1992 年 AMC 8 真题（P版）

1 答案 B

分子两两分组得 \((10-9)\) \(+ (8-7)\) \(+ (6-5)\) \(+ (4-3)\) \(+ (2-1) = 5\)。

分母分组为 \((1-2)\) \(+ (3-4)\) \(+ (5-6)\) \(+ (7-8)\) \(+ 9\)，有四组 \(-1\)，再加 \(9\)，结果也是 \(5\)。

所以原式为 \(\dfrac{5}{5} = 1\)。

所以正确答案是 B。

English

Grouping the numerator in pairs gives \((10-9)\) \(+ (8-7)\) \(+ (6-5)\) \(+ (4-3)\) \(+ (2-1) = 5.\)

Grouping the denominator as \((1-2)\) \(+ (3-4)\) \(+ (5-6)\) \(+ (7-8)\) \(+ 9\) gives four pairs of \(-1\) plus \(9\), which is \(5.\)

So the expression is \(\dfrac{5}{5} = 1.\)

Thus, the correct answer is B .

2 答案 D

\(\dfrac54 = 1\dfrac14\)。逐项检查：\(\dfrac{10}{8} = \dfrac54\)，\(1\dfrac14 = \dfrac54\)，\(1\dfrac{3}{12} = 1\dfrac14 = \dfrac54\)，且 \(1\dfrac{10}{40} = 1\dfrac14 = \dfrac54\)。

但 \(1\dfrac15 = \dfrac65 \neq \dfrac54\)。

所以正确答案是 D。

English

Note \(\dfrac54 = 1\dfrac14.\) Now check each choice: \(\dfrac{10}{8} = \dfrac54,\) \(1\dfrac14 = \dfrac54,\) \(1\dfrac{3}{12} = 1\dfrac14 = \dfrac54,\) and \(1\dfrac{10}{40} = 1\dfrac14 = \dfrac54.\)

But \(1\dfrac15 = \dfrac65 \neq \dfrac54.\)

Thus, the correct answer is D .

3 答案 D

最大差应使用最大数 \(12\) 减去最小数 \(-16\)。

这给出 \(12 - (-16) = 28\)。

所以正确答案是 D。

English

The largest difference uses the largest number, \(12\), minus the smallest number, \(-16.\)

This gives \(12 - (-16) = 28.\)

Thus, the correct answer is D .

4 答案 E

非一垒安打共有 \(1 + 1 + 5 = 7\) 次，所以一垒安打有 \(35 - 7 = 28\) 次。

一垒安打所占比例为 \(\dfrac{28}{35} = \dfrac45 = 80\%\)。

所以正确答案是 E。

English

The non-single hits number \(1 + 1 + 5 = 7,\) so the singles number \(35 - 7 = 28.\)

The fraction of singles is \(\dfrac{28}{35} = \dfrac45 = 80\%.\)

Thus, the correct answer is E .

5 答案 E

长方形面积为 \(6\)，去掉的圆面积为 \(\pi\left(\dfrac12\right)^2 = \dfrac{\pi}{4} \approx 0.79\)。

阴影区域面积约为 \(6 - 0.79 \approx 5.2\)，最接近的整数是 \(5\)。

所以正确答案是 E。

English

The rectangle has area \(6,\) and the removed circle has area \(\pi\left(\dfrac12\right)^2 = \dfrac{\pi}{4} \approx 0.79.\)

So the shaded region has area \(6 - 0.79 \approx 5.2,\) whose closest whole number is \(5.\)

Thus, the correct answer is E .

6 答案 D

第一组三个数给出 \(1 + 3 - 4 = 0\)，第二组三个数给出 \(2 + 5 - 6 = 1\)。

它们的和是 \(0 + 1 = 1\)。

所以正确答案是 D。

English

The first triple gives \(1 + 3 - 4 = 0,\) and the second gives \(2 + 5 - 6 = 1.\)

Their sum is \(0 + 1 = 1.\)

Thus, the correct answer is D .

7 答案 A

三个数字的和为 \(26\)，只能是数字 \(9\)、\(9\)、\(8\)。由它们组成的 \(3\) 位数是 \(899\)、\(989\) 和 \(998\)。

其中只有 \(998\) 是偶数，所以这样的数恰好有 \(1\) 个。

所以正确答案是 A。

English

A digit-sum of \(26\) with three digits requires the digits \(9,\) \(9,\) and \(8.\) The \(3\)-digit numbers using them are \(899,\) \(989,\) and \(998.\)

Of these, only \(998\) is even, so there is exactly \(1\) such number.

Thus, the correct answer is A .

8 答案 C

铅笔成本为 \(1500 \times \$0.10 = \$150\)。要获利 \(\$100\)，收入必须是 \(\$150 + \$100 = \$250\)。

每支售价 \(\$0.25\)，所以必须卖出 \(\dfrac{\$250}{\$0.25} = 1000\) 支。

所以正确答案是 C。

English

The pencils cost \(1500 \times \$0.10 = \$150.\) To make a \(\$100\) profit, the revenue must be \(\$150 + \$100 = \$250.\)

At \(\$0.25\) each, he must sell \(\dfrac{\$250}{\$0.25} = 1000\) pencils.

Thus, the correct answer is C .

9 答案 B

若男性人数为 \(M\)，则女性人数为 \(2M\)，总人口满足 \(M + 2M = 480\)。

所以 \(3M = 480\)，得到 \(M = 160\)。

所以正确答案是 B。

English

If there are \(M\) males, then there are \(2M\) females, and together \(M + 2M = 480.\)

So \(3M = 480,\) giving \(M = 160.\)

Thus, the correct answer is B .

10 答案 B

大三角形面积为 \(\dfrac12 \times 8 \times 8 = 32\)，所以 \(16\) 个全等小三角形中每个面积为 \(\dfrac{32}{16} = 2\)。

图中有十个小三角形被涂色，所以阴影面积是 \(10 \times 2 = 20\)。

所以正确答案是 B。

English

The large triangle has area \(\dfrac12 \times 8 \times 8 = 32,\) so each of the \(16\) congruent small triangles has area \(\dfrac{32}{16} = 2.\)

Ten of the small triangles are shaded, so the shaded area is \(10 \times 2 = 20.\)

Thus, the correct answer is B .

11 答案 B

调查总人数是 \(50 + 60 + 40 + 60 + 40 = 250\)。

偏好蓝色的比例为 \(\dfrac{60}{250} = 24\%\)。

所以正确答案是 B。

English

The total number surveyed is \(50 + 60 + 40 + 60 + 40 = 250.\)

The percent preferring blue is \(\dfrac{60}{250} = 24\%.\)

Thus, the correct answer is B .

12 答案 C

在 \(30{,}000\) 英里的行驶中，始终有 \(4\) 个轮胎在使用，所以总轮胎里程为 \(4 \times 30{,}000 = 120{,}000\) 轮胎英里。

平均分给 \(5\) 个轮胎，每个轮胎使用 \(\dfrac{120{,}000}{5} = 24{,}000\) 英里。

所以正确答案是 C。

English

During the \(30{,}000\) miles, \(4\) tires are always in use, so the total tire-mileage is \(4 \times 30{,}000 = 120{,}000\) tire-miles.

Split equally among \(5\) tires, each tire is used \(\dfrac{120{,}000}{5} = 24{,}000\) miles.

Thus, the correct answer is C .

13 答案 B

五个分数总和为 \(5 \times 90 = 450\)。中位数是第三个分数 \(91\)。因为 \(94\) 是众数，它至少出现两次，且这两次都在中位数以上，所以最高两个分数分别是 \(94\) 和 \(94\)。

三个最高分是 \(91\)、\(94\)、\(94\)，和为 \(279\)。因此最低两个分数之和是 \(450 - 279 = 171\)。

所以正确答案是 B。

English

The five scores sum to \(5 \times 90 = 450.\) The median is the third score, \(91.\) Since \(94\) is the mode, it must appear at least twice, and both copies lie above the median, so the two highest scores are \(94\) and \(94.\)

The three highest scores are \(91,\) \(94,\) and \(94,\) summing to \(279.\) So the two lowest sum to \(450 - 279 = 171.\)

Thus, the correct answer is B .

14 答案 D

加入的 \(4\) 加仑对应从 \(\dfrac13\) 满到 \(\dfrac12\) 满的增加量，即容量的 \(\dfrac12 - \dfrac13 = \dfrac16\)。

如果容量的 \(\dfrac16\) 是 \(4\) 加仑，那么总容量是 \(4 \times 6 = 24\) 加仑。

所以正确答案是 D。

English

The \(4\) gallons account for the change from \(\dfrac13\) full to \(\dfrac12\) full, which is \(\dfrac12 - \dfrac13 = \dfrac16\) of the capacity.

If \(\dfrac16\) of the capacity is \(4\) gallons, the full capacity is \(4 \times 6 = 24\) gallons.

Thus, the correct answer is D .

15 答案 C

这个序列是 \(9\) 个字母的循环块 \(ABCDEDCBA\) 不断重复。

因为 \(1992 = 9 \times 221 + 3\)，所以第 \(1992\) 个字母是一个循环块中的第 \(3\) 个字母，即 \(C\)。

所以正确答案是 C。

English

The sequence is the \(9\)-letter block \(ABCDEDCBA\) repeated again and again.

Since \(1992 = 9 \times 221 + 3,\) the \(1992\)nd letter is the \(3\)rd letter of a block, which is \(C.\)

Thus, the correct answer is C .

16 答案 B

给定圆柱体的体积是 \(\pi \times 10^2 \times 5 = 500\pi\)，所以两倍体积是 \(1000\pi\)。

各选项体积为：A 是 \(\pi \times 20^2 \times 5 = 2000\pi\)，B 是 \(\pi \times 10^2 \times 10 = 1000\pi\)，C 是 \(\pi \times 5^2 \times 20 = 500\pi\)，D 是 \(\pi \times 20^2 \times 10 = 4000\pi\)。只有 B 等于 \(1000\pi\)。

所以正确答案是 B。

English

The given cylinder has volume \(\pi \times 10^2 \times 5 = 500\pi,\) so a cylinder of twice the volume has volume \(1000\pi.\)

The choices have volumes \(\pi \times 20^2 \times 5 = 2000\pi\) (A), \(\pi \times 10^2 \times 10 = 1000\pi\) (B), \(\pi \times 5^2 \times 20 = 500\pi\) (C), and \(\pi \times 20^2 \times 10 = 4000\pi\) (D). Only (B) equals \(1000\pi.\)

Thus, the correct answer is B .

17 答案 B

由三角形不等式，\(6.5 + s\) 必须大于最长边 \(10\)，所以 \(s \gt 3.5\)。

大于 \(3.5\) 的最小整数是 \(4\)，而它确实能组成三角形。

所以正确答案是 B。

English

By the triangle inequality, \(6.5 + s\) must exceed the longest side \(10,\) so \(s \gt 3.5.\)

The smallest whole number greater than \(3.5\) is \(4,\) and it does form a valid triangle.

Thus, the correct answer is B .

18 答案 A

这辆车总共行驶 \(80 + 100 = 180\) 英里，总时间为 \(1.5 + 0.5 + 2 = 4\) 小时，其中堵车停止时间也计入。

平均速度为 \(\dfrac{180}{4} = 45\) 英里每小时。

所以正确答案是 A。

English

The car covered \(80 + 100 = 180\) miles in total, over \(1.5 + 0.5 + 2 = 4\) hours (the traffic stop still counts as time).

The average speed is \(\dfrac{180}{4} = 45\) miles per hour.

Thus, the correct answer is A .

19 答案 C

从第 \(5\) 个出口到第 \(26\) 个出口，共有 \(26 - 5 = 21\) 个相邻出口间隔，每个至少 \(5\) 英里。

要最大化其中一个间隔，就让其他 \(20\) 个间隔都等于 \(5\) 英里，共占 \(20 \times 5 = 100\) 英里。剩下的间隔是 \(118 - 100 = 18\) 英里。

所以正确答案是 C。

English

From the \(5\)th exit to the \(26\)th exit there are \(26 - 5 = 21\) gaps between consecutive exits, each at least \(5\) miles.

To maximize one gap, make the other \(20\) gaps exactly \(5\) miles, using \(20 \times 5 = 100\) miles. The remaining gap is \(118 - 100 = 18\) miles.

Thus, the correct answer is C .

20 答案 D

五个图案都由 \(6\) 个正方形组成。把 A、B、C 和 E 折叠时，各正方形能分别覆盖立方体的六个面。

对图案 D，无论怎样折叠，都会迫使两个正方形落在同一个面上而重叠，所以不能形成立方体。

所以正确答案是 D。

English

Each of the five patterns has \(6\) squares. Folding patterns (A), (B), (C), and (E) wraps the squares neatly onto the six faces of a cube.

For pattern (D), any attempt to fold forces two of the squares to land on the same face, so they overlap and no cube can be formed.

Thus, the correct answer is D .

21 答案 B

从图上读出（鼓手，号手）的销售额：一月 \((7, 9)\)，二月 \((5, 3)\)，三月 \((9, 6)\)，四月 \((9, 12)\)，五月 \((8, 10)\)。

较大值超过较小值的百分比分别为：一月 \(\tfrac{2}{7} \approx 29\%\)，二月 \(\tfrac{2}{3} \approx 67\%\)，三月 \(\tfrac{3}{6} = 50\%\)，四月 \(\tfrac{3}{9} \approx 33\%\)，五月 \(\tfrac{2}{8} = 25\%\)。最大的是二月，此时 \(5\) 比 \(3\) 大约 \(67\%\)。

所以正确答案是 B。

English

Reading the graph gives (drums, bugles): January \((7, 9),\) February \((5, 3),\) March \((9, 6),\) April \((9, 12),\) and May \((8, 10).\)

The percent excess of the larger over the smaller is \(\tfrac{2}{7} \approx 29\%\) in January, \(\tfrac{2}{3} \approx 67\%\) in February, \(\tfrac{3}{6} = 50\%\) in March, \(\tfrac{3}{9} \approx 33\%\) in April, and \(\tfrac{2}{8} = 25\%\) in May. The greatest is February, where \(5\) exceeds \(3\) by about \(67\%.\)

Thus, the correct answer is B .

22 答案 C

一块只共享一条边的新增瓷砖会使周长增加 \(2\)：四条新边中有两条因共享边而不在外边界上。若共享两条边，则周长增加 \(0\)。

加入两块瓷砖后，周长可能增加 \(0\)、\(2\) 或 \(4\)，新周长可能为 \(14\)、\(16\) 或 \(18\)。选项中只有 \(18\) 可能。

所以正确答案是 C。

English

A tile that shares exactly one side adds \(2\) to the perimeter (four new edges minus two hidden), while a tile sharing two sides adds \(0.\)

With two added tiles the perimeter can change by \(0,\) \(2,\) or \(4,\) giving new perimeters of \(14,\) \(16,\) or \(18.\) Of the choices, only \(18\) is possible.

Thus, the correct answer is C .

23 答案 B

共有 \(36\) 个等可能结果。按第一颗骰子的点数，计算乘积大于 \(10\) 的有序数对：若为 \(2\)，有 \(1\) 个结果满足，即 \(2 \times 6\)；若为 \(3\)，有 \(3\) 个；若为 \(4\)，有 \(4\) 个；若为 \(5\)，有 \(4\) 个；若为 \(6\)，有 \(5\) 个。

有利结果总数为 \(1 + 3 + 4 + 4 + 5 = 17\)，所以概率是 \(\dfrac{17}{36}\)。

所以正确答案是 B。

English

There are \(36\) equally likely outcomes. Counting the ordered pairs whose product exceeds \(10\): with a first die of \(2\) there is \(1\) (namely \(2 \times 6\)); of \(3\), there are \(3\); of \(4\), there are \(4\); of \(5\), there are \(4\); of \(6\), there are \(5.\)

That is \(1 + 3 + 4 + 4 + 5 = 17\) favorable outcomes, so the probability is \(\dfrac{17}{36}.\)

Thus, the correct answer is B .

24 答案 A

相邻圆相切，所以圆心组成的正方形边长为 \(2 \times 3 = 6\)，面积为 \(36\)。

在正方形内，每个圆贡献一个四分之一圆，四个四分之一圆合成一个完整圆，面积为 \(9\pi \approx 28.3\)。阴影面积为 \(36 - 9\pi \approx 7.7\)。

所以正确答案是 A。

English

Because adjacent circles are tangent, the square through the centers has side \(2 \times 3 = 6\) and area \(36.\)

Inside the square, each circle contributes a quarter-circle, and the four quarters make one full circle of area \(9\pi \approx 28.3.\) The shaded region is \(36 - 9\pi \approx 7.7.\)

Thus, the correct answer is A .

25 答案 D

第 \(n\) 次倒出后，剩余比例为 \(\dfrac12 \times \dfrac23 \times \dfrac34 \times \cdots \times \dfrac{n}{n+1}\)，逐项约分后等于 \(\dfrac{1}{n+1}\)。

令 \(\dfrac{1}{n+1} = \dfrac{1}{10}\)，得到 \(n = 9\)。

所以正确答案是 D。

English

After the \(n\)th pouring, the fraction remaining is \(\dfrac12 \times \dfrac23 \times \dfrac34 \times \cdots \times \dfrac{n}{n+1},\) which telescopes to \(\dfrac{1}{n+1}.\)

Setting \(\dfrac{1}{n+1} = \dfrac{1}{10}\) gives \(n = 9.\)

Thus, the correct answer is D .