1992 AMC 8真题及答案详细解析

1992 AMC 8 真题

答案详细解析请参考文末

Problem 1

Problem 2

Which of the following is not equal to ?

Problem 3

What is the largest difference that can be formed by subtracting two numbers chosen from the set ?

Problem 4

During the softball season, Judy had  hits. Among her hits were  home run,  triple and  doubles. The rest of her hits were singles. What percent of her hits were singles?

Problem 5

A circle of diameter  is removed from a  rectangle, as shown. Which whole number is closest to the area of the shaded region?

Problem 6

Suppose thatmeans . For example,is . Then the sumis

Problem 7

The digit-sum of  is . How many 3-digit whole numbers, whose digit-sum is , are even?

Problem 8

A store owner bought  pencils at  each. If he sells them for  each, how many of them must he sell to make a profit of exactly ?

Problem 9

The population of a small town is . The graph indicates the number of females and males in the town, but the vertical scale-values are omitted. How many males live in the town?

Problem 10

An isosceles right triangle with legs of length  is partitioned into  congruent triangles as shown. The shaded area is

Problem 11

The bar graph shows the results of a survey on color preferences. What percent preferred blue?

Problem 12

The five tires of a car (four road tires and a full-sized spare) were rotated so that each tire was used the same number of miles during the first  miles the car traveled. For how many miles was each tire used?

Problem 13

Five test scores have a mean (average score) of , a median (middle score) of  and a mode (most frequent score) of . The sum of the two lowest test scores is

Problem 14

When four gallons are added to a tank that is one-third full, the tank is then one-half full. The capacity of the tank in gallons is

Problem 15

What is the  letter in this sequence?

Problem 16

Which cylinder has twice the volume of the cylinder shown above?

Problem 17

The sides of a triangle have lengths , , and , where  is a whole number. What is the smallest possible value of ?

Problem 18

On a trip, a car traveled  miles in an hour and a half, then was stopped in traffic for  minutes, then traveled  miles during the next hours. What was the car's average speed in miles per hour for the -hour trip?

Problem 19

The distance between the  and  exits on an interstate highway is  miles. If any two exits are at least  miles apart, then what is the largest number of miles there can be between two consecutive exits that are between the  and  exits?

Problem 20

Which pattern of identical squares could NOT be folded along the lines shown to form a cube?

Problem 21

Northside's Drum and Bugle Corps raised money for a trip. The drummers and bugle players kept separate sales records. According to the double bar graph, in what month did one group's sales exceed the other's by the greatest percent?

Problem 22

Eight  square tiles are arranged as shown so their outside edges form a polygon with a perimeter of  units. Two additional tiles of the same size are added to the figure so that at least one side of each tile is shared with a side of one of the squares in the original figure. Which of the following could be the perimeter of the new figure?

Problem 23

If two dice are tossed, the probability that the product of the numbers showing on the tops of the dice is greater than  is

Problem 24

Four circles of radius  are arranged as shown. Their centers are the vertices of a square. The area of the shaded region is closest to

Problem 25

One half of the water is poured out of a full container. Then one third of the remainder is poured out. Continue the process: one fourth of the remainder for the third pouring, one fifth of the remainder for the fourth pouring, etc. After how many pourings does exactly one tenth of the original water remain?

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1992 AMC8真答案详细解析

1.

2.

The fraction in question is equal to . Expressing all of the choices we have,

.

It's clear then that choice  is the answer.

3.To maximize anything of the form , we maximize  and minimize . The maximal element of the set is  and the minimal element is , so the maximal difference is

4.From the information given,  hits were single. Thus, the percentage was

5.

The area of the shaded region is the area of the circle subtracted from the area of the rectangle.

The diameter of the circle is , so the radius is  and the area is

The rectangle obviously has area , so the area of the shaded region is . This is closest to .

6.

The first triangle represents  The 2nd triangle represents 

Solving the first triangle, we get  Solving the 2nd triangle, we get 

Since we have to add the 2 triangles the final answer is , which is .

7.The highest digit sum for three-digit numbers is . Therefore, the only possible digit combination is . Of course, of the three possible numbers, only  works. Thus, the answer is .

8., so the store owner is  below profit. Therefore he needs to sell  dollars worth of pencils. Selling them at  each gives .

9.The graph show that the ratio of men to total population is , so the total number of men is .

10.

Solution 1

Because the smaller triangles are congruent, the shaded area take  of the largest triangles area, which is , so the shaded area is.

Solution 2

Each of the triangle has side length of , so the area is . Because there are  triangles is the shaded area, its area is .

11.The total frequency is , with the blue frequency of . Therefore, the precentage that preferred blue is .

12.In the  miles, four tires were always used at one time, so the amount of miles the five tires were used in total is . Five tires were used and each was used equally, so each tire was used for .

13.Because there was an odd number of scores,  must be the middle score. Since there are two scores above  and  appears the most frequent (so at least twice) and ,  appears twice. Also, the sum of the five numbers is . Thus, the sum of the lowest two scores is .

14.Four gallons is . Thus, capacity of the tank in gallons is .

15.The sequence consists of repeated strings of 9 letters, . When 1992 is divided by 9, the remainder is 3, and so the 1992nd letter in the sequence is the 3rd letter of the smaller string, .

16.The volume of a cylinder is given by the formula . The given cylinder therefore has a volume of . Plugging in the radius and height of the answer choices, we find the cylinder that has a volume of double the volume is .

17.By Triangle Inequality,  and therefore . The smallest whole number that satisfies this is .

18.The average speed is given by the total distance traveled divided by the total time traveled.

19.There are  pairs of consecutive exits. To find the maximum number of miles of one of these, the other  must be equal to the minimum number yielding a total of  miles. The longest distance must be .

20.If pattern  were to be folded into a cube, the topmost square would overlap the rightmost square.

21.The ratio of the sales of the group with more sales and the sales of the group with less sales is largest in  which is .

22.

One such figure would be

The perimeter of this figure is .

23.

When the first dice is , there are no possibilities for the second dice.

When the first dice is , the second dice can be , giving one possibility.

When the first dice is , the second dice can be , giving three possibilities.

When the first dice is , the second dice can be , giving four possibilities.

When the first dice is , the second dice can be , giving four possibilities.

When the first dice is , the second dice can be , giving five possibilities.

The total number of ways for the product to greater than 10 is  and the total number of possibilities is , yielding a probability of .

24.The side of the square of made up of two radii, and the area of the square is . The unshaded region of the square is composed of four quarter circles, and has the area of one circle, which is . The area of the shaded region is  which less than , closest to .

25.

1)Model the amount left in the container as follows:

After the first pour  remains, after the second  remains, etc.

This becomes the product .

Note that the terms cancel out leaving .

Now all that remains is to count the number of terms, as the numerators form an arithmetic sequence with a common difference of 1 and endpoints (1,9), the number of pourings is .

2) First pour will leave 1/2 The second pour 1/3 The third pour 1/4 Noticing the pattern that each pour, the water left = 1/(the number of times that we pour+1) So to get 1/10, the number of times that we pour is 9, therefore is 

以上解析方式仅供参考

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