AMC10美国数学竞赛真题2005B卷

Problem 1

A scout troop buys candy bars at a price of five for $. They sell all the candy bars at a price of two for $. What was the profit, in dollars?

Solution

Problem 2

A positive number has the property that

of is . What is ?

Solution

Problem 3

A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is used. What fraction of the original amount of paint is available to use on the third day?

Solution

Problem 4

For real numbers

and , define

?

. What is the value of

Solution

Problem 5

Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?

Solution

Problem 6

At the beginning of the school year, Lisa's goal was to earn an A on at least of her quizzes for the year. She earned an A on of the first quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?

Solution

Problem 7

A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square?

Solution

Problem 8

An -foot by

-foot ?oor is tiled with square tiles of size foot by foot. Each

foot

tile has a pattern consisting of four white quarter circles of radius

centered at each corner of the tile. The remaining portion of the tile is shaded.

How many square feet of the ?oor are shaded?

Solution

Problem 9

One fair die has faces , , , , , and another has faces , , , , , . The dice are rolled and the numbers on the top faces are added. What is the probability that the sum will be odd?

Solution

Problem 10

In

on line

, we have such that

and

lies between and

. Suppose that is a point and . What is ?

Solution

Problem 11

The first term of a sequence is . Each succeeding term is the sum of the

cubes of the digits of the previous term. What is the term of the sequence?

Solution

Problem 12

Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime?

Solution

Problem 13

How many numbers between and ?

are integer multiples of or but not

Solution

Problem 14

Equilateral midpoint of

has side length , . What is the area of

is the midpoint of

?

, and

is the

Solution

Problem 15

An envelope contains eight bills: ones, fives, tens, and twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $ or more?

Solution

Problem 16

The quadratic equation

, and none of

?

Solution

,

has roots that are twice those of , and

is zero. What is the value of

Problem 17

Suppose that

,

,

, and

. What is

?

Solution

Problem 18

All of David's telephone numbers have the form , where , , , , , and are distinct digits and in increasing order, and none is either or . How many different telephone numbers can David have?

Solution

,

Problem 19

On a certain math exam, of the students got points, got points,

got points, got points, and the rest got points. What is the difference between the mean and the median score on this exam?

Solution

Problem 20

What is the average (mean) of all -digit numbers that can be formed by using each of the digits , , , , and exactly once?

Solution

Problem 21

Forty slips are placed into a hat, each bearing a number , , , , , , , , , or , with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let be the probability that all four slips bear the same number. Let be the probability that two of the slips bear a number and the other two bear a number

. What is the value of

Solution

?

Problem 22

For how many positive integers

?

less than or equal to

is evenly divisible by

Solution

Problem 23

In trapezoid we have parallel to and as the midpoint of . The area of

. What is

?

Solution

,

as the midpoint of

is twice the area of

,

Problem 24

Let

and

be two-digit integers such that

and ?

Solution

satisfy

is obtained by reversing the digits of for some positive integer

.

. The integers What is

Problem 25

A subset of the set of integers from to , inclusive, has the property that no two elements of sum to . What is the maximum possible number of elements in ?

For how many positive integers

?

less than or equal to

is evenly divisible by

Solution

Problem 23

In trapezoid we have parallel to and as the midpoint of . The area of

. What is

?

Solution

,

as the midpoint of

is twice the area of

,

Problem 24

Let

and

be two-digit integers such that

and ?

Solution

satisfy

is obtained by reversing the digits of for some positive integer

.

. The integers What is

Problem 25

A subset of the set of integers from to , inclusive, has the property that no two elements of sum to . What is the maximum possible number of elements in ?

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