\mathrm{(A)}\ {{{0.063}}} \qquad \mathrm{(B)}\ {{{0.189}}} \qquad \mathrm{(C)}\ {{{0.233}}} \qquad \mathrm{(D)}\ {{{0.333}}} \qquad \mathrm{(E)}\ {{{0.441}}}</math> | ||
[[2008 AMC 10B Problems/Problem 17|Solution]] | [[2008 AMC 10B Problems/Problem 17|Solution]] | ||
==Problem 18== | ==Problem 18== | ||
− | {{ | + | Bricklayer Brenda would take nine hours to build a chimney alone, and Bricklayer Brandon would take <math>10</math> hours to build it alone. When they work together, they talk a lot, and their combined output decreases by <math>10</math> bricks per hour. Working together, they build the chimney in <math>5</math> hours. How many bricks are in the chimney? |
+ | |||
+ | <math>\mathrm{(A)}\ 500\qquad\mathrm{(B)}\ 900\qquad\mathrm{(C)}\ 950\qquad\mathrm{(D)}\ 1000\qquad\mathrm{(E)}\ 1900</math> | ||
[[2008 AMC 10B Problems/Problem 18|Solution]] | [[2008 AMC 10B Problems/Problem 18|Solution]] | ||
==Problem 19== | ==Problem 19== | ||
− | {{ | + | A cylindrical tank with radius <math>4</math> feet and height <math>9</math> feet is lying on its side. The tank is filled with water to a depth of <math>2</math> feet. What is the volume of water, in cubic feet? |
+ | |||
+ | <math> | ||
+ | \mathrm{(A)}\ 24\pi - 36 \sqrt {2} | ||
+ | \qquad | ||
+ | \mathrm{(B)}\ 24\pi - 24 \sqrt {3} | ||
+ | \qquad | ||
+ | \mathrm{(C)}\ 36\pi - 36 \sqrt {3} | ||
+ | \qquad | ||
+ | \mathrm{(D)}\ 36\pi - 24 \sqrt {2} | ||
+ | \qquad | ||
+ | \mathrm{(E)}\ 48\pi - 36 \sqrt {3} | ||
+ | </math> | ||
[[2008 AMC 10B Problems/Problem 19|Solution]] | [[2008 AMC 10B Problems/Problem 19|Solution]] | ||
==Problem 20== | ==Problem 20== | ||
− | {{ | + | The faces of a cubical die are marked with the numbers <math>1</math>, <math>2</math>, <math>2</math>, <math>3</math>, <math>3</math>, and <math>4</math>. The faces of another die are marked with the numbers <math>1</math>, <math>3</math>, <math>4</math>, <math>5</math>, <math>6</math>, and <math>8</math>. Both dice are thrown. What is the probability that the sum of the top two numbers will be <math>5</math>, <math>7</math>, or <math>9</math>? |
+ | |||
+ | <math>\mathrm{(A)}\ 5/18\qquad\mathrm{(B)}\ 7/18\qquad\mathrm{(C)}\ 11/18\qquad\mathrm{(D)}\ 3/4\qquad\mathrm{(E)}\ 8/9</math> | ||
[[2008 AMC 10B Problems/Problem 20|Solution]] | [[2008 AMC 10B Problems/Problem 20|Solution]] | ||
==Problem 21== | ==Problem 21== | ||
− | {{ | + | |
+ | Ten chairs are evenly spaced around a round table and numbered clockwise from <math>1</math> through <math>10</math>. Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse. How many seating arrangements are possible? | ||
+ | |||
+ | <math>\mathrm{(A)}\ 240\qquad\mathrm{(B)}\ 360\qquad\mathrm{(C)}\ 480\qquad\mathrm{(D)}\ 540\qquad\mathrm{(E)}\ 720</math> | ||
[[2008 AMC 10B Problems/Problem 21|Solution]] | [[2008 AMC 10B Problems/Problem 21|Solution]] | ||
==Problem 22== | ==Problem 22== | ||
− | {{ | + | Three red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that no two neighboring beads are the same color? |
+ | |||
+ | <math>\mathrm{(A)}\ 1/12\qquad\mathrm{(B)}\ 1/10\qquad\mathrm{(C)}\ 1/6\qquad\mathrm{(D)}\ 1/3\qquad\mathrm{(E)}\ 1/2</math> | ||
[[2008 AMC 10B Problems/Problem 22|Solution]] | [[2008 AMC 10B Problems/Problem 22|Solution]] | ||
==Problem 23== | ==Problem 23== | ||
− | {{ | + | A rectangular floor measures <math>a</math> by <math>b</math> feet, where <math>a</math> and <math>b</math> are positive integers with <math>b > a</math>. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width 1 foot around the painted |
+ | rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair <math>(a, b)</math>? | ||
+ | |||
+ | <math>\mathrm{(A)}\ 1\qquad\mathrm{(B)}\ 2\qquad\mathrm{(C)}\ 3\qquad\mathrm{(D)}\ 4\qquad\mathrm{(E)}\ 5</math> | ||
[[2008 AMC 10B Problems/Problem 23|Solution]] | [[2008 AMC 10B Problems/Problem 23|Solution]] | ||
+ | |||
==Problem 24== | ==Problem 24== | ||
− | {{ | + | Quadrilateral <math>ABCD</math> has <math>AB = BC = CD</math>, angle <math>ABC = 70</math> and angle <math>BCD = 170</math>. What is the measure of angle <math>BAD</math>? |
+ | |||
+ | <math>\mathrm{(A)}\ 75\qquad\mathrm{(B)}\ 80\qquad\mathrm{(C)}\ 85\qquad\mathrm{(D)}\ 90\qquad\mathrm{(E)}\ 95</math> | ||
[[2008 AMC 10B Problems/Problem 24|Solution]] | [[2008 AMC 10B Problems/Problem 24|Solution]] | ||
==Problem 25== | ==Problem 25== | ||
− | {{ | + | Michael walks at the rate of <math>5</math> feet per second on a long straight path. Trash pails are located every <math>200</math> feet along the path. A garbage truck travels at <math>10</math> feet per second in the same direction as Michael and stops for <math>30</math> seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet? |
+ | |||
+ | <math>\mathrm{(A)}\ 4\qquad\mathrm{(B)}\ 5\qquad\mathrm{(C)}\ 6\qquad\mathrm{(D)}\ 7\qquad\mathrm{(E)}\ 8</math> | ||
[[2008 AMC 10B Problems/Problem 25|Solution]] | [[2008 AMC 10B Problems/Problem 25|Solution]] | ||
==See also== | ==See also== | ||
+ | {{AMC10 box|year=2008|ab=B|before=[[2008 AMC 10A Problems]]|after=[[2009 AMC 10A Problems]]}} | ||
* [[AMC 10]] | * [[AMC 10]] | ||
* [[AMC 10 Problems and Solutions]] | * [[AMC 10 Problems and Solutions]] | ||
Line 138: | Line 225: | ||
* [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=219 2008 AMC B Math Jam Transcript] | * [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=219 2008 AMC B Math Jam Transcript] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} |
Revision as of 16:52, 7 October 2022
2008 AMC 10B (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
A basketball player made baskets during a game. Each basket was worth either
or
points. How many different numbers could represent the total points scored by the player?
Problem 2
A block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums?
Problem 3
Assume that is a positive real number. Which is equivalent to
?
Problem 4
A semipro baseball league has teams with players each. League rules state that a player must be paid at least
and that the total of all players' salaries for each team cannot exceed
What is the maximum possible salary, in dollars, for a single player?
Problem 5
For real numbers and
, define
. What is
?
Problem 6
Points and
lie on
. The length of
is
times the length of
, and the length of
is
times the length of
. The length of
is what fraction of the length of
?
Problem 7
An equilateral triangle of side length is completely filled in by non-overlapping equilateral triangles of side length
. How many small triangles are required?
Problem 8
A class collects to buy flowers for a classmate who is in the hospital. Roses cost
each, and carnations cost
each. No other flowers are to be used. How many different bouquets could be purchased for exactly
?
Problem 9
A quadratic equation has two real solutions. What is the average of these two solutions?
Problem 10
Points and
are on a circle of radius
and
. Point
is the midpoint of the minor arc
. What is the length of the line segment
?
Problem 11
Suppose that is a sequence of real numbers satisfying
,
and that and
. What is
?
Problem 12
Postman Pete has a pedometer to count his steps. The pedometer records up to 99999 steps, then flips over to 00000 on the next step. Pete plans to determine his mileage for a year. On January 1 Pete sets the pedometer to 00000. During the year, the pedometer flips from 99999 to 00000 forty-four times. On December 31 the pedometer reads 50000. Pete takes 1800 steps per mile. Which of the following is closest to the number of miles Pete walked during the year?
Problem 13
For each positive integer , the mean of the first
terms of a sequence is
. What is the 2008th term of the sequence?
Problem 14
Triangle has
,
, and
in the first quadrant. In addition,
and
. Suppose that
is rotated
counterclockwise about
. What are the coordinates of the image of
?
Problem 15
How many right triangles have integer leg lengths and
and a hypotenuse of length
, where
?
Problem 16
Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, the sum is .)
Problem 17
A poll shows that of all voters approve of the mayor's work. On three separate occasions a pollster selects a voter at random. What is the probability that on exactly one of these three occasions the voter approves of the mayor's work?
Problem 18
Bricklayer Brenda would take nine hours to build a chimney alone, and Bricklayer Brandon would take hours to build it alone. When they work together, they talk a lot, and their combined output decreases by
bricks per hour. Working together, they build the chimney in
hours. How many bricks are in the chimney?
Problem 19
A cylindrical tank with radius feet and height
feet is lying on its side. The tank is filled with water to a depth of
feet. What is the volume of water, in cubic feet?
Problem 20
The faces of a cubical die are marked with the numbers ,
,
,
,
, and
. The faces of another die are marked with the numbers
,
,
,
,
, and
. Both dice are thrown. What is the probability that the sum of the top two numbers will be
,
, or
?
Problem 21
Ten chairs are evenly spaced around a round table and numbered clockwise from through
. Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse. How many seating arrangements are possible?
Problem 22
Three red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that no two neighboring beads are the same color?
Problem 23
A rectangular floor measures by
feet, where
and
are positive integers with
. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width 1 foot around the painted
rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair
?
Problem 24
Quadrilateral has
, angle
and angle
. What is the measure of angle
?
Problem 25
Michael walks at the rate of feet per second on a long straight path. Trash pails are located every
feet along the path. A garbage truck travels at
feet per second in the same direction as Michael and stops for
seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet?
See also
2008 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by 2008 AMC 10A Problems |
Followed by 2009 AMC 10A Problems | |
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 AMC 10 Problems and Solutions |
- AMC 10
- AMC 10 Problems and Solutions
- 2008 AMC 10B
- 2008 AMC B Math Jam Transcript
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.