2006 AMC 8 真题练习

1

第 1 题

应用题

Mindy made three purchases for \textdollar 1.98 dollars, \textdollar 5.04 dollars, and \textdollar 9.89 dollars. What was her total, to the nearest dollar?

💡 解题思路

The three prices round to $\textdollar 2$ , $\textdollar 5$ , and $\textdollar 10$ , which have a sum of $\boxed{\textbf{(D)}\ 17}$ .

2

第 2 题

综合

== Problem ==(if you like jayesh read the problem) On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5. What is his score?

💡 解题思路

As the AMC 8 only rewards 1 point for each correct answer, everything is irrelevant except the number Billy answered correctly, $\boxed{\textbf{(C)}\ 13}$ .

3

第 3 题

行程问题

Elisa swims laps in the pool. When she first started, she completed 10 laps in 25 minutes. Now, she can finish 12 laps in 24 minutes. By how many minutes has she improved her lap time?

💡 解题思路

When Elisa started, she finished a lap in $\frac{25}{10}=2.5$ minutes. Now, she finishes a lap is $\frac{24}{12}= 2$ minutes. The difference is $2.5-2=\boxed{\textbf{(A)}\ \frac{1}{2}}$ .

4

第 4 题

概率

Initially, a spinner points west. Chenille moves it clockwise 2 \dfrac{1}{4} revolutions and then counterclockwise 3 \dfrac{3}{4} revolutions. In what direction does the spinner point after the two moves? [图]

💡 解题思路

If the spinner goes clockwise $2 \dfrac{1}{4}$ revolutions and then counterclockwise $3 \dfrac{3}{4}$ revolutions, it ultimately goes counterclockwise $1 \dfrac{1}{2}$ which brings the spinner pointin

5

第 5 题

几何·面积

Points A, B, C and D are midpoints of the sides of the larger square. If the larger square has area 60, what is the area of the smaller square? [图]

💡 解题思路

Drawing segments $AC$ and $BD$ , the number of triangles outside square $ABCD$ is the same as the number of triangles inside the square. Thus areas must be equal so the area of $ABCD$ is half the area

6

第 6 题

几何·面积

The letter T is formed by placing two 2 × 4 inch rectangles next to each other, as shown. What is the perimeter of the T, in inches? [图]

💡 解题思路

If the two rectangles were seperate, the perimeter would be $2(2(2+4)=24$ . It easy to see that their connection erases 2 from each of the rectangles, so the final perimeter is $24-2 \times 2 = \boxed

7

第 7 题

几何·面积

== Problem 7 ==( Jayesh is king ) Circle X has a radius of π . Circle Y has a circumference of 8 π . Circle Z has an area of 9 π . List the circles in order from smallest to the largest radius.

💡 解题思路

Not using the formulas of circles, $C=2 \pi r$ and $A= \pi r^2$ , we find that circle $Y$ has a radius of $4$ and circle $Z$ has a radius of $3$ . Also, circle X has a radius of $\pi$ . Thus, the orde

8

第 8 题

分数与比例

The table shows some of the results of a survey by radiostation KACL. What percentage of the males surveyed listen to the station? \begin{tabular}{|c|c|c|c|}\hline & Listen & Don't Listen & Total ; \hline Males & ? & 26 & ? ; \hline Females & 58 & ? & 96 ; \hline Total & 136 & 64 & 200 ; \hline\end{tabular}

💡 解题思路

Filling out the chart, it becomes

9

第 9 题

行程问题

What is the product of \frac{3}{2}×\frac{4}{3}×\frac{5}{4}×·s×\frac{2006}{2005} ?

💡 解题思路

The numerator in each fraction cancels out with the denominator of the next fraction. There are only two numbers that didn't cancel: $\frac{2006}{2}=\boxed{\textbf{(C)}\ 1003}$ .

10

第 10 题

几何·面积

Jorge's teacher asks him to plot all the ordered pairs (w. l) of positive integers for which w is the width and l is the length of a rectangle with area 12. What should his graph look like? (A) [图] (B) [图] (C) [图] (D) [图] (E) [图]

💡 解题思路

The length of the rectangle will relate invertly to the width, specifically using the theorem $l=\frac{12}{w}$ . The only graph that could represent a inverted relationship is $\boxed{\textbf{(A)}}$ .

11

第 11 题

几何·面积

How many two-digit numbers have digits whose sum is a perfect square?

💡 解题思路

There is $1$ integer whose digits sum to $1$ : $10$ .

12

第 12 题

分数与比例

Antonette gets 70 \% on a 10-problem test, 80 \% on a 20-problem test and 90 \% on a 30-problem test. If the three tests are combined into one 60-problem test, which percent is closest to her overall score?

💡 解题思路

$70 \% \cdot 10=7$

13

第 13 题

行程问题

Cassie leaves Escanaba at 8:30 AM heading for Marquette on her bike. She bikes at a uniform rate of 12 miles per hour. Brian leaves Marquette at 9:00 AM heading for Escanaba on his bike. He bikes at a uniform rate of 16 miles per hour. They both bike on the same 62-mile route between Escanaba and Marquette. At what time in the morning do they meet?

💡 解题思路

If Cassie leaves $\frac{1}{2}$ an hour earlier then Brian, when Brian starts, the distance between them will be $62-\frac{12}{2}=56$ . Every hour, they will get $12+16=28$ miles closer. $\frac{56}{28}

14

第 14 题

综合

Problems 14, 15 and 16 involve Mrs. Reed's English assignment. A Novel Assignment The students in Mrs. Reed's English class are reading the same 760 -page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds. If Bob and Chandra both read the whole book, Bob will spend how many more seconds reading than Chandra?

💡 解题思路

The information is the same for Problems 14,15, and 16. Therefore, we shall only use the information we need. All we need for this problem is that there's 760 pages, Bob reads a page in 45 seconds and

15

第 15 题

行程问题

Problems 14, 15 and 16 involve Mrs. Reed's English assignment. A Novel Assignment The students in Mrs. Reed's English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds. Chandra and Bob, who each have a copy of the book, decide that they can save time by "team reading" the novel. In this scheme, Chandra will read from page 1 to a certain page and Bob will read from the next page through page 760, finishing the book. When they are through they will tell each other about the part they read. What is the last page that Chandra should read so that she and Bob spend the same amount of time reading the novel?

💡 解题思路

Same as the previous problem, we only use the information we need. Note that it's not just Chandra reads half of it and Bob reads the rest since they have different reading rates. In this case, we set

16

第 16 题

行程问题

Problems 14, 15 and 16 involve Mrs. Reed's English assignment. A Novel Assignment The students in Mrs. Reed's English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds. Before Chandra and Bob start reading, Alice says she would like to team read with them. If they divide the book into three sections so that each reads for the same length of time, how many seconds will each have to read?

💡 解题思路

The amount of pages Bob, Chandra, and Alice will read is in the ratio 4:6:9. Therefore, Bob, Chandra, and Alice read 160, 240, and 360 pages respectively. They would also be reading for the same amoun

17

第 17 题

概率

Jeff rotates spinners P , Q and R and adds the resulting numbers. What is the probability that his sum is an odd number? [图]

💡 解题思路

In order for Jeff to have an odd number sum, the numbers must either be Odd + Odd + Odd or Even + Even + Odd. We easily notice that we cannot obtain Odd + Odd + Odd because spinner $Q$ contains only e

18

第 18 题

几何·面积

A cube with 3-inch edges is made using 27 cubes with 1-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white?

💡 解题思路

The surface area of the cube is $6(3)(3)=54$ . Each of the eight black cubes has 3 faces on the outside, making $3(8)=24$ black faces. Therefore there are $54-24=30$ white faces. To find the ratio, we

19

第 19 题

几何·面积

Triangle ABC is an isosceles triangle with \overline{AB}=\overline{BC} . Point D is the midpoint of both \overline{BC} and \overline{AE} , and \overline{CE} is 11 units long. Triangle ABD is congruent to triangle ECD . What is the length of \overline{BD} ? [图]

💡 解题思路

Since triangle $ABD$ is congruent to triangle $ECD$ and $\overline{CE} =11$ , $\overline{AB}=11$ . Since $\overline{AB}=\overline{BC}$ , $\overline{BC}=11$ . Because point $D$ is the midpoint of $\ove

20

第 20 题

综合

A singles tournament had six players. Each player played every other player only once, with no ties. If Helen won 4 games, Ines won 3 games, Janet won 2 games, Kendra won 2 games and Lara won 2 games, how many games did Monica win?

💡 解题思路

Since there are 6 players, a total of $\frac{6(6-1)}{2}=15$ games are played. So far, $4+3+2+2+2=13$ games finished (one person won from each game), so Monica needs to win $15-13 = \boxed{\textbf{(C)}

21

第 21 题

行程问题

An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. The aquarium is filled with water to a depth of 37 cm. A rock with volume 1000cm^3 is then placed in the aquarium and completely submerged. By how many centimeters does the water level rise?

💡 解题思路

The water level will rise $1$ cm for every $100 \cdot 40 = 4000\text{cm}^2$ . Since $1000$ is $\frac{1}{4}$ of $4000$ , the water will rise $\frac{1}{4}\cdot1 = \boxed{\textbf{(A)}\ 0.25}$

22

第 22 题

规律与数列

Three different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell. What is the difference between the largest and smallest numbers possible in the top cell? [图]

💡 解题思路

If the lower cells contain $A, B$ and $C$ , then the second row will contain $A + B$ and $B + C$ , and the top cell will contain $A + 2B + C$ . To obtain the smallest sum, place $1$ in the center cell

23

第 23 题

概率

A box contains gold coins. If the coins are equally divided among six people, four coins are left over. If the coins are equally divided among five people, three coins are left over. If the box holds the smallest number of coins that meets these two conditions, how many coins are left when equally divided among seven people?

💡 解题思路

This is a modular arithmetic problem.

24

第 24 题

行程问题

In the multiplication problem below A , B , C , D are different digits. What is A+B ? \[\begin{array}{cccc}& A & B & A ; × & & C & D ; \hline C & D & C & D ; \end{array}\]

💡 解题思路

$CDCD = CD \cdot 101$ , so $ABA = 101$ . Therefore, $A = 1$ and $B = 0$ , so $A+B=1+0=\boxed{\textbf{(A)}\ 1}$ .

25

第 25 题

数论

Barry wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table, as shown. The sums of the two numbers on each of the three cards are equal. The three numbers on the hidden sides are prime numbers. What is the average of the hidden prime numbers? [图]

💡 解题思路

Notice that 44 and 38 are both even, while 59 is odd. If any odd prime is added to 59, an even number will be obtained. However, the only way to obtain this even number(common sum) would be to add ano

📄 2006 AMC 8 真题

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