2007 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
📋 答题说明
共 25 道题，每题从 A、B、C、D、E 五个选项中选一个答案，点击选项即可选择
答题过程中可随时更改选项，选完后点击底部「提交答案」统一批改
提交后显示对错、正确答案和简短解题思路
点击题目右侧 ⭐ 可收藏难题，方便后续复习
题目涉及图形的部分，原题以文字描述代替（图形题建议配合原版试卷使用）
1
第 1 题
统计
Theresa's parents have agreed to buy her tickets to see her favorite band if she spends an average of 10 hours per week helping around the house for 6 weeks. For the first 5 weeks she helps around the house for 8 , 11 , 7 , 12 and 10 hours. How many hours must she work for the final week to earn the tickets? (A)\ 9 (B)\ 10 (C)\ 11 (D)\ 12 (E)\ 13
💡 解题思路
Let $x$ be the number of hours she must work for the final week. We are looking for the average, so \[\frac{8 + 11 + 7 + 12 + 10 + x}{6} = 10\] Solving gives: \[\frac{48 + x}{6} = 10\]
2
第 2 题
坐标几何
650 students were surveyed about their pasta preferences. The choices were lasagna, manicotti, ravioli and spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti? [图] (A) \frac{2}{5} (B) \frac{1}{2} (C) \frac{5}{4} (D) \frac{5}{3} (E) \frac{5}{2}
💡 解题思路
The answer is $\dfrac{\text{number of students who preferred spaghetti}}{\text{number of students who preferred manicotti}}$
3
第 3 题
数论
What is the sum of the two smallest prime factors of 250 ? (A)\ 2 (B)\ 5 (C)\ 7 (D)\ 10 (E)\ 12 https://youtu.be/7an5wU9Q5hk?t=272
💡 解题思路
The prime factorization of $250$ is $2 \cdot 5^3$ . The smallest two are $2$ and $5$ . $2+5 = \boxed{\text{(C) }7}$ .
4
第 4 题
计数
A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window? (A)\ 12 (B)\ 15 (C)\ 18 (D)\ 30 (E)\ 36
💡 解题思路
Georgie can enter the haunted house through any of the six windows. Then, he can leave through any of the remaining five windows.
5
第 5 题
应用题
Chandler wants to buy a 500 dollar mountain bike. For his birthday, his grandparents send him 50 dollars, his aunt sends him 35 dollars and his cousin gives him 15 dollars. He earns 16 dollars per week for his paper route. He will use all of his birthday money and all of the money he earns from his paper route. In how many weeks will he be able to buy the mountain bike? (A)\ 24 (B)\ 25 (C)\ 26 (D)\ 27 (E)\ 28
💡 解题思路
Let $x$ be the number of weeks.
6
第 6 题
分数与比例
The average cost of a long-distance call in the USA in 1985 was 41 cents per minute, and the average cost of a long-distance call in the USA in 2005 was 7 cents per minute. Find the approximate percent decrease in the cost per minute of a long- distance call. (A)\ 7 (B)\ 17 (C)\ 34 (D)\ 41 (E)\ 80
💡 解题思路
The percent decrease is (the amount of decrease)/(original amount)
7
第 7 题
统计
The average age of 5 people in a room is 30 years. An 18 -year-old person leaves the room. What is the average age of the four remaining people? (A)\ 25 (B)\ 26 (C)\ 29 (D)\ 33 (E)\ 36
💡 解题思路
Let $x$ be the average of the remaining $4$ people.
8
第 8 题
几何·面积
In trapezoid ABCD , \overline{AD} is perpendicular to \overline{DC} , AD = AB = 3 , and DC = 6 . In addition, E is on \overline{DC} , and \overline{BE} is parallel to \overline{AD} . Find the area of \triangle BEC . [图]
💡 解题思路
Clearly, $ABED$ is a square with side-length $3.$ By segment subtraction, we have $EC = DC - DE = 6 - 3 = 3.$
9
第 9 题
几何·面积
To complete the grid below, each of the digits 1 through 4 must occur once in each row and once in each column. What number will occupy the lower right-hand square? \[\begin{tabular}{|c|c|c|c|}\hline 1 & & 2 & ; \hline 2 & 3 & & ; \hline & &&4 ; \hline & && ; \hline\end{tabular}\] (A)\ 1 (B)\ 2 (C)\ 3 (D)\ 4 (E)\ cannot be determined
💡 解题思路
The number in the first row, last column must be a $3$ due to the fact if a $3$ was in the first row, second column, there would be two threes in that column. By the same reasoning, the number in the
10
第 10 题
数论
For any positive integer n , define [n] to be the sum of the positive factors of n . For example, [6] = 1 + 2 + 3 + 6 = 12 . Find [\boxed{11]} .
💡 解题思路
We have \begin{align*} \boxed{\boxed{11}}&=\boxed{1+11} \\ &=\boxed{12} \\ &=1+2+3+4+6+12 \\ &=28, \end{align*} from which the answer is $\boxed{\textbf{(D)}\ 28}.$
11
第 11 题
几何·面积
Tiles I, II, III and IV are translated so one tile coincides with each of the rectangles A, B, C and D . In the final arrangement, the two numbers on any side common to two adjacent tiles must be the same. Which of the tiles is translated to Rectangle C ? [图] (A)\ I (B)\ II (C)\ III (D)\ IV (E) cannot be determined
💡 解题思路
We first notice that tile III has a $0$ on the bottom and a $5$ on the right side. Since no other tile has a $0$ or a $5$ , Tile III must be in rectangle $D$ . Tile III also has a $1$ on the left, so
12
第 12 题
几何·面积
A unit hexagram is composed of a regular hexagon of side length 1 and its 6 equilateral triangular extensions, as shown in the diagram. What is the ratio of the area of the extensions to the area of the original hexagon? [图] (A)\ 1:1 (B)\ 6:5 (C)\ 3:2 (D)\ 2:1 (E)\ 3:1
💡 解题思路
The six equilateral triangular extensions fit perfectly into the hexagon meaning the answer is $\boxed{\textbf{(A) }1:1}$
13
第 13 题
综合
Sets A and B , shown in the Venn diagram, have the same number of elements. Their union has 2007 elements and their intersection has 1001 elements. Find the number of elements in A . [图] (A)\ 503 (B)\ 1006 (C)\ 1504 (D)\ 1507 (E)\ 1510
💡 解题思路
Let $x$ be the number of elements in $A$ and $B$ which is equal.
14
第 14 题
几何·面积
The base of isosceles \triangle ABC is 24 and its area is 60 . What is the length of one of the congruent sides? (A)\ 5 (B)\ 8 (C)\ 13 (D)\ 14 (E)\ 18
💡 解题思路
The area of a triangle is shown by $\frac{1}{2}bh$ . We set the base equal to $24$ , and the area equal to $60$ , and we get the triangle's height, or altitude, to be $5$ . In this isosceles triangle,
15
第 15 题
综合
Let a, b and c be numbers with 0 < a < b < c . Which of the following is impossible? (A) \ a + c < b (B) \ a · b < c (C) \ a + b < c (D) \ a · c < b (E)\frac{b}{c} = a
💡 解题思路
According to the given rules, every number needs to be positive. Since $c$ is always greater than $b$ , adding a positive number ( $a$ ) to $c$ will always make it greater than $b$ .
16
第 16 题
几何·面积
Amanda Reckonwith draws five circles with radii 1, 2, 3, 4 and 5 . Then for each circle she plots the point (C,A) , where C is its circumference and A is its area. Which of the following could be her graph? (A) [图] (B) [图] (C) [图] (D) [图] (E) [图]
💡 解题思路
The circumference of a circle is obtained by simply multiplying the radius by $2\pi$ . So, the C-coordinate (in this case, it is the x-coordinate) will increase at a steady rate. The area, however, is
17
第 17 题
分数与比例
A mixture of 30 liters of paint is 25\% red tint, 30\% yellow tint and 45\% water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture? (A)\ 25 (B)\ 35 (C)\ 40 (D)\ 45 (E)\ 50
💡 解题思路
Since $30\%$ of the original $30$ liters of paint was yellow, and $5$ liters of yellow paint were added to make the new mixture, there are $9+5=14$ liters of yellow tint in the new mixture. Since only
18
第 18 题
规律与数列
The product of the two 99 -digit numbers 303,030,303,...,030,303 and 505,050,505,...,050,505 has thousands digit A and units digit B . What is the sum of A and B ? (A)\ 3 (B)\ 5 (C)\ 6 (D)\ 8 (E)\ 10
💡 解题思路
We can first make a small example to find out $A$ and $B$ . So,
19
第 19 题
几何·面积
Pick two consecutive positive integers whose sum is less than 100 . Square both of those integers and then find the difference of the squares. Which of the following could be the difference? (A)\ 2 (B)\ 64 (C)\ 79 (D)\ 96 (E)\ 131
💡 解题思路
Let the smaller of the two numbers be $x$ . Then, the problem states that $(x+1)+x<100$ . $(x+1)^2-x^2=x^2+2x+1-x^2=2x+1$ . $2x+1$ is obviously odd, so only answer choices C and E need to be considere
20
第 20 题
综合
Before the district play, the Unicorns had won 45\% of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all?
💡 解题思路
At the beginning of the problem, the Unicorns had played $y$ games and they had won $x$ of these games. From the information given in the problem, we can say that $\frac{x}{y}=0.45.$ Next, the Unicorn
21
第 21 题
概率
Two cards are dealt from a deck of four red cards labeled A , B , C , D and four green cards labeled A , B , C , D . A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair?
💡 解题思路
Notice that, no matter which first card you choose, there are exactly $4$ cards that either have the same color or letter as it. Since there are $7$ cards left to choose from, the probability is $\box
22
第 22 题
几何·面积
A lemming sits at a corner of a square with side length 10 meters. The lemming runs 6.2 meters along a diagonal toward the opposite corner. It stops, makes a 90^{\circ} right turn and runs 2 more meters. A scientist measures the shortest distance between the lemming and each side of the square. What is the average of these four distances in meters?
💡 解题思路
The shortest segments would be perpendicular to the square. The lemming went $x$ meters horizontally and $y$ meters vertically. No matter how much it went, the lemming would have been $x$ and $y$ mete
23
第 23 题
几何·面积
What is the area of the shaded pinwheel shown in the 5 × 5 grid? [图]
💡 解题思路
The area of the square around the pinwheel is 25. The area of the pinwheel is equal to $\text{the square } - \text{ the white space.}$ Each of the four triangles have a base of 3 units and a height of
24
第 24 题
综合
💡 解题思路
The number of ways to form a 3-digit number is $4 \cdot 3 \cdot 2 = 24$ . The combination of digits that give us multiples of 3 are (1,2,3) and (2,3,4), as the integers in the subsets have a sum which
25
第 25 题
几何·面积
On the dart board shown in the figure below, the outer circle has radius 6 and the inner circle has radius 3 . Three radii divide each circle into three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to the area of the region. When two darts hit this board, the score is the sum of the point values of the regions hit. What is the probability that the score is odd? (A) \frac{17}{36} (B) \frac{35}{72} (C) \frac{1}{2} (D) \frac{37}{72} (E) \frac{19}{36}
💡 解题思路
To get an odd sum, we must add an even number and an odd number (a 1 and a 2). So we have a little casework to do. Before we do that, we also have to figure out some relative areas. You could either f