AMC 美国数学竞赛 2001 AMC 10 试题及答案解析

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USA AMC 10 2001

1

The median of the list

is

mean?



Solution

2

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



Solution

3

The sum of two numbers is . Suppose 3 is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?

Solution

4

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

Solution

5

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

. What is the








Solution 6 Let

and

denote the product and the sum, respectively, of the

and

. Suppose

. What is the units

digits of the integer . For example, is a two-digit number such that digit of

?



Solution

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

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



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they next be together tutoring in the lab?



Solution

9

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

of annual income plus

of

of any amount above

. Kristin noticed that the state income tax she paid amounted to

of her annual income. What was her annual income?



Solution 10

If , , and are positive with

is



Solution

11

Consider the dark square in an array of unit squares, part of which is shown. The first 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





, , and , then





Solution



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12

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

Solution

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

14

A charity sells 140 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

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 16

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The mean of three numbers is 10 more than the least of the numbers and 15 less than the greatest. The median of the three numbers is 5. What is their sum?

Solution

17

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

sector of a



A cone with slant height of A cone with height of

and radius

and radius

and radius

A cone with slant height of A cone with height of

and radius

and radius

A cone with slant height of Solution

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



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Solution

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

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?





Solution

21

A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter and

altitude , and the axes of the cylinder and cone coincide. Find the radius of the cylinder.



Solution 22



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In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by , , , , and . Find .







Solution

23

A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?



Solution 24

In trapezoid

,





Solution

25

How many positive integers not exceeding but not ?

,

and are perpendicular to , with , and . What is ?

are multiples of or



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