AMC10美国数学竞赛真题 2001年

Problem 1

The median of the list

is

the mean?

Solution

. What is

Problem 2

A number is more than the product of its reciprocal and its additive inverse. In which interval does the number lie?

Solution

Problem 3

The sum of two numbers is . Suppose is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?

Solution

Problem 4

What is the maximum number for the possible points of intersection of a circle and a triangle?

Solution

Problem 5

How many of the twelve pentominoes pictured below have at least one line of symmetry?

Solution

Problem 6

Let

and

denote the product and the sum, respectively, of the digits of

and

. Suppose

is a

?

the integer . For example,

two-digit number such that . What is the units digit of

Solution

Problem 7

When the decimal point of a certain positive decimal number is moved four places to the right, the new number is four times the reciprocal of the original number. What is the original number?

Solution

Problem 8

Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Their

schedule is as follows: Darren works every third school day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working in the math lab. In how many school days from today will they next be together tutoring in the lab?

Solution

Problem 9

The state income tax where Kristin lives is levied at the rate of

of annual income plus

of any amount above

of the first

. Kristin of her

noticed that the state income tax she paid amounted to annual income. What was her annual income?

Solution

Problem 10

If ,

, and are positive with

is

,

, and

Solution

, then

Problem 11

Consider the dark square in an array of unit squares, part of which is shown. The ?rst ring of squares around this center square contains unit squares. The second ring contains unit squares. If we continue this process, the number of unit squares in the ring is

Solution

Problem 12

Suppose that is the product of three consecutive integers and that by . Which of the following is not necessarily a divisor of ?

is divisible

Solution

Problem 13

A telephone number has the form , where each letter represents a different digit. The digits in each part of the numbers are in decreasing order; that is, , , and . Furthermore, , , and are consecutive even digits; , , , and are consecutive odd digits; and . Find .

Solution

Problem 14

A charity sells benefit tickets for a total of . Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?

Solution

Problem 15

A street has parallel curbs feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is feet and each stripe is feet long. Find the distance, in feet, between the stripes?

Solution

Problem 16

The mean of three numbers is more than the least of the numbers and less than the greatest. The median of the three numbers is . What is their sum?

Solution

Problem 17

Which of the cones listed below can be formed from a radius by aligning the two straight sides?

sector of a circle of

Solution

Problem 18

The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to

Solution

Problem 19

Pat wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. How many different selections are possible?

Solution

Problem 20

A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with sides of length . What is the length of each side of the octagon?

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