2001 AMC 8 — Official Competition Problems (November 2019)
📅 2011 年11月📝 25题选择题⏱ 40分钟🎯 满分25分✅ 含解题思路👥 612 人已练习
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1
第 1 题
时间问题
Casey's shop class is making a golf trophy. He has to paint 300 dimples on a golf ball. If it takes him 2 seconds to paint one dimple, how many minutes will it take for him to finish (A) \ 4 (B) \ 6 (C) \ 8 (D) \ 10 (E) \ 12
💡 解题思路
It will take him $300\cdot2=600$ seconds to paint all the dimples.
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第 2 题
规律与数列
I'm thinking of two whole numbers. Their product is 24 and their sum is 11. What is the larger number? (A)\ 3 (B)\ 4 (C)\ 6 (D)\ 8 (E)\ 12
💡 解题思路
Let the numbers be $x$ and $y$ . Then we have $x+y=11$ and $xy=24$ . Solving for $x$ in the first equation yields $x=11-y$ , and substituting this into the second equation gives $(11-y)(y)=24$ . Simpl
3
第 3 题
应用题
Granny Smith has \63. Elberta has \2 more than Anjou and Anjou has one-third as much as Granny Smith. How many dollars does Elberta have? (A)\ 17 (B)\ 18 (C)\ 19 (D)\ 21 (E)\ 23
💡 解题思路
Since Anjou has $\frac{1}{3}$ the amount of money as Granny Smith and Granny Smith has $ $63$ , Anjou has $\frac{1}{3}\times63=21$ dollars. Elberta has $ $2$ more than this, so she has $ $23$ , or $\b
4
第 4 题
数字运算
The digits 1, 2, 3, 4 and 9 are each used once to form the smallest possible even five-digit number. The digit in the tens place is (A)\ 1 (B)\ 2 (C)\ 3 (D)\ 4 (E)\ 9
💡 解题思路
Since the number is even, the last digit must be $2$ or $4$ . To make the smallest possible number, the ten-thousands digit must be as small as possible, so the ten-thousands digit is $1$ . Simillarly
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第 5 题
行程问题
On a dark and stormy night Snoopy suddenly saw a flash of lightning. Ten seconds later he heard the sound of thunder. The speed of sound is 1088 feet per second and one mile is 5280 feet. Estimate, to the nearest half-mile, how far Snoopy was from the flash of lightning. (A)\ 1 (B)\ 1\frac{1}{2} (C)\ 2 (D)\ 2\frac{1}{2} (E)\ 3
💡 解题思路
During the $10$ seconds, the sound traveled $1088\times10=10880$ feet from the lightning to Snoopy. This is equivalent to $\frac{10880}{5280}\approx2$ miles, $\boxed{\text{C}}$ .
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第 6 题
行程问题
Six trees are equally spaced along one side of a straight road. The distance from the first tree to the fourth is 60 feet. What is the distance in feet between the first and last trees? (A)\ 90 (B)\ 100 (C)\ 105 (D)\ 120 (E)\ 140
💡 解题思路
There are $3$ spaces between the 1st and 4th trees, so each of these spaces has $\frac{60}{3}=20$ feet. Between the first and last trees there are $5$ spaces, so the distance between them is $20\times
7
第 7 题
几何·面积
To promote her school's annual Kite Olympics, Genevieve makes a small kite and a large kite for a bulletin board display. The kites look like the one in the diagram. For her small kite Genevieve draws the kite on a one-inch grid. For the large kite she triples both the height and width of the entire grid. [图] What is the number of square inches in the area of the small kite? (A)\ 21 (B)\ 22 (C)\ 23 (D)\ 24 (E)\ 25
💡 解题思路
The area of a kite is half the product of its diagonals. The diagonals have lengths of $6$ and $7$ , so the area is $\frac{(6)(7)}{2}=21, \boxed{\text{A}}$ .
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第 8 题
综合
To promote her school's annual Kite Olympics, Genevieve makes a small kite and a large kite for a bulletin board display. The kites look like the one in the diagram. For her small kite Genevieve draws the kite on a one-inch grid. For the large kite she triples both the height and width of the entire grid. [图] Genevieve puts bracing on her large kite in the form of a cross connecting opposite corners of the kite. How many inches of bracing material does she need? (A)\ 30 (B)\ 32 (C)\ 35 (D)\ 38 (E)\ 39
💡 解题思路
Each diagonal of the large kite is $3$ times the length of the corresponding diagonal of the short kite since it was made with a grid $3$ times as long in both height and width. The diagonals of the s
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第 9 题
几何·面积
To promote her school's annual Kite Olympics, Genevieve makes a small kite and a large kite for a bulletin board display. The kites look like the one in the diagram. For her small kite Genevieve draws the kite on a one-inch grid. For the large kite she triples both the height and width of the entire grid. [图] The large kite is covered with gold foil. The foil is cut from a rectangular piece that just covers the entire grid. How many square inches of waste material are cut off from the four corners?
💡 解题思路
The large grid has dimensions three times that of the small grid, so its dimensions are $3(6)\times3(7)$ , or $18\times21$ , so the area is $(18)(21)=378$ . The area of the kite is half of the area of
10
第 10 题
行程问题
A collector offers to buy state quarters for 2000% of their face value. At that rate how much will Bryden get for his four state quarters? (A)\ 20 dollars (B)\ 50 dollars (C)\ 200 dollars (D)\ 500 dollars (E)\ 2000 dollars
💡 解题思路
$2000\%$ is equivalent to $20\times100\%$ . Therefore, $2000\%$ of a number is the same as $20$ times that number. $4$ quarters is $1$ dollar, so Bryden will get $20\times1={20}$ dollars, $\boxed{\tex
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第 11 题
几何·面积
Points A , B , C and D have these coordinates: A(3,2) , B(3,-2) , C(-3,-2) and D(-3, 0) . The area of quadrilateral ABCD is [图] (A)\ 12 (B)\ 15 (C)\ 18 (D)\ 21 (E)\ 24
💡 解题思路
[asy] for (int i = -4; i <= 4; ++i) { for (int j = -4; j <= 4; ++j) { dot((i,j)); } } draw((0,-4)--(0,4),linewidth(1)); draw((-4,0)--(4,0),linewidth(1)); for (int i = -4; i <= 4; ++i) { draw((i,-1/3)-
12
第 12 题
行程问题
If a\otimes b = \dfrac{a + b}{a - b} , then (6\otimes 4)\otimes 3 =(A)\ 4 (B)\ 13 (C)\ 15 (D)\ 30 (E)\ 72
Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle's pie graph showing this data, how many degrees should she use for cherry pie? (A)\ 10 (B)\ 20 (C)\ 30 (D)\ 50 (E)\ 72
💡 解题思路
There are $36$ students in the class: $12$ prefer chocolate pie, $8$ prefer apple pie, and $6$ prefer blueberry pie. Therefore, $36-12-8-6=10$ students prefer cherry pie or lemon pie. Half of these pr
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第 14 题
计数
Tyler has entered a buffet line in which he chooses one kind of meat, two different vegetables and one dessert. If the order of food items is not important, how many different meals might he choose? (A)\ 4 (B)\ 24 (C)\ 72 (D)\ 80 (E)\ 144
💡 解题思路
There are $3$ possibilities for the meat and $4$ possibilites for the dessert, for a total of $4\times3=12$ possibilities for the meat and the dessert. There are $4$ possibilities for the first vegeta
15
第 15 题
行程问题
Homer began peeling a pile of 44 potatoes at the rate of 3 potatoes per minute. Four minutes later Christen joined him and peeled at the rate of 5 potatoes per minute. When they finished, how many potatoes had Christen peeled? (A)\ 20 (B)\ 24 (C)\ 32 (D)\ 33 (E)\ 40
💡 解题思路
After the $4$ minutes of Homer peeling alone, he had peeled $4\times3=12$ potatoes. This means that there are $44-12=32$ potatoes left. Once Christen joins him, the two are peeling potatoes at a rate
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第 16 题
几何·面积
A square piece of paper, 4 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle? [图] (A)\ \dfrac{1}{3} (B)\ \dfrac{1}{2} (C)\ \dfrac{3}{4} (D)\ \dfrac{4}{5} (E)\ \dfrac{5}{6}
💡 解题思路
The smaller rectangles each have the same height as the original square, but have $\frac{1}{4}$ the length, since the paper is folded in half and then cut in half the same way. The larger rectangle ha
17
第 17 题
分数与比例
For the game show Who Wants To Be A Millionaire? , the dollar values of each question are shown in the following table (where K = 1000). \[\begin{tabular}{rccccccccccccccc} Question & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 ; Value & 100 & 200 & 300 & 500 & 1K & 2K & 4K & 8K & 16K & 32K & 64K & 125K & 250K & 500K & 1000K \end{tabular}\] Between which two questions is the percent increase of the value the smallest? (A)\ From 1 to 2 (B)\ From 2 to 3 (C)\ From 3 to 4 (D)\ From 11 to 12 (E)\ From 14 to 15
💡 解题思路
Notice that in two of the increases, the dollar amount doubles. The increases in which this is not true is $2$ to $3$ , $3$ to $4$ , and $11$ to $12$ . We can disregard $11$ to $12$ since that increas
18
第 18 题
数论
Two dice are thrown. What is the probability that the product of the two numbers is a multiple of 5? (A)\ \dfrac{1}{36} (B)\ \dfrac{1}{18} (C)\ \dfrac{1}{6} (D)\ \dfrac{11}{36} (E)\ \dfrac{1}{3}
💡 解题思路
This is equivalent to asking for the probability that at least one of the numbers is a multiple of $5$ , since if one of the numbers is a multiple of $5$ , then the product with it and another integer
19
第 19 题
坐标几何
Car M traveled at a constant speed for a given time. This is shown by the dashed line. Car N traveled at twice the speed for the same distance. If Car M and Car N's speed and time are shown as solid line, which graph illustrates this? [图]
💡 解题思路
Since car N has twice the speed, it must be twice as high on the speed axis. Also, since cars M and N travel at the same distance but car N has twice the speed, car N must take half the time. Therefor
20
第 20 题
综合
Kaleana shows her test score to Quay, Marty and Shana, but the others keep theirs hidden. Quay thinks, "At least two of us have the same score." Marty thinks, "I didn't get the lowest score." Shana thinks, "I didn't get the highest score." List the scores from lowest to highest for Marty (M), Quay (Q) and Shana (S). (A)\ S,Q,M (B)\ Q,M,S (C)\ Q,S,M (D)\ M,S,Q (E)\ S,M,Q
💡 解题思路
Since the only other score Quay knows is Kaleana's, and he knows that two of them have the same score, Quay and Kaleana must have the same score, and $K=Q$ . Marty knows that he didn't get the lowest
21
第 21 题
统计
The mean of a set of five different positive integers is 15. The median is 18. The maximum possible value of the largest of these five integers is (A)\ 19 (B)\ 24 (C)\ 32 (D)\ 35 (E)\ 40
💡 解题思路
Since there is an odd number of terms, the median is the number in the middle, specifically, the third largest number is $18$ , and there are $2$ numbers less than $18$ and $2$ numbers greater than $1
22
第 22 题
综合
On a twenty-question test, each correct answer is worth 5 points, each unanswered question is worth 1 point and each incorrect answer is worth 0 points. Which of the following scores is NOT possible? (A)\ 90 (B)\ 91 (C)\ 92 (D)\ 95 (E)\ 97
💡 解题思路
The highest possible score is if you get every answer right, to get $5(20)=100$ . The second highest possible score is if you get $19$ questions right and leave the remaining one blank, to get a $5(19
23
第 23 题
几何·面积
Points R , S and T are vertices of an equilateral triangle, and points X , Y and Z are midpoints of its sides. How many noncongruent triangles can be drawn using any three of these six points as vertices? [图] (A)\ 1 (B)\ 2 (C)\ 3 (D)\ 4 (E)\ 20
💡 解题思路
There are $6$ points in the figure, and $3$ of them are needed to form a triangle, so there are ${6\choose{3}} =20$ possible triplets of the $6$ points. However, some of these created congruent triang
24
第 24 题
几何·面积
Each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. There are 2 red-white pairs. How many white pairs coincide? [图]
💡 解题思路
Each half has $3$ red triangles, $5$ blue triangles, and $8$ white triangles. There are also $2$ pairs of red triangles, so $2$ red triangles on each side are used, leaving $1$ red triangle, $5$ blue
25
第 25 题
数论
There are 24 four-digit whole numbers that use each of the four digits 2, 4, 5 and 7 exactly once. Only one of these four-digit numbers is a multiple of another one. Which of the following is it?
💡 解题思路
There are only 5 options for the problem so we can just try them. This is easy since that we only need try to use $2$ , $3$ to divide them. Even dividing by $4$ leads to a solution that starts with $1