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
GET READY FOR THE AMC 10 WITH AoPS
Learn with outstanding instructors and top-scoring students from around the world in our AMC 10 Problem Series online course.
CHECK SCHEDULE

Difference between revisions of "2008 AMC 10B Problems"

(Problem 7 added to 2008 AMC 10B)
(copied problem statements from their individual pages)
Line 35: Line 35:
  
 
==Problem 6==
 
==Problem 6==
−
{{problem}}
+
Points B and C lie on AD. The length of AB is 4 times the length of BD, and the length of AC is 9 times the length of CD. The length of BC is what fraction of the length of AD?
 +
 
 +
<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]]
Line 51: Line 53:
  
 
==Problem 9==
 
==Problem 9==
−
{{problem}}
+
A quadratic equation ax^2 - 2ax + b = 0 has two real solutions. What is the average of these two solutions?
 +
A) 1 B) 2 C) b/a D) 2b/a
  
 
[[2008 AMC 10B Problems/Problem 9|Solution]]
 
[[2008 AMC 10B Problems/Problem 9|Solution]]
  
 
==Problem 10==
 
==Problem 10==
−
{{problem}}
+
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==
−
{{problem}}
+
Suppose that <math>(u_n)</math> is a [[sequence]] of real numbers satifying <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==
−
{{problem}}
+
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 mileae 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==
−
{{problem}}
+
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]]
Line 81: Line 95:
  
 
==Problem 15==
 
==Problem 15==
−
{{problem}}
+
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==
−
{{problem}}
+
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 is no die is rolled, the sum is 0.)
 +
 
 +
<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==
−
{{problem}}
+
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==
−
{{problem}}
+
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?
 +
 
{{problem}}
 
{{problem}}
  
Line 106: Line 130:
  
 
==Problem 20==
 
==Problem 20==
−
{{problem}}
+
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>. 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. 5 married couples are to sit in the chairs with men and women alternating, and no one is to sit either across from or next to his/her spousse. How many seating arrangements are possible?
+
 
 +
Ten chairs are evenly spaced around a round table. 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==
−
{{problem}}
+
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]]
Line 125: Line 156:
 
[[2008 AMC 10B Problems/Problem 23|Solution]]
 
[[2008 AMC 10B Problems/Problem 23|Solution]]
 
==Problem 24==
 
==Problem 24==
−
{{problem}}
+
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==
−
{{problem}}
+
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 for 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]]

Revision as of 15:15, 25 January 2009

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?

$\mathrm{(A)}\ 2\qquad\mathrm{(B)}\ 3\qquad\mathrm{(C)}\ 4\qquad\mathrm{(D)}\ 5\qquad\mathrm{(E)}\ 6$

Solution

Problem 2

A $4\times 4$ block of calendar dates has the numbers $1$ through $4$ in the first row, $8$ though $11$ in the second, $15$ though $18$ in the third, and $22$ through $25$ in the fourth. The order of the numbers in the second and the fourth rows are reversed. The numbers on each diagonal are added. What will be the positive difference between the diagonal sums?

$\mathrm{(A)}\ 2\qquad\mathrm{(B)}\ 4\qquad\mathrm{(C)}\ 6\qquad\mathrm{(D)}\ 8\qquad\mathrm{(E)}\ 10$

Solution

Problem 3

Assume that $x$ is a positive real number. Which is equivalent to $\sqrt[3]{x\sqrt{x}}$?

$\mathrm{(A)}\ x^{1/6}\qquad\mathrm{(B)}\ x^{1/4}\qquad\mathrm{(C)}\ x^{3/8}\qquad\mathrm{(D)}\ x^{1/2}\qquad\mathrm{(E)}\ x$

Solution

Problem 4

A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least <dollar/>15,000 and that the total of all players' salaries for each team cannot exceed <dollar/>700,000. What is the maximum possible salary, in dollars, for a single player?

$\mathrm{(A)}\ 270,000\qquad\mathrm{(B)}\ 385,000\qquad\mathrm{(C)}\ 400,000\qquad\mathrm{(D)}\ 430,000\qquad\mathrm{(E)}\ 700,000$

Solution

Problem 5

For real numbers $a$ and $b$, define $a$$b=(a-b)^2$. What is $(x-y)^2$$(y-x)^2$?

$\mathrm{(A)}\ 0\qquad\mathrm{(B)}\ x^2+y^2\qquad\mathrm{(C)}\ 2x^2\qquad\mathrm{(D)}\ 2y^2\qquad\mathrm{(E)}\ 4xy$

Solution

Problem 6

Points B and C lie on AD. The length of AB is 4 times the length of BD, and the length of AC is 9 times the length of CD. The length of BC is what fraction of the length of AD?

$\mathrm{(A)}\ 1/36\qquad\mathrm{(B)}\ 1/13\qquad\mathrm{(C)}\ 1/10\qquad\mathrm{(D)}\ 5/36\qquad\mathrm{(E)}\ 1/5$

Solution

Problem 7

An equilateral triangle of side length 10 is completely filled in by non-overlapping equilateral triangles of side length 1. How many small triangles are required?

$\mathrm{(A)}\ 10\qquad\mathrm{(B)}\ 25\qquad\mathrm{(C)}\ 100\qquad\mathrm{(D)}\ 250\qquad\mathrm{(E)}\ 1000$ Solution

Problem 8

This problem has not been edited in. If you know this problem, please help us out by adding it.

Solution

Problem 9

A quadratic equation ax^2 - 2ax + b = 0 has two real solutions. What is the average of these two solutions? A) 1 B) 2 C) b/a D) 2b/a

Solution

Problem 10

Points $A$ and $B$ are on a circle of radius $5$ and $AB=6$. Point $C$ is the midpoint of the minor arc $AB$. What is the length of the line segment $AC$?

$\mathrm{(A)}\ \sqrt{10}\qquad\mathrm{(B)}\ \frac{7}{2}\qquad\mathrm{(C)}\ \sqrt{14}\qquad\mathrm{(D)}\ \sqrt{15}\qquad\mathrm{(E)}\ 4$

Solution

Problem 11

Suppose that $(u_n)$ is a sequence of real numbers satifying $u_{n+2}=2u_{n+1}+u_n$,

and that $u_3=9$ and $u_6=128$. What is $u_5$?

$\mathrm{(A)}\ 40\qquad\mathrm{(B)}\ 53\qquad\mathrm{(C)}\ 68\qquad\mathrm{(D)}\ 88\qquad\mathrm{(E)}\ 104$

Solution

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 mileae 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?

$\mathrm{(A)}\ 2500\qquad\mathrm{(B)}\ 3000\qquad\mathrm{(C)}\ 3500\qquad\mathrm{(D)}\ 4000\qquad\mathrm{(E)}\ 4500$

Solution

Problem 13

For each positive integer $n$, the mean of the first $n$ terms of a sequence is $n$. What is the 2008th term of the sequence?

$\mathrm{(A)}\ {{{2008}}} \qquad \mathrm{(B)}\ {{{4015}}} \qquad \mathrm{(C)}\ {{{4016}}} \qquad \mathrm{(D)}\ {{{4,030,056}}} \qquad \mathrm{(E)}\ {{{4,032,064}}}$

Solution

Problem 14

This problem has not been edited in. If you know this problem, please help us out by adding it.

Solution

Problem 15

How many right triangles have integer leg lengths $a$ and $b$ and a hypotenuse of length $b+1$, where $b<100$?

$\mathrm{(A)}\ 6\qquad\mathrm{(B)}\ 7\qquad\mathrm{(C)}\ 8\qquad\mathrm{(D)}\ 9\qquad\mathrm{(E)}\ 10$

Solution

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 is no die is rolled, the sum is 0.)

$\mathrm{(A)}\ {{{\frac{3} {8}}}} \qquad \mathrm{(B)}\ {{{\frac{1} {2}}}} \qquad \mathrm{(C)}\ {{{\frac{43} {72}}}} \qquad \mathrm{(D)}\ {{{\frac{5} {8}}}} \qquad \mathrm{(E)}\ {{{\frac{2} {3}}}}$

Solution

Problem 17

A poll shows that $70\%$ 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?

$\mathrm{(A)}\ {{{0.063}}} \qquad \mathrm{(B)}\ {{{0.189}}} \qquad \mathrm{(C)}\ {{{0.233}}} \qquad \mathrm{(D)}\ {{{0.333}}} \qquad \mathrm{(E)}\ {{{0.441}}}$

Solution

Problem 18

Bricklayer Brenda would take nine hours to build a chimney alone, and bricklayer Brandon would take $10$ hours to build it alone. When they work together, they talk a lot, and their combined output decreases by $10$ bricks per hour. Working together, they build the chimney in $5$ hours. How many bricks are in the chimney?

$\mathrm{(A)}\ 500\qquad\mathrm{(B)}\ 900\qquad\mathrm{(C)}\ 950\qquad\mathrm{(D)}\ 1000\qquad\mathrm{(E)}\ 1900$

Solution

Problem 19

A cylindrical tank with radius $4$ feet and height $9$ feet is lying on its side. The tank is filled with water to a depth of $2$ feet. What is the volume of water, in cubic feet?

This problem has not been edited in. If you know this problem, please help us out by adding it.

Solution

Problem 20

The faces of a cubical die are marked with the numbers $1$, $2$, $2$, $3$, $3$, and $4$. The faces of another die are marked with the numbers $1$, $3$, $4$, $5$, $6$, and $8$. What is the probability that the sum of the top two numbers will be $5$, $7$, or $9$?

$\mathrm{(A)}\ 5/18\qquad\mathrm{(B)}\ 7/18\qquad\mathrm{(C)}\ 11/18\qquad\mathrm{(D)}\ 3/4\qquad\mathrm{(E)}\ 8/9$

Solution

Problem 21

Ten chairs are evenly spaced around a round table. 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?

$\mathrm{(A)}\ 240\qquad\mathrm{(B)}\ 360\qquad\mathrm{(C)}\ 480\qquad\mathrm{(D)}\ 540\qquad\mathrm{(E)}\ 720$

Solution

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?

$\mathrm{(A)}\ 1/12\qquad\mathrm{(B)}\ 1/10\qquad\mathrm{(C)}\ 1/6\qquad\mathrm{(D)}\ 1/3\qquad\mathrm{(E)}\ 1/2$

Solution

Problem 23

This problem has not been edited in. If you know this problem, please help us out by adding it.

Solution

Problem 24

Quadrilateral $ABCD$ has $AB = BC = CD$, angle $ABC = 70$ and angle $BCD = 170$. What is the measure of angle $BAD$?

$\mathrm{(A)}\ 75\qquad\mathrm{(B)}\ 80\qquad\mathrm{(C)}\ 85\qquad\mathrm{(D)}\ 90\qquad\mathrm{(E)}\ 95$

Solution

Problem 25

Michael walks at the rate of $5$ feet per second on a long straight path. Trash pails are located every $200$ feet along the path. A garbage truck travels at $10$ feet per second in the same direction as Michael and stops for $30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving for the next pail. How many times will Michael and the truck meet?

$\mathrm{(A)}\ 4\qquad\mathrm{(B)}\ 5\qquad\mathrm{(C)}\ 6\qquad\mathrm{(D)}\ 7\qquad\mathrm{(E)}\ 8$

Solution

See also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2008_AMC_10B_Problems&oldid=29750"