2000 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
综合
Aunt Anna is 42 years old. Caitlin is 5 years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin? (A)\ 15 (B)\ 16 (C)\ 17 (D)\ 21 (E)\ 37
💡 解题思路
If Brianna is half as old as Aunt Anna, then Brianna is $\frac{42}{2}$ years old, or $21$ years old.
2
第 2 题
综合
Which of these numbers is less than its reciprocal? (A)\ -2 (B)\ -1 (C)\ 0 (D)\ 1 (E)\ 2
💡 解题思路
The number $0$ has no reciprocal, and $1$ and $-1$ are their own reciprocals. This leaves only $2$ and $-2$ . The reciprocal of $2$ is $1/2$ , but $2$ is not less than $1/2$ . The reciprocal of $-2$ i
3
第 3 题
综合
How many whole numbers lie in the interval between \frac{5}{3} and 2π ? (A)\ 2 (B)\ 3 (C)\ 4 (D)\ 5 (E)\ infinitely many
💡 解题思路
The smallest whole number in the interval is $2$ because $5/3$ is more than $1$ but less than $2$ . The largest whole number in the interval is $6$ because $2\pi$ is more than $6$ but less than $7$ .
4
第 4 题
坐标几何
In 1960 only 5% of the working adults in Carlin City worked at home. By 1970 the "at-home" work force had increased to 8%. In 1980 there were approximately 15% working at home, and in 1990 there were 30%. The graph that best illustrates this is: [图]
💡 解题思路
The data are $1960 (5\%)$ , $1970 (8\%)$ , $1980 (15\%)$ , and $1990 (30\%)$ . Only one of these graphs has the answer and that is choice $\boxed{E}$ .
5
第 5 题
规律与数列
Each principal of Lincoln High School serves exactly one 3 -year term. What is the maximum number of principals this school could have during an 8 -year period? (A)\ 2 (B)\ 3 (C)\ 4 (D)\ 5 (E)\ 8
💡 解题思路
If the first year of the $8$ -year period was the final year of a principal's term, then in the next six years two more principals would serve, and the last year of the period would be the first year
6
第 6 题
几何·面积
Figure ABCD is a square. Inside this square three smaller squares are drawn with the side lengths as labeled. The area of the shaded L -shaped region is [图] (A)\ 7 (B)\ 10 (C)\ 12.5 (D)\ 14 (E)\ 15
💡 解题思路
The side of the large square is $1 + 3 + 1 = 5$ , so the area of the large square is $5^2 = 25$ .
7
第 7 题
综合
What is the minimum possible product of three different numbers of the set \{-8,-6,-4,0,3,5,7\} ? (A)\ -336 (B)\ -280 (C)\ -210 (D)\ -192 (E)\ 0
💡 解题思路
The only way to get a negative product using three numbers is to multiply one negative number and two positives or three negatives. Only two reasonable choices exist: $(-8)\times(-6)\times(-4) = (-8)\
8
第 8 题
概率
Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). The total number of dots NOT visible in this view is [图] (A)\ 21 (B)\ 22 (C)\ 31 (D)\ 41 (E)\ 53
💡 解题思路
The numbers on one die total $1+2+3+4+5+6 = 21$ , so the numbers on the three dice total $63$ . Numbers $1, 1, 2, 3, 4, 5, 6$ are visible, and these total $22$ . This leaves $63 - 22 = \boxed{\text{(D
9
第 9 题
几何·面积
Three-digit powers of 2 and 5 are used in this "cross-number" puzzle. What is the only possible digit for the outlined square? \[\begin{array}{lcl} ACROSS & & DOWN ; 2.~ 2^m & & 1.~ 5^n \end{array}\] [图] (A)\ 0 (B)\ 2 (C)\ 4 (D)\ 6 (E)\ 8
💡 解题思路
The $3$ -digit powers of $5$ are $125$ and $625$ , so space $2$ is filled with a $2$ . The only $3$ -digit power of $2$ beginning with $2$ is $256$ , so the outlined block is filled with a $\boxed{\te
10
第 10 题
综合
Ara and Shea were once the same height. Since then Shea has grown 20% while Ara has grown half as many inches as Shea. Shea is now 60 inches tall. How tall, in inches, is Ara now? (A)\ 48 (B)\ 51 (C)\ 52 (D)\ 54 (E)\ 55
💡 解题思路
Shea has grown $20\%$ , if x was her original height, then $1.2x = 60$ , so she was originally $\frac{60}{1.2}=50$ inches tall which is a $60 - 50 = 10$ inch increase. Ara also started off at $50$ inc
11
第 11 题
数论
The number 64 has the property that it is divisible by its unit digit. How many whole numbers between 10 and 50 have this property?
💡 解题思路
Casework by the units digit $u$ will help organize the answer.
12
第 12 题
整数运算
A block wall 100 feet long and 7 feet high will be constructed using blocks that are 1 foot high and either 2 feet long or 1 foot long (no blocks may be cut). The vertical joins in the blocks must be staggered as shown, and the wall must be even on the ends. What is the smallest number of blocks needed to build this wall? [图] (A)\ 344 (B)\ 347 (C)\ 350 (D)\ 353 (E)\ 356
💡 解题思路
Since the bricks are $1$ foot high, there will be $7$ rows. To minimize the number of blocks used, rows $1, 3, 5,$ and $7$ will look like the bottom row of the picture, which takes $\frac{100}{2} = 50
13
第 13 题
几何·面积
In triangle CAT , we have \angle ACT =\angle ATC and \angle CAT = 36^\circ . If \overline{TR} bisects \angle ATC , then \angle CRT = [图] (A)\ 36^\circ (B)\ 54^\circ (C)\ 72^\circ (D)\ 90^\circ (E)\ 108^\circ
💡 解题思路
In $\triangle ACT$ , the three angles sum to $180^\circ$ , and $\angle C = \angle T$
14
第 14 题
数字运算
What is the units digit of 19^{19} + 99^{99} ? (A)\ 0 (B)\ 1 (C)\ 2 (D)\ 8 (E)\ 9 https://youtu.be/7an5wU9Q5hk?t=1552
💡 解题思路
Finding a pattern for each half of the sum, even powers of $19$ have a units digit of $1$ , and odd powers of $19$ have a units digit of $9$ . So, $19^{19}$ has a units digit of $9$ .
15
第 15 题
几何·面积
Triangles ABC , ADE , and EFG are all equilateral. Points D and G are midpoints of \overline{AC} and \overline{AE} , respectively. If AB = 4 , what is the perimeter of figure ABCDEFG ? [图] (A)\ 12 (B)\ 13 (C)\ 15 (D)\ 18 (E)\ 21
💡 解题思路
The large triangle $ABC$ has sides of length $4$ . The medium triangle has sides of length $2$ . The small triangle has sides of length $1$ . There are $3$ segment sizes, and all segments depicted are
16
第 16 题
几何·面积
In order for Mateen to walk a kilometer (1000m) in his rectangular backyard, he must walk the length 25 times or walk its perimeter 10 times. What is the area of Mateen's backyard in square meters? (A)\ 40 (B)\ 200 (C)\ 400 (D)\ 500 (E)\ 1000
💡 解题思路
The length $L$ of the rectangle is $\frac{1000}{25}=40$ meters. The perimeter $P$ is $\frac{1000}{10}=100$ meters. Since $P_{rect} = 2L + 2W$ , we plug values in to get:
17
第 17 题
分数与比例
The operation \otimes is defined for all nonzero numbers by a\otimes b =\frac{a^{2}}{b} . Determine [(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)] . (A)\ -\frac{2}{3} (B)\ -\frac{1}{4} (C)\ 0 (D)\ \frac{1}{4} (E)\ \frac{2}{3}
💡 解题思路
Follow PE(MD)(AS), doing the innermost parentheses first.
18
第 18 题
几何·面积
Consider these two geoboard quadrilaterals. Which of the following statements is true? [图] (A)\ The area of quadrilateral I is more than the area of quadrilateral II.(B)\ The area of quadrilateral I is less than the area of quadrilateral II.(C)\ The quadrilaterals have the same area and the same perimeter.(D)\ The quadrilaterals have the same area, but the perimeter of I is more than the perimeter of II.(E)\ The quadrilaterals have the same area, but the perimeter of I is less than the perimeter of II.
💡 解题思路
First consider the area of the two figures. Assume that the pegs are $1$ unit apart. Divide region I into two triangles by drawing a horizontal line on the second row from the top. Shifting the bottom
19
第 19 题
几何·面积
Three circular arcs of radius 5 units bound the region shown. Arcs AB and AD are quarter-circles, and arc BCD is a semicircle. What is the area, in square units, of the region? [图] (A)\ 25 (B)\ 10+5π (C)\ 50 (D)\ 50+5π (E)\ 25π
💡 解题思路
Draw two squares: one that has opposing corners at $A$ and $B$ , and one that has opposing corners at $A$ and $D$ . These squares share side $\overline{AO}$ , where $O$ is the center of the large semi
20
第 20 题
概率
You have nine coins: a collection of pennies, nickels, dimes, and quarters having a total value of 1.02 , with at least one coin of each type. How many dimes must you have? (A)\ 1 (B)\ 2 (C)\ 3 (D)\ 4 (E)\ 5$
💡 解题思路
Since you have one coin of each type, $1 + 5 + 10 + 25 = 41$ cents are already determined, leaving you with a total of $102 - 41 = 61$ cents remaining for $5$ coins.
21
第 21 题
概率
Keiko tosses one penny and Ephraim tosses two pennies. The probability that Ephraim gets the same number of heads that Keiko gets is (A)\ \frac{1}{4} (B)\ \frac{3}{8} (C)\ \frac{1}{2} (D)\ \frac{2}{3} (E)\ \frac{3}{4}
💡 解题思路
Divide it into $2$ cases:
22
第 22 题
几何·面积
A cube has edge length 2 . Suppose that we glue a cube of edge length 1 on top of the big cube so that one of its faces rests entirely on the top face of the larger cube. The percent increase in the surface area (sides, top, and bottom) from the original cube to the new solid formed is closest to [图] (A)\ 10 (B)\ 15 (C)\ 17 (D)\ 21 (E)\ 25
💡 解题思路
The original cube has $6$ faces, each with an area of $2\cdot 2 = 4$ square units. Thus the original figure had a total surface area of $24$ square units.
23
第 23 题
统计
There is a list of seven numbers. The average of the first four numbers is 5 , and the average of the last four numbers is 8 . If the average of all seven numbers is 6\frac{4}{7} , then the number common to both sets of four numbers is (A)\ 5\frac{3}{7} (B)\ 6 (C)\ 6\frac{4}{7} (D)\ 7 (E)\ 7\frac{3}{7}
💡 解题思路
Remember that if a list of $n$ numbers has an average of $k$ , then the sum $S$ of all the numbers on the list is $S = nk$ .
24
第 24 题
几何·角度
If \angle A = 20^\circ and \angle AFG =\angle AGF , then \angle B+\angle D = [图] (A)\ 48^\circ (B)\ 60^\circ (C)\ 72^\circ (D)\ 80^\circ (E)\ 90^\circ
💡 解题思路
As a strategy, think of how $\angle B + \angle D$ would be determined, particularly without determining either of the angles individually, since it may not be possible to determine $\angle B$ or $\ang
25
第 25 题
几何·面积
The area of rectangle ABCD is 72 units squared. If point A and the midpoints of \overline{BC} and \overline{CD} are joined to form a triangle, the area of that triangle is [图] (A)\ 21 (B)\ 27 (C)\ 30 (D)\ 36 (E)\ 40
💡 解题思路
To quickly solve this multiple choice problem, make the (not necessarily valid, but very convenient) assumption that $ABCD$ can have any dimension. Give the rectangle dimensions of $AB = CD = 12$ and