2008 年 AMC 8 真题（P版）

1 答案 B

苏珊在游乐项目上花了 \(2\cdot 12=24\) 美元，所以她一共花了 \(12+24=36\) 美元。

她还剩 \(50-36=14\) 美元。

所以正确答案是 B。

English

Susan spent \(2\cdot 12=24\) dollars on rides, so she spent \(12+24=36\) dollars total.

She had \(50-36=14\) dollars left.

Thus, B is the correct answer.

2 答案 A

代码 BEST OF LUCK 依次表示数字 \(0\) 到 \(9\)，所以 \(C=8\)、\(L=6\)、\(U=7\)、\(E=1\)。

因此 \(\text{CLUE}=8671\)。

所以正确答案是 A。

English

The code BEST OF LUCK represents the digits \(0\) through \(9\) in order, so \(C=8\), \(L=6\), \(U=7\), and \(E=1\).

Therefore, \(\text{CLUE}=8671\).

Thus, A is the correct answer.

3 答案 A

因为 \(13\) 号是星期五，所以 \(6\) 号也是星期五。\(1\) 号比这个星期五早 \(5\) 天，因此是星期日。

所以正确答案是 A。

English

Since the \(13\)th is a Friday, we know that the \(6\)th is also a Friday. The \(1\)st is \(5\) days before the Friday, making it a Sunday.

Thus, the answer is A .

4 答案 C

外三角形面积为 \(16\)，内三角形面积为 \(1\)。

因此三个全等梯形的总面积为 \(16-1=15\)，每个梯形的面积为 \(15\div3=5\)。

所以正确答案是 C。

English

The outer triangle has area \(16\), and the inner triangle has area \(1\).

So the three congruent trapezoids have total area \(16-1=15\), and each trapezoid has area \(15\div3=5\).

Thus, C is the correct answer.

5 答案 E

里程表增加了 \(1661-1441=220\) 英里。

巴尼一共骑了 \(4+6=10\) 小时，所以平均速度为 \(220\div10=22\) 英里每小时。

所以正确答案是 E。

English

The odometer increased by \(1661-1441=220\) miles.

Barney rode for \(4+6=10\) hours, so his average speed was \(220\div10=22\) miles per hour.

Thus, E is the correct answer.

6 答案 D

把中央阴影正方形细分后，图中共有 \(16\) 个相等的小正方形。

其中 \(6\) 个是阴影，\(10\) 个不是阴影，所以所求比值为 \(6:10=3:5\)。

所以正确答案是 D。

English

After subdividing the central shaded square, the figure has \(16\) equal small squares.

Of these, \(6\) are shaded and \(10\) are unshaded, so the desired ratio is \(6:10=3:5\).

Thus, D is the correct answer.

7 答案 E

由 \(\dfrac{3}{5}=\dfrac{M}{45}\)，得 \(M=\dfrac{3\cdot45}{5}=27\)。

由 \(\dfrac{3}{5}=\dfrac{60}{N}\)，得 \(3N=300\)，所以 \(N=100\)。

因此 \(M+N=27+100=127\)。

所以正确答案是 E。

English

From \(\dfrac{3}{5}=\dfrac{M}{45}\), we get \(M=\dfrac{3\cdot45}{5}=27\).

From \(\dfrac{3}{5}=\dfrac{60}{N}\), we get \(3N=300\), so \(N=100\).

Therefore, \(M+N=27+100=127\).

Thus, E is the correct answer.

8 答案 D

四个月的销售额分别为 \(100\)、\(60\)、\(40\) 和 \(120\) 美元。

它们的平均值为 \(\dfrac{100+60+40+120}{4}\) \(=\dfrac{320}{4}\) \(=80\) 美元。

所以正确答案是 D。

English

The four monthly sales are \(100,\) \(60,\) \(40,\) and \(120\) dollars.

Their average is \(\dfrac{100+60+40+120}{4}\) \(=\dfrac{320}{4}\) \(=80\) dollars.

Thus, D is the correct answer.

9 答案 D

\(15\%\) 的亏损使投资从 \(\$100\) 变为 \(0.85\cdot\$100=\$85\)。

\(20\%\) 的增长再使其变为 \(1.2\cdot\$85=\$102\)。

投资从 \(\$100\) 变为 \(\$102\)，也就是增长 \(2\%\)。

所以正确答案是 D。

English

The \(15\%\) loss changes the investment from \(\$100\) to \(0.85\cdot\$100=\$85\).

The \(20\%\) gain then changes it to \(1.2\cdot\$85=\$102\).

The investment went from \(\$100\) to \(\$102\), which is a \(2\%\) gain.

Thus, D is the correct answer.

10 答案 D

A 房间的年龄总和为 \[6\cdot 40 = 240\text{。}\]

B 房间的年龄总和为 \[4\cdot 25 = 100\text{。}\]

总和为 \[240+100 = 340\text{。}\]

因此平均年龄为 \[\dfrac{340}{10} =34\text{。}\]

所以正确答案是 D。

English

The sum of the ages in Room A is \[6\cdot 40 = 240.\]

The sum of the ages in Room B is \[4\cdot 25 = 100.\]

The total sum is \[240+100 = 340.\]

The average is therefore \[\dfrac{340}{10} =34.\]

Thus, the answer is D .

11 答案 A

同时养两种动物的人数等于养猫人数加养狗人数，再减去至少养一种动物的人数。

因此同时养两种动物的人数为 \[20+26-39 = 7\text{。}\]

所以正确答案是 A。

English

The number of people that have both animals is equal to the number of people that own a cat plus the number of people that own a dog minus the number of people that own either.

Therefore, the number of people who own both is \[20+26-39 = 7.\]

Thus, the answer is A .

12 答案 C

第一次反弹升到 \(2\) 米。之后每次反弹都是前一次反弹高度的 \(\dfrac{2}{3}\)。

第四次反弹升到 \(2\left(\dfrac{2}{3}\right)^3=\dfrac{16}{27}\)，大于 \(\dfrac{1}{2}\)。

第五次反弹升到 \(2\left(\dfrac{2}{3}\right)^4=\dfrac{32}{81}\)，小于 \(\dfrac{1}{2}\)。

所以正确答案是 C。

English

The first bounce rises to \(2\) meters. Each later bounce is \(\dfrac{2}{3}\) of the previous bounce.

The fourth bounce rises to \(2\left(\dfrac{2}{3}\right)^3=\dfrac{16}{27}\), which is greater than \(\dfrac{1}{2}\).

The fifth bounce rises to \(2\left(\dfrac{2}{3}\right)^4=\dfrac{32}{81}\), which is less than \(\dfrac{1}{2}\).

Thus, C is the correct answer.

13 答案 C

设三个箱子的重量为 \(a\)、\(b\)、\(c\)。已知 \[\begin{cases}a+b = 122\\a+c = 125\\b+c = 127\end{cases}\text{。}\]把这些式子相加得到 \[2(a+b+c) =\] \[ 122+125+127=374\text{。}\]所以 \[a+b+c = 187\text{。}\]

所以正确答案是 C。

English

Let the weights be \(a,\) \(b,\) \(c.\) We know \[\begin{cases}a+b = 122\\a+c = 125\\b+c = 127.\end{cases}\] Adding all of this yields \[2(a+b+c) =\]\[ 122+125+127=374.\] This makes \[a+b+c = 187.\]

Thus, the answer is C .

14 答案 C

另外两个 A 必须各占剩余两行和剩余两列中的一个格子。三个 A 的位置有两种可能的对角模式。

对于任意一种 A 的位置模式，第一行剩余两个位置可以填成 B,C 或 C,B，之后其余格子都被确定。

因此共有 \(2\cdot2=4\) 种排列。

所以正确答案是 C。

English

The other two A’s must occupy one square in each of the remaining two rows and columns. There are two possible diagonal patterns for the three A’s.

For either A-pattern, the two remaining positions in the top row can be filled as B,C or C,B, and then the rest of the grid is forced.

So there are \(2\cdot2=4\) arrangements.

Thus, C is the correct answer.

15 答案 B

她前 \(8\) 场的得分和为 \(37\)。因为前 \(9\) 场的平均分是整数，所以前 \(9\) 场的总分是 \(9\) 的倍数。

由于第九场得分少于 \(10\)，前 \(9\) 场的总分在 \(37\) 到 \(47\) 之间，并且是 \(9\) 的倍数，因此总分为 \(45\)。所以第 \(9\) 场得分为 \(45-37=8\)。

特蕾莎前 \(9\) 场的总分是 \(45\)。因为前 \(10\) 场的平均分是整数，所以前 \(10\) 场的总分是 \(10\) 的倍数。

由于第十场得分少于 \(10\)，前 \(10\) 场的总分在 \(45\) 到 \(55\) 之间，并且是 \(10\) 的倍数，因此总分为 \(50\)。所以第 \(10\) 场得分为 \(50-45=5\)。

因此乘积为 \(8\cdot 5=40\)。

所以正确答案是 B。

English

The sum of her first \(8\) scores is \(37.\) Since the average of the first \(9\) scores is an integer, the sum of the first \(9\) scores is a multiple of \(9.\)

Since the score is less than \(10,\) the sum of the scores after \(9\) games is between \(37\) and \(47,\) and is a multiple of \(9,\) making the sum \(45.\) Thus, the score of the \(9\)th game is \(45-37=8.\)

The sum of Theresa’s first \(9\) scores is \(45.\) Since the average of the first \(10\) scores is an integer, the sum of the first \(10\) scores is a multiple of \(10.\)

Since the score is less than \(10,\) the sum of the scores after \(10\) games is between \(45\) and \(55,\) and is a multiple of \(10,\) making the sum \(50.\) Thus, the score of the \(10\)th game is \(50-45=5.\)

Therefore, their product is \(8\cdot 5=40.\)

Thus, the answer is B .

16 答案 D

这个立体由 \(7\) 个单位立方体组成，所以体积是 \(7\) 立方单位。

中央立方体没有外露面。六个外侧立方体各有 \(5\) 个外露面，所以表面积为 \(6\cdot5=30\) 平方单位。

体积与表面积之比为 \(7:30\)。

所以正确答案是 D。

English

The shape uses \(7\) unit cubes, so its volume is \(7\) cubic units.

The center cube has no exposed faces. Each of the six outer cubes has \(5\) exposed faces, for a total surface area of \(6\cdot5=30\) square units.

The ratio of volume to surface area is \(7:30\).

Thus, D is the correct answer.

17 答案 D

若边长为 \(l\) 和 \(w\)，则 \(2l+2w=50\)，所以 \(l+w=25\)。

最小可能面积来自边长 \(1\) 和 \(24\)，面积为 \(24\)。

最大可能面积来自和为 \(25\) 且最接近的整数对，即 \(12\) 和 \(13\)，面积为 \(156\)。

差为 \(156-24=132\)。

所以正确答案是 D。

English

If the side lengths are \(l\) and \(w\), then \(2l+2w=50\), so \(l+w=25\).

The smallest possible area comes from side lengths \(1\) and \(24\), giving area \(24\).

The largest possible area comes from the closest integer pair with sum \(25\), namely \(12\) and \(13\), giving area \(156\).

The difference is \(156-24=132\).

Thus, D is the correct answer.

18 答案 E

圆的周长是 \(\pi d = 2r\pi \)，所以走四分之一圆周的长度是 \(\dfrac {\pi r }2\)。

它先沿大圆走四分之一圈，这部分是 \(10 \pi\) 米。然后从大圆到小圆，长度为 \(20-10=10\) 米。

接着沿小圆走四分之一圈，这部分是 \(5 \pi\) 米。然后穿过小圆的直径，长度为 \(10\cdot 2=20\) 米。

再沿小圆走四分之一圈，这部分是 \(5 \pi\) 米。最后从小圆到大圆，长度为 \(20-10=10\) 米。

总长度为 \[10\pi + 10+5\pi + 20 + 5\pi + 10 \] \[=20\pi + 40\text{。}\]

所以正确答案是 E。

English

The circumference of a circle is \(\pi d = 2r\pi ,\) so going a quarter of the way around is \(\dfrac {\pi r }2.\)

He goes a quarter of the way around the large circle, so this part is \(10 \pi \) meters. He then goes from the larger circle to the smaller circle, which is \(20-10=10\) meters.

He goes a quarter of the way around the smaller circle, so this part is \(5 \pi \) meters. He then goes through the diameter of the smaller circle, which is \(10\cdot 2=20\) meters.

He then goes a quarter of the way around the smaller circle, so this part is \(5 \pi \) meters. He finally goes from the smaller circle to the larger circle, which is \(20-10=10\) meters.

The total length is \[10\pi + 10+5\pi + 20 + 5\pi + 10 \]\[=20\pi + 40.\]

Thus, the answer is E .

19 答案 B

每个点都有 \(2\) 个点与它相距一个单位。

因此无论第一个点怎么选，其余 \(7\) 个点中有 \(2\) 个与它相距一个单位，所以概率为 \(\dfrac 27\)。

所以正确答案是 B。

English

Each dot has \(2\) dots that are one unit away from it.

Therefore, regardless of the choice of the first dot, \(2\) of the other \(7\) dots would be within one unit, so the probability is \(\dfrac 27.\)

Thus, the answer is B .

20 答案 B

设通过测试的男生数和女生数都为 \(p\)。

因为男生中的 \(\dfrac{2}{3}\) 通过了测试，男生人数为 \(\dfrac{3p}{2}\)。因为女生中的 \(\dfrac{3}{4}\) 通过了测试，女生人数为 \(\dfrac{4p}{3}\)。

使两个人数都为整数的最小正数 \(p\) 是 \(6\)。此时有 \(9\) 名男生和 \(8\) 名女生，最小总人数为 \(17\)。

所以正确答案是 B。

English

Let \(p\) be the common number of boys and girls who passed.

Since \(\dfrac{2}{3}\) of the boys passed, the number of boys is \(\dfrac{3p}{2}\). Since \(\dfrac{3}{4}\) of the girls passed, the number of girls is \(\dfrac{4p}{3}\).

The smallest positive \(p\) that makes both counts whole is \(6\). Then there are \(9\) boys and \(8\) girls, for a minimum total of \(17\) students.

Thus, B is the correct answer.

21 答案 C

圆柱直径是 \(8\) 厘米，所以半径是 \(4\) 厘米。它的长度是 \(6\) 厘米。

整个圆柱的体积是 \(\pi r^2h=\pi\cdot4^2\cdot6=96\pi\)。

虚线切割得到圆柱的一半，所以楔形部分体积为 \(48\pi\)，约为 \(151\) 立方厘米。

所以正确答案是 C。

English

The cylinder has diameter \(8\) cm, so its radius is \(4\) cm. Its length is \(6\) cm.

The whole cylinder has volume \(\pi r^2h=\pi\cdot4^2\cdot6=96\pi\).

The dashed cut takes half of the cylinder, so the wedge has volume \(48\pi\), which is about \(151\) cubic centimeters.

Thus, C is the correct answer.

22 答案 A

令 \(x=\dfrac{n}{3}\)。则 \(n=3x\)，所以 \(3n=9x\)。

\(x\) 和 \(9x\) 都必须是三位整数。因此 \(100\le x\)，并且 \(9x\le999\)，所以 \(100\le x\le111\)。

共有 \(111-100+1=12\) 个整数 \(x\)，每个都对应一个正整数 \(n\)。

所以正确答案是 A。

English

Let \(x=\dfrac{n}{3}\). Then \(n=3x\), so \(3n=9x\).

Both \(x\) and \(9x\) must be three-digit whole numbers. Thus \(100\le x\) and \(9x\le999\), so \(100\le x\le111\).

There are \(111-100+1=12\) integer values of \(x\), and each gives one positive integer value of \(n\).

Thus, A is the correct answer.

23 答案 C

因为答案是比值，取正方形边长为 \(3\)。则 \(AF=2\)、\(FE=1\)、\(CD=2\)、\(DE=1\)。

正方形面积为 \(9\)。三角形 \(ABF\) 和 \(BCD\) 的面积都为 \(\dfrac{1}{2}\cdot3\cdot2=3\)。

三角形 \(DEF\) 的面积为 \(\dfrac{1}{2}\cdot1\cdot1=\dfrac{1}{2}\)。

所以 \([BFD]=9-3-3-\dfrac{1}{2}=\dfrac{5}{2}\)，所求比值为 \(\dfrac{\frac{5}{2}}{9}=\dfrac{5}{18}\)。

所以正确答案是 C。

English

Because the answer is a ratio, choose the side length of the square to be \(3\). Then \(AF=2\), \(FE=1\), \(CD=2\), and \(DE=1\).

The square has area \(9\). The areas of triangles \(ABF\) and \(BCD\) are each \(\dfrac{1}{2}\cdot3\cdot2=3\).

Triangle \(DEF\) has area \(\dfrac{1}{2}\cdot1\cdot1=\dfrac{1}{2}\).

So \([BFD]=9-3-3-\dfrac{1}{2}=\dfrac{5}{2}\), and the desired ratio is \(\dfrac{\frac{5}{2}}{9}=\dfrac{5}{18}\).

Thus, C is the correct answer.

24 答案 C

如果骰子点数为 \(1\)，牌可以是 \(1\)、\(4\)、\(9\)，有 \(3\) 种组合。

如果骰子点数为 \(2\)，牌可以是 \(2\)、\(8\)，有 \(2\) 种组合。

如果骰子点数为 \(3\)，牌可以是 \(3\)，有 \(1\) 种组合。

如果骰子点数为 \(4\)，牌可以是 \(1\)、\(4\)、\(9\)，有 \(3\) 种组合。

如果骰子点数为 \(5\)，牌可以是 \(5\)，有 \(1\) 种组合。

如果骰子点数为 \(6\)，牌可以是 \(6\)，有 \(1\) 种组合。

组合总数为 \[3+2+1+3+1+1=11\text{。}\]共有 \(60\) 个等可能组合，所以概率是 \(\dfrac{11}{60} \)。

所以正确答案是 C。

English

If the rolled number was \(1,\) then the tile can be \(1,\) \(4,\) \(9,\) yielding \(3\) combinations.

If the rolled number was \(2,\) then the tile can be \(2,\) \(8,\) yielding \(2\) combinations.

If the rolled number was \(3,\) then the tile can be \(3,\) yielding \(1\) combination.

If the rolled number was \(4,\) then the tile can be \(1,\) \(4,\) \(9,\) yielding \(3\) combinations.

If the rolled number was \(5,\) then the tile can be \(5,\) yielding \(1\) combination.

If the rolled number was \(6,\) then the tile can be \(6,\) yielding \(1\) combination.

The total number of combinations is \[3+2+1+3+1+1=11.\] There are \(60\) combinations each with equal likelihood, so the probability is \(\dfrac{11}{60} .\)

Thus, the answer is C .

25 答案 A

最大圆的半径为 \(12\)，所以整个设计的面积为 \(12^2\pi=144\pi\)。

阴影部分的面积为 \(2^2\pi=4\pi\)、\((6^2-4^2)\pi=20\pi\) 和 \((10^2-8^2)\pi=36\pi\)。

阴影总面积为 \(4\pi+20\pi+36\pi=60\pi\)，所以阴影比例为 \(\dfrac{60\pi}{144\pi}=\dfrac{5}{12}\)，约为 \(42\%\)。

所以正确答案是 A。

English

The largest circle has radius \(12\), so the entire design has area \(12^2\pi=144\pi\).

The shaded parts have areas \(2^2\pi=4\pi\), \((6^2-4^2)\pi=20\pi\), and \((10^2-8^2)\pi=36\pi\).

The total shaded area is \(4\pi+20\pi+36\pi=60\pi\), so the shaded fraction is \(\dfrac{60\pi}{144\pi}=\dfrac{5}{12}\), which is about \(42\%\).

Thus, A is the correct answer.