2011 年 AMC 8 真题（P版）

1 答案 E

苹果总共花了 \(3 \cdot 50 = 150\) 美分，也就是 \(\$1.50\)。因此 Margie 收到的找零是 \(\$5.00-\$1.50=\$3.50\)。

所以正确答案是 E。

English

The apples cost a total of \(3 \cdot 50 = 150\) cents, which equals \(\$1.50.\) This means that Margie received \(\$5.00-\$1.50=\$3.50\) in change.

Thus, E is the correct answer.

2 答案 E

Karl 菜园的面积为 \[20 \text{ft} \cdot 45 \text{ft} = 900 \text{ft}^2\text{。}\]Makenna 菜园的面积为 \[25 \text{ft} \cdot 40 \text{ft} = 1000 \text{ft}^2\text{。}\]

两者面积差为 \[1000 \text{ft}^2 - 900 \text{ft}^2 = 100 \text{ft}^2\text{。}\]因此 Makenna 的菜园比 Karl 的大 \(100 \text{ft}^2\)。

所以正确答案是 E。

English

The area of Karl’s garden is \[20 \text{ft} \cdot 45 \text{ft} = 900 \text{ft}^2.\] The area of Makenna’s garden is \[25 \text{ft} \cdot 40 \text{ft} = 1000 \text{ft}^2.\]

The difference of these areas is \[1000 \text{ft}^2 - 900 \text{ft}^2 = 100 \text{ft}^2.\] Therefore, Makenna’s garden is \(100 \text{ft}^2\) larger than Karl’s.

Thus, E is the correct answer.

3 答案 D

原图案是一个 \(5\times5\) 的正方形。每边向外扩展一块方砖后，得到一个 \(7\times7\) 的正方形，所以新增加的阴影边框有 \(7^2-5^2=24\) 块方砖。加上原来的 \(8\) 块阴影方砖，共有 \(24+8=32\) 块阴影方砖。原来的 \(17\) 块非阴影方砖不变，因此阴影方砖与非阴影方砖之比为 \(32:17\)。

所以正确答案是 D。

English

The original pattern is a \(5\times5\) square. Extending it by one tile on every side makes a \(7\times7\) square, so the new shaded border contains \(7^2-5^2=24\) tiles. Together with the original \(8\) shaded tiles, there are \(24+8=32\) shaded tiles. The original \(17\) unshaded tiles do not change, so the ratio of shaded tiles to unshaded tiles is \(32:17.\)

Thus, D is the correct answer.

4 答案 C

为了更容易求这些值，先得到有序列表：\[ 0, 0, 1, 1, 2, 2, 3, 3, 3\text{。}\]

由此可知众数是 \(3\)，中位数是 \(2\)，平均数是 \(\frac{15}{9} = \frac{5}{3}\)。

因为 \(\dfrac{5}{3} < 2 < 3\)，所以有：

平均数 \(<\) 中位数 \(<\) 众数。

所以正确答案是 C。

English

To find these values more easily, we can get the following ordered list: \[ 0, 0, 1, 1, 2, 2, 3, 3, 3. \]

From this, we see that the mode is \(3,\) the median is \(2,\) and the mean is \(\frac{15}{9} = \frac{5}{3}.\)

Since \(\dfrac{5}{3} < 2 < 3,\) we get that:

mean \(<\) median \(<\) mode.

Thus, C is the correct answer.

5 答案 D

\(2011\) 除以 \(60\) 的余数是 \(31\)。也就是说 \(2011 = 60 \cdot 33 + 31\)，所以 \(2011\) 分钟等于 \(33\) 小时 \(31\) 分钟。

经过 \(24\) 小时到达一月 \(2\) 日，所以此时已过去 \(9\) 小时 \(31\) 分钟，也就是 \(2\) 日上午。

所以正确答案是 D。

English

The remainder when \(2011\) is divided by \(60\) is \(31.\) This means that \(2011 = 60 \cdot 33 + 31,\) which means that \(2011\) minutes is the same as \(33\) hours and \(31\) minutes.

\(24\) hours takes us to January \(2,\) so we get that we are \(9\) hours and \(31\) minutes into January \(2.\)

Thus, D is the correct answer.

6 答案 D

已知有 \(45\) 人拥有摩托车，所以 \(351 - 45 = 306\) 人不拥有摩托车。

所以正确答案是 D。

English

We know that \(45\) people own motorcycles, so \(351 - 45 = 306\) people do not own motorcycles.

Thus, D is the correct answer.

7 答案 C

左上和右下的阴影区域各占所在正方形的四分之一。右上占八分之一，左下占八分之三。它们的总面积为 \[\dfrac{1}{4} + \dfrac{1}{4} + \dfrac{1}{8} + \dfrac{3}{8} = 1\text{。}\]

因此阴影区域合起来等于一个正方形的面积，占四个正方形总面积的 \(25 \%\)。

所以正确答案是 C。

English

The top left and the bottom right shaded regions are both a quarter of each square. The top right is one-eighth, and the bottom left is three-eighths. Their combined area is \[\dfrac{1}{4} + \dfrac{1}{4} + \dfrac{1}{8} + \dfrac{3}{8} = 1.\]

Therefore, the shaded regions combined equal the area of one square, so they are \(25 \%\) of the total area.

Thus, C is the correct answer.

8 答案 B

表格列出了全部九种结果。不同的和为 \(3,5,7,9\) 和 \(11\)，所以共有 \(5\) 个可能的值。

所以正确答案是 B。

English

The table shows all nine outcomes. The distinct sums are \(3,5,7,9,\) and \(11\), so there are \(5\) possible values.

Thus, B is the correct answer.

9 答案 E

Carmen 共行驶 \(35\) 英里，用时 \(7\) 小时，所以平均速度为 \(\frac{35}{7} = 5\) 英里每小时。

所以正确答案是 E。

English

Carmen travels \(35\) miles in \(7\) hours, so her average speed is \(\frac{35}{7} = 5\) miles per hour.

Thus, E is the correct answer.

10 答案 C

小费固定为 \(\$2\)，从总额中减去后剩 \(\$8\)。这大于 \(\$2.40\)，所以再减去前半英里的车费，并把 \(\dfrac{1}{2}\) 英里计入总距离。

现在还有 \(\$5.60\) 可用于额外里程。收费标准为 \(\$0.20\) 每 \(0.1\) 英里，等价于 \(\$2\) 每 \(1\) 英里。因此这笔钱还能乘坐 \(\frac{5.60}{2} = 2.8\) 英里。总距离为 \(2.8 + \dfrac{1}{2} = 3.3\) 英里。

所以正确答案是 C。

English

There is a guaranteed \(\$2\) tip, so we can subtract that from the total, leaving \(\$8.\) This is greater than \(\$2.40,\) so we can subtract that and add \(\dfrac{1}{2}\) miles to the total distance.

We now have \(\$5.60\) to use for additional miles. \(\$0.20\) per \(0.1\) mile is the same as \(\$2\) for \(1\) mile. That means one can ride for \(\frac{5.60}{2} = 2.8\) more miles with this much money. This leaves a total of \(2.8 + \dfrac{1}{2} = 3.3\) miles.

Thus, C is the correct answer.

11 答案 A

可以通过每天的差值来计算平均分钟数之差。

从星期一开始，Sasha 与 Asha 的差值分别为 \(10\)、\(-10\)、\(20\)、\(30\)、\(-20\)。总差为 \(30\) 分钟，因此平均差为 \(30 \div 5 = 6\)。

所以正确答案是 A。

English

We can calculate the difference in average minutes by looking at the differences per day.

Starting with Monday, the differences between Sasha and Asha are \(10,\) \(-10,\) \(20,\) \(30,\) and \(-20.\) This is a total of \(30\) minutes. Therefore, the average difference is \(30 \div 5 = 6.\)

Thus, A is the correct answer.

12 答案 B

设 Angie 的座位已经确定。Carlos 等可能坐在另外 \(3\) 个座位中的任意一个，但其中只有一个座位与 Angie 相对。因此概率为 \(\dfrac{1}{3}\)。

所以正确答案是 B。

English

Consider that Angie’s seat is chosen. Carlos has an equal probability of being in any of the other \(3\) seats. Only one of them is opposite Angie, however. Therefore, the probability is \(\dfrac{1}{3}.\)

Thus, B is the correct answer.

13 答案 C

有 \[\begin{align*} SC &= DC + SR - DR \\ &= 15 + 15 - 25 \\ &= 5 \end{align*}\text{。}\]

因此 \(PBSC\) 的面积为 \(5 \cdot 15 = 75\)。\(AQRD\) 的面积为 \(25 \cdot 15 = 375\)。

\(\dfrac{75}{375} = \dfrac{1}{5}\)，也就是 \(20 \%\)。

所以正确答案是 C。

English

We get that \[\begin{align*} SC &= DC + SR - DR \\ &= 15 + 15 - 25 \\ &= 5. \end{align*}\]

This means that the area of \(PBSC\) is \(5 \cdot 15 = 75.\) The area of \(AQRD\) is \(25 \cdot 15 = 375.\)

\(\dfrac{75}{375} = \dfrac{1}{5},\) which is \(20 \%.\)

Thus, C is the correct answer.

14 答案 C

女生总数为 \[\dfrac{4}{9} \cdot 270 + \dfrac{5}{9} \cdot 180 = \] \[ 120 + 100 = 220\text{。}\]

学生总数为 \(270 + 180 = 450\)，所以女生所占比例为 \(\dfrac{220}{450} = \dfrac{22}{45}\)。

所以正确答案是 C。

English

The total number of girls is \[\dfrac{4}{9} \cdot 270 + \dfrac{5}{9} \cdot 180 = \] \[ 120 + 100 = 220. \]

There are \(270 + 180 = 450\) students total, so the fraction of girls is \(\dfrac{220}{450} = \dfrac{22}{45}.\)

Thus, C is the correct answer.

15 答案 D

为了求位数，可以尝试把这个数写成 \(10\) 的幂。

有 \[\begin{align*} 4^5 \cdot 5^{10} &= 2^{10} \cdot 5^{10} \\ &= 10^{10} \end{align*}\text{。}\]

这说明所求数是 \(1\) 后面跟着 \(10\) 个零，共有 \(11\) 位数字。

所以正确答案是 D。

English

To find the number of digits, we can try to express this number in terms of powers of \(10.\)

We get that \[\begin{align*} 4^5 \cdot 5^{10} &= 2^{10} \cdot 5^{10} \\ &= 10^{10}. \end{align*}\]

This shows that the desired number is \(1\) followed by \(10\) zeros, for a total of \(11\) digits.

Thus, D is the correct answer.

16 答案 C

因为这两个三角形都是等腰三角形，图中从顶点作出的高会平分底边，并把每个三角形分成两个全等的直角三角形。

由勾股定理，面积为 \(A\) 的三角形的高为 \[\sqrt{25^2 - 15^2} = 20\text{。}\]类似地，面积为 \(B\) 的三角形的高为 \[\sqrt{25^2 - 20^2} = 15\text{。}\]

因此 \[ A = \dfrac{1}{2} \cdot 20 \cdot 30 = 300\text{。}\]同样，\[ B = \dfrac{1}{2} \cdot 15 \cdot 40 = 300\text{。}\]

所以 \(A = B\)。

所以正确答案是 C。

English

Since these triangles are isosceles, the altitude shown from the apex of each triangle bisects its base and creates two congruent right triangles.

Using the Pythagorean theorem, we get that the altitude of the triangle with area \(A\) equals \[\sqrt{25^2 - 15^2} = 20.\] Similarly, we get that the altitude of the triangle with area \(B\) equals \[\sqrt{25^2 - 20^2} = 15.\]

With these altitudes, we can calculate the areas of the triangles. We get that \[ A = \dfrac{1}{2} \cdot 20 \cdot 30 = 300. \] Similarly, \[ B = \dfrac{1}{2} \cdot 15 \cdot 40 = 300. \]

Therefore, \(A = B.\)

Thus, C is the correct answer.

17 答案 A

为了找到所需指数，注意所有底数都是质数。因此质因数分解会有帮助。

有 \[588 = 2^2 \cdot 3^1 \cdot 7^2\text{。}\]

由此可知 \(w = 2\)、\(x = 1\)、\(y = 0\) 且 \(z = 2\)，其中 \(y = 0\) 使 \(5^y\) 这一项等于 \(1\)。

代入这些指数可得 \[ \begin{gather*} 2w + 3x + 5y + 7z \\ = 2 \cdot 2 + 3 \cdot 1 + 5 \cdot 0 + 7 \cdot 2 \\ = 21 \end{gather*}\text{。}\]

所以正确答案是 A。

English

To find the desired exponents, note that all the bases are prime numbers. This means that finding the prime factorization will be helpful.

We get that \[588 = 2^2 \cdot 3^1 \cdot 7^2.\]

From this, it is clear that \(w = 2,\) \(x = 1,\) \(y = 0,\) and \(z = 2\) (\(y = 0\) since that makes the \(5^y\) term equal \(1\)).

Therefore, \[ \begin{gather*} 2w + 3x + 5y + 7z \\ = 2 \cdot 2 + 3 \cdot 1 + 5 \cdot 0 + 7 \cdot 2 \\ = 21. \end{gather*} \]

Thus, A is the correct answer.

18 答案 D

掷骰子两次时有 \(3\) 类结果：第一次大于第二次、两次相等、第一次小于第二次。第一类和第三类由于对称而概率相同。

第二类发生的概率是 \(\dfrac{1}{6}\)，因为第一次可以是任意数，而第二次必须等于第一次。其余两类的总概率为 \(1 - \dfrac{1}{6} = \dfrac{5}{6}\)。因此每一类的概率为 \(\dfrac{5}{6} \div 2 = \dfrac{5}{12}\)。

所求概率是第一类加第二类，共为 \(\dfrac{5}{12} + \dfrac{1}{6} = \dfrac{7}{12}\)。

所以正确答案是 D。

English

There are \(3\) possible outcomes when rolling a die twice: the first number is greater than the second, both numbers are equal, or the first number is less than the second number. The first and third outcomes have the same probability since they are symmetric.

The second outcome has a \(\dfrac{1}{6}\) chance of happening, since the first number can be anything, and the second number must equal the first number. The other two outcomes have a combined probability of \(1 - \dfrac{1}{6} = \dfrac{5}{6}.\) This means that each outcome has a \(\dfrac{5}{6} \div 2 = \dfrac{5}{12}\) chance of happening.

The desired probability is the first outcome plus the second outcome, for a total probability of \(\dfrac{5}{12} + \dfrac{1}{6} = \dfrac{7}{12}.\)

Thus, D is the correct answer.

19 答案 D

可以把图形分成这些区域，使统计长方形更容易。

图中的长方形是 \(b\)、\(c\)、\(d\)、\(ab\)、\(bc\)、\(cd\)、\(cf\)、\(de\)、\(abcd\)、\(cdef\) 和 \(bcfg\)。这些共有 \(11\) 个长方形。

所以正确答案是 D。

English

We can split the figure into these regions to make it easier to count the rectangles.

The rectangles in this figure are \(b,\) \(c,\) \(d,\) \(ab,\) \(bc,\) \(cd,\) \(cf,\) \(de,\) \(abcd,\) \(cdef,\) and \(bcfg.\) These form \(11\) rectangles.

Thus, D is the correct answer.

20 答案 D

可以作如下高，更容易求面积。

用勾股定理可得 \[ DE = \sqrt{15^2 - 12^2} = 9 \] 且 \[ FC = \sqrt{20^2 - 12^2} = 16\text{。}\]

还知道 \[ EF = AB = 50\text{，}\]所以 \[ DC = DE + EF + FC = 75\text{。}\]

因此 \(ABCD\) 的面积为 \[ \dfrac{1}{2} \cdot (DC + 50) \cdot 12 = 6 \cdot 125 \] \[ = 750\text{。}\]

所以正确答案是 D。

English

We can drop the following altitudes to more easily find the area.

We can use the Pythagorean Theorem to get that \[ DE = \sqrt{15^2 - 12^2} = 9 \] and \[ FC = \sqrt{20^2 - 12^2} = 16. \]

We also know that \[ EF = AB = 50, \] so \[ DC = DE + EF + FC = 75. \]

Then the area of \(ABCD\) is \[ \dfrac{1}{2} \cdot (DC + 50) \cdot 12 = 6 \cdot 125 \] \[ = 750. \]

Thus, D is the correct answer.

21 答案 C

陈述的第一部分表示 Norb 的年龄大于 \(36\)。

第二部分表示 Norb 的年龄要么在 \(36\) 和 \(38\) 之间，要么在 \(47\) 和 \(49\) 之间。

因为 \(37\) 是质数而 \(48\) 不是，所以 Norb 的年龄是 \(37\)。

所以正确答案是 C。

English

The first part of the statement means that Norb’s age is greater than \(36.\)

The second part means that Norb’s age is either between \(36\) and \(38\) or between \(47\) and \(49.\)

Since \(37\) is prime and \(48\) is not, Norb’s age is \(37.\)

Thus, C is the correct answer.

22 答案 D

要找十位数字，只需跟踪 \(7\) 的幂模 \(100\) 的结果。因为 \(7^4=2401\equiv1\pmod{100}\)，末两位每四次方重复。

因为 \(2011=4\cdot502+3\)，所以 \[\begin{aligned}7^{2011}&\equiv7^3=343\\&\equiv43\pmod{100}\end{aligned}\] 因此十位数字是 \(4\)。

所以正确答案是 D。

English

To find the tens digit, it is enough to track powers of \(7\) modulo \(100\). Since \(7^4=2401\equiv1\pmod{100}\), the last two digits repeat every four powers.

Because \(2011=4\cdot502+3\), we have \[\begin{aligned}7^{2011}&\equiv7^3=343\\&\equiv43\pmod{100}\end{aligned}\] Thus the tens digit is \(4\).

Thus, D is the correct answer.

23 答案 D

一个数能被 \(5\) 整除，个位数字必须是 \(0\) 或 \(5\)。

如果个位数字是 \(0\)，另外三位中必须有一位是 \(5\)。剩下两位必须从 \(\{1, 2, 3, 4\}\) 中选择。有 \(6\) 种方式选这两个数字，并有 \(6\) 种方式排列这三位数字，共 \(6 \cdot 6 = 36\) 个数。

如果个位数字是 \(5\)，千位数字有 \(4\) 种选择。选定千位后，其他数字共有 \(4 \cdot 3 = 12\) 种方式填入中间 \(2\) 位。因此此情况共有 \(4 \cdot 12 = 48\) 个数。

合并两种情况，总数为 \(36 + 48 = 84\)。

所以正确答案是 D。

English

For a number to be divisible by \(5,\) the units digit must be either \(0\) or \(5.\)

If the units digit is \(0,\) one of the other three digits must be \(5.\) The remaining two digits must be chosen from \(\{1, 2, 3, 4\}.\) There are \(6\) ways to choose the pair, and there are \(6\) ways to arrange the three digits for a total of \(6 \cdot 6 = 36\) numbers.

If the units digit is \(5,\) there are \(4\) ways to choose the thousands digit. There are \(4 \cdot 3 = 12\) ways to choose the other \(2\) digits. This leaves a total of \(4 \cdot 12 = 48\) numbers for this case.

Combining both cases, we get the total number of such integers is \(36 + 48 = 84.\)

Thus, D is the correct answer.

24 答案 A

两个数相加得到奇数时，一个必须是奇数，另一个必须是偶数。唯一的偶质数是 \(2\)，所以另一个数只能是 \(9999\)。但 \(9999\) 不是质数，因此 \(10001\) 不能写成两个质数之和。

所以正确答案是 A。

English

For two numbers to add to an odd number, one of them must be odd and the other even. Thus the only even prime is \(2,\) so the other number is forced to be \(9999.\) \(9999\) is not prime, however, so \(10001\) cannot be written as the sum of two primes.

Thus, A is the correct answer.

25 答案 A

圆内阴影面积等于圆面积减去小正方形面积。内正方形的边长可由勾股定理求得：\[\sqrt{1^2 + 1^2} = \sqrt{2}\text{。}\]

因此内正方形面积为 \(\sqrt{2}^2 = 2\)。圆内阴影面积为 \(1^2\pi - 2 = \pi - 2\)。

外正方形面积为 \(2^2 = 4\)，所以两个正方形之间的阴影面积为 \(4 - 2 = 2\)。

所求比值为 \[\dfrac{\pi - 2}{2} \approx \dfrac{3.14 - 2}{2} \approx \dfrac{1}{2}\text{。}\]

所以正确答案是 A。

English

The circle’s shaded area is equal to the area of the circle minus the area of the smaller square. The side length of the inner square can be calculated using the Pythagorean Theorem to get \[\sqrt{1^2 + 1^2} = \sqrt{2}.\]

Therefore, the area of the inner square is \(\sqrt{2}^2 = 2.\) The circle’s shaded area is then \(1^2\pi - 2 = \pi - 2.\)

The area of the outside square is \(2^2 = 4,\) so the shaded area between the two squares is \(4 - 2 = 2.\)

The desired fraction is \[\dfrac{\pi - 2}{2} \approx \dfrac{3.14 - 2}{2} \approx \dfrac{1}{2}.\]

Thus, A is the correct answer.