2008 AMC 10B Problems

2008 AMC 10B Problems

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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?

Solution

The number of points could have been 10, 11, 12, 13, 14, or 15. Thus, the

answer is

.

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?

Solution

After reversing the numbers on the second and fourth rows, the block will look

like this:

The difference between the two diagonal sums is:

.

Problem 3

Assume that

is a

positive

real number

. Which is equivalent to

?

Solution

Problem 4

A semipro baseball league has teams with 21 players each. League rules state

that a player must be paid at least $$15,000 and that the total of all players'

salaries for each team cannot exceed $$700,000. What is the maximum

possible salary, in dollars, for a single player?

Solution

The maximum occurs when 20 players get the minimum wage and the total of

all players' salaries is 700000. That is when one player gets

.

Problem 5

For

real numbers

and

, define

. What is

?

Solution

Since

, it follows that

, and

Problem 6

A triathlete competes in a triathlon in which the swimming, biking, and running

segments are all of the same length. The triathlete swims at a rate of 3

kilometers per hour, bikes at a rate of 20 kilometers per hour, and runs at a

rate of 10 kilometers per hour. Which of the following is closest to the

triathlete's average speed, in kilometers per hour, for the entire race?

Solution

Let

be the length of one segment of the race.

Average speed is total distance divided by total time. The total distance is

and the total time is

.

,

Thus, the average speed is

answer is

. This is closest to

, so the

.

Problem 7

The fraction

simplifies to which of the following?

Solution

Notice that

can be factored out of the numerator:

Thus,

the expression is equal to

, and the answer is

.

Problem 8

Heather compares the price of a new computer at two different stores. Store

offers

off the sticker price followed by a

rebate, and store

offers

off the same sticker price with no rebate. Heather saves

by buying the

computer at store

instead of store

. What is the sticker price of the

computer, in dollars?

Solution

Let the sticker price be

.

The price of the computer is

Heather saves

Solving, we find

at store

, and

.

.

at store

.

at store

, so

, and the thus answer is

Problem 9

Suppose that

true about

?

is an integer. Which of the following statements must be

Solution

For

to be an integer,

must be even, but not necessarily divisible by

. Thus, the answer is

.

Problem 10

Points

and

are on a circle of radius

and

. Point

of the minor arc

. What is the length of the line segment

is the

midpoint

?

Solution

Let the center of the circle be

, and let

(then

is the midpoint of

).

By the

Pythagorean Theorem

,

.

Using the Pythagorean Theorem again,

.

be the intersection of

and

, since they are both radii.

, and by subtraction,

Problem 11

Suppose that

is a

sequence

of real numbers satifying

,

and that

and

. What is

?

Solution

Plugging in

, we get

Plugging in

, we get

This is simply a system of two equations with two unknowns. Substituting gives

, and

.

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?

(A) 2500 (B) 3000 (C) 3500 (D) 4000 (E) 4500

Solution

Every time the pedometer flips from

Pete has walked

So, if the pedometer flipped

Pete walked

Dividing by

gives

.

steps.

times

steps.

to

This is closest to answer

Problem 13

For each positive integer

, the mean of the first

terms of a sequence is

.

What is the 2008th term of the sequence?

Solution

Since the mean of the first

terms is

, the sum of the first

terms is

Thus, the sum of the first

terms is

and the sum of the first

terms is

. Hence, the 2008th term is

.

Problem 14

Older television screens have an aspect ratio of

. That is, the ratio of the

width to the height is

. The aspect ratio of many movies is not

, so

they are sometimes shown on a television screen by

strips of equal height at the top and bottom of the screen, as shown. Suppose

a movie has an aspect ratio of

and is shown on an older television screen

with a

-inch diagonal. What is the height, in inches, of each darkened strip?

Solution

Let the width and height of the screen be

and

respectively, and let the

width and height of the movie be

and

respectively.

By the

Pythagorean Theorem

, the diagonal is

.

Since the movie and the screen have the same width,

Thus, the height of each strip is

.

.

. So

Problem 15

How many right triangles have integer leg lengths a and b and a hypotenuse of

length b+1, where b<100?

(A) 6 (B) 7 (C) 8 (D) 9 (E) 10

Solution

By the pytahagorean theorem,

This means that

We know that

.

, and that

.

We also know that a must be odd, since the right

side is odd.

So

, and the answer is

.

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.)

Solution

We consider 3 cases based on the outcome of the coin:

Case 1, 0 heads: The probability of this occuring on the coin flip is

. The

probability that 0 rolls of a die will result in an odd sum is

.

Case 2, 1 head: The probability of this case occuring is

. The proability that

one die results as an odd number is

.

Case 3, 2 heads: The probability of this occuring is

. The probability that 2 die

result in an odd sum is

Thus, the probability of having an odd sum rolled is