1985 年 AMC 8 真题（P版）

1 答案 A

合并两个分数后，分子是 \(3 \cdot 5 \cdot 7 \cdot 9 \cdot 11\)，分母是 \(9 \cdot 11 \cdot 3 \cdot 5 \cdot 7\)。它们是同一个乘积。

因此这个值是 \(1\)。

所以正确答案是 A。

English

Combining the two fractions, the numerator is \(3 \cdot 5 \cdot 7 \cdot 9 \cdot 11\) and the denominator is \(9 \cdot 11 \cdot 3 \cdot 5 \cdot 7.\) These are the same product.

So the value is \(1.\)

Thus, the correct answer is A .

2 答案 B

一共有 \(10\) 项。首项和末项的平均数是 \(\dfrac{90 + 99}{2} = 94.5\)。

所以总和是 \(10 \cdot 94.5 = 945\)。

所以正确答案是 B。

English

There are \(10\) terms. The average of the first and last is \(\dfrac{90 + 99}{2} = 94.5.\)

So the sum is \(10 \cdot 94.5 = 945.\)

Thus, the correct answer is B .

3 答案 D

因为 \(\dfrac{10^7}{10^4} = 10^3 = 1000\)，所以原式是 \(\dfrac{1000}{5} = 200\)。

所以正确答案是 D。

English

Since \(\dfrac{10^7}{10^4} = 10^3 = 1000,\) the expression is \(\dfrac{1000}{5} = 200.\)

Thus, the correct answer is D .

4 答案 C

把图形补成完整的 \(6 \times 9\) 矩形，面积是 \(54\)。左下角被去掉的是一个 \(2\) 乘 \(4\) 的矩形，面积是 \(8\)。

所以多边形面积是 \(54 - 8 = 46\)。

所以正确答案是 C。

English

Completing the figure to the full \(6 \times 9\) rectangle gives an area of \(54.\) The piece removed from the lower-left corner is a rectangle measuring \(2\) by \(4,\) with area \(8.\)

So the polygon’s area is \(54 - 8 = 46.\)

Thus, the correct answer is C .

5 答案 C

及格成绩共有 \(5 + 4 + 3 + 3 = 15\) 个，总成绩数是 \(15 + 5 = 20\)。

所以及格成绩所占的分数是 \(\dfrac{15}{20} = \dfrac{3}{4}\)。

所以正确答案是 C。

English

The satisfactory grades number \(5 + 4 + 3 + 3 = 15,\) and the total number of grades is \(15 + 5 = 20.\)

So the fraction that is satisfactory is \(\dfrac{15}{20} = \dfrac{3}{4}.\)

Thus, the correct answer is C .

6 答案 D

因为 \(7.5\) 厘米是 \(5\) 厘米的 \(1.5\) 倍，所以纸张数也是 \(1.5\) 倍。

也就是 \(1.5 \times 500 = 750\) 张。

所以正确答案是 D。

English

Since \(7.5\) cm is \(1.5\) times \(5\) cm, the stack holds \(1.5\) times as many sheets.

That is \(1.5 \times 500 = 750\) sheets.

Thus, the correct answer is D .

7 答案 C

第 \(n\) 行有 \(2n - 1\) 个方格。由于两端都是未涂色方格且两种方格交替，每行未涂色方格比涂色方格多一个，所以涂色方格数是 \(n - 1\)。

对第 \(37\) 行，这就是 \(37 - 1 = 36\)。

所以正确答案是 C。

English

Row \(n\) contains \(2n - 1\) squares. Because both ends are unshaded and the squares alternate, each row has one more unshaded square than shaded square, so the number of shaded squares is \(n - 1.\)

For the \(37\)th row, this is \(37 - 1 = 36.\)

Thus, the correct answer is C .

8 答案 A

代入 \(a = -2\) 得到 \(-3a = 6\)，\(4a = -8\)，\(\dfrac{24}{a} = -12\)，\(a^2 = 4\)，以及 \(1\)。

其中最大的是 \(6\)，也就是 \(-3a\)。

所以正确答案是 A。

English

Substituting \(a = -2\) gives the values \(-3a = 6,\) \(4a = -8,\) \(\dfrac{24}{a} = -12,\) \(a^2 = 4,\) and \(1.\)

The largest of these is \(6,\) which is \(-3a.\)

Thus, the correct answer is A .

9 答案 A

每个因数 \(1 - \dfrac1k\) 都等于 \(\dfrac{k-1}{k}\)，所以乘积是

\[\dfrac12 \cdot \dfrac23 \cdot \dfrac34 \cdots \dfrac{9}{10}\text{。}\]

每个分子都会和前一个分母相消，最后留下 \(\dfrac{1}{10}\)。

所以正确答案是 A。

English

Each factor \(1 - \dfrac1k\) equals \(\dfrac{k-1}{k},\) so the product is

\[\dfrac12 \cdot \dfrac23 \cdot \dfrac34 \cdots \dfrac{9}{10}.\]

Every numerator cancels the previous denominator, leaving \(\dfrac{1}{10}.\)

Thus, the correct answer is A .

10 答案 C

\(\dfrac15\) 和 \(\dfrac13\) 的中点是它们的平均数。

\[\dfrac{\frac15 + \frac13}{2} = \dfrac{\frac{8}{15}}{2} = \dfrac{4}{15}\text{。}\]

所以正确答案是 C。

English

The midpoint of \(\dfrac15\) and \(\dfrac13\) is their average.

\[\dfrac{\frac15 + \frac13}{2} = \dfrac{\frac{8}{15}}{2} = \dfrac{4}{15}.\]

Thus, the correct answer is C .

11 答案 E

把这个展开图折成让 \(X\) 成为底面。那么 \(U\)、\(V\)、\(W\)、\(Z\) 会围成四个侧面。

唯一剩下的正方形 \(Y\) 成为顶面，与底面 \(X\) 相对。

所以正确答案是 E。

English

Fold the net so that \(X\) is the bottom face. Then \(U,\) \(V,\) \(W,\) and \(Z\) wrap around to become the four side faces.

The only remaining square, \(Y,\) becomes the top face, which is opposite the bottom face \(X.\)

Thus, the correct answer is E .

12 答案 B

三角形周长是 \(6.2 + 8.3 + 9.5 = 24\) 厘米，所以正方形周长也是 \(24\) 厘米。

正方形边长是 \(\dfrac{24}{4} = 6\) 厘米，因此面积是 \(6^2 = 36 \text{ cm}^2\)。

所以正确答案是 B。

English

The triangle’s perimeter is \(6.2 + 8.3 + 9.5 = 24\) cm, so the square also has perimeter \(24\) cm.

The square’s side is \(\dfrac{24}{4} = 6\) cm, so its area is \(6^2 = 36 \text{ cm}^2.\)

Thus, the correct answer is B .

13 答案 B

步行距离：\(4 \times \dfrac34 = 3\) 英里。跑步距离：\(10 \times \dfrac12 = 5\) 英里。

总距离是 \(3 + 5 = 8\) 英里。

所以正确答案是 B。

English

Walking: \(4 \times \dfrac34 = 3\) miles. Running: \(10 \times \dfrac12 = 5\) miles.

The total distance is \(3 + 5 = 8\) miles.

Thus, the correct answer is B .

14 答案 B

两种税相差价格的 \(0.5\%\)。

也就是 \(0.005 \times \$20 = \$0.10\)。

所以正确答案是 B。

English

The two taxes differ by \(0.5\%\) of the price.

That is \(0.005 \times \$20 = \$0.10.\)

Thus, the correct answer is B .

15 答案 C

从 \(200\) 到 \(299\) 的 \(100\) 个数全部含有 \(2\)。

在 \(100\) 到 \(199\) 中，十位是 \(2\) 的有 \(10\) 个，个位是 \(2\) 的有 \(10\) 个，但 \(122\) 被重复计算了一次，所以有 \(10 + 10 - 1 = 19\) 个。\(300\) 到 \(399\) 同样有 \(19\) 个。

总数是 \(100 + 19 + 19 = 138\)。

所以正确答案是 C。

English

All \(100\) numbers from \(200\) to \(299\) contain a \(2.\)

Among \(100\) to \(199,\) there are \(10\) with a \(2\) in the tens place and \(10\) with a \(2\) in the units place, but \(122\) is counted twice, giving \(10 + 10 - 1 = 19.\) The range \(300\) to \(399\) similarly contributes \(19.\)

The total is \(100 + 19 + 19 = 138.\)

Thus, the correct answer is C .

16 答案 D

\(30\) 名学生分成 \(5\) 个相等的份，每份 \(6\) 人。男生占 \(2\) 份，即 \(12\) 人；女生占 \(3\) 份，即 \(18\) 人。

所以女生比男生多 \(18 - 12 = 6\) 人。

所以正确答案是 D。

English

The \(30\) students split into \(5\) equal parts of \(6.\) Boys make up \(2\) parts (\(12\)) and girls \(3\) parts (\(18\)).

So there are \(18 - 12 = 6\) more girls than boys.

Thus, the correct answer is D .

17 答案 D

七次测验的平均分是 \(85\)，总分为 \(7 \times 85 = 595\)；六次测验的平均分是 \(84\)，总分为 \(6 \times 84 = 504\)。

第七次测验的分数是 \(595 - 504 = 91\)。

所以正确答案是 D。

English

Seven tests averaging \(85\) total \(7 \times 85 = 595\) points; six tests averaging \(84\) total \(6 \times 84 = 504\) points.

The seventh score is \(595 - 504 = 91.\)

Thus, the correct answer is D .

18 答案 E

由 \(9p \lt 10\) 得 \(p \lt 1.111\ldots\)，由 \(10p \gt 11\) 得 \(p \gt 1.10\)。

在 \(\$1.10\) 和 \(\$1.111\ldots\) 之间唯一的整美分价格是 \(\$1.11\)。

所以正确答案是 E。

English

From \(9p \lt 10\) we get \(p \lt 1.111\ldots,\) and from \(10p \gt 11\) we get \(p \gt 1.10.\)

The only price in whole cents between \(\$1.10\) and \(\$1.111\ldots\) is \(\$1.11.\)

Thus, the correct answer is E .

19 答案 B

新周长是 \(2(1.1\ell + 1.1w) = 1.1 \cdot 2(\ell + w)\)，也就是旧周长的 \(1.1\) 倍。

这表示增加了 \(10\%\)。

所以正确答案是 B。

English

The new perimeter is \(2(1.1\ell + 1.1w) = 1.1 \cdot 2(\ell + w),\) which is \(1.1\) times the old perimeter.

That is a \(10\%\) increase.

Thus, the correct answer is B .

20 答案 C

因为 \(31 = 4 \cdot 7 + 3\)，一月 \(1\) 日、\(2\) 日和 \(3\) 日所对应的三个星期几各出现五次，其余星期几各出现四次。

要让星期二和星期六都只出现四次，它们都不能是一月 \(1\) 日、\(2\) 日或 \(3\) 日。唯一可行的开始日是星期三：这样星期三、星期四、星期五出现五次，而星期二和星期六各出现四次。

所以正确答案是 C。

English

Since \(31 = 4 \cdot 7 + 3,\) the weekdays falling on January \(1,\) \(2,\) and \(3\) each occur five times that month, and every other weekday occurs four times.

For Tuesday and Saturday to occur only four times, neither may be January \(1,\) \(2,\) or \(3.\) The only starting day that works is Wednesday: then Wednesday, Thursday, and Friday occur five times, while Tuesday and Saturday each occur four times.

Thus, the correct answer is C .

21 答案 E

四次加薪后薪水被乘以 \(1.1^4 = 1.4641\)。

这表示增加了 \(46.41\%\)，大于 \(45\%\)。

所以正确答案是 E。

English

After four raises the salary is multiplied by \(1.1^4 = 1.4641.\)

That is an increase of \(46.41\%,\) which is more than \(45\%.\)

Thus, the correct answer is E .

22 答案 B

第一位是 \(8\) 个允许数字之一（\(2\) 到 \(9\)），所以 \(\dfrac18\) 的号码以 \(9\) 开头。最后一位可以是 \(10\) 个数字之一，所以 \(\dfrac{1}{10}\) 的号码以 \(0\) 结尾。

这两个条件相互独立，所以所求比例是 \(\dfrac18 \cdot \dfrac{1}{10} = \dfrac{1}{80}\)。

所以正确答案是 B。

English

The first digit is one of \(8\) allowed digits (\(2\) through \(9\)), so \(\dfrac18\) of the numbers begin with \(9.\) The last digit is any of \(10\) digits, so \(\dfrac{1}{10}\) end in \(0.\)

These conditions are independent, so the fraction is \(\dfrac18 \cdot \dfrac{1}{10} = \dfrac{1}{80}.\)

Thus, the correct answer is B .

23 答案 E

每天共有 \(1200 \times 5 = 6000\) 次学生上课。每节课有 \(30\) 名学生，所以共有 \(\dfrac{6000}{30} = 200\) 节课。

每位老师教 \(4\) 节课，所以有 \(\dfrac{200}{4} = 50\) 位老师。

所以正确答案是 E。

English

Each day there are \(1200 \times 5 = 6000\) student-class attendances. Since each class holds \(30\) students, there are \(\dfrac{6000}{30} = 200\) classes.

Each teacher teaches \(4\) classes, so there are \(\dfrac{200}{4} = 50\) teachers.

Thus, the correct answer is E .

24 答案 D

设 \(C\) 为三个顶点数之和。把三条边的和相加可得 \(3S = 75 + C\)，因为六个数之和为 \(75\)，且每个顶点数都被多计算了一次。要使 \(S\) 最大，应把三个最大的数 \(13\)、\(14\)、\(15\) 放在顶点，此时 \(C = 42\)。

于是 \(3S = 75 + 42 = 117\)，所以 \(S = 39\)。这个值可以达到：在顶点 \(13\) 与 \(14\) 之间放 \(12\)，在 \(14\) 与 \(15\) 之间放 \(10\)，在 \(15\) 与 \(13\) 之间放 \(11\)。

所以正确答案是 D。

English

Let \(C\) be the sum of the corner numbers. Adding the three side sums gives \(3S = 75 + C,\) because the six numbers sum to \(75\) and every corner is counted one extra time. To maximize \(S,\) place the three largest numbers \(13,\) \(14,\) \(15\) at the corners, giving \(C = 42.\)

Then \(3S = 75 + 42 = 117,\) so \(S = 39.\) This is achievable: put \(12\) between corners \(13\) and \(14,\) put \(10\) between \(14\) and \(15,\) and put \(11\) between \(15\) and \(13.\)

Thus, the correct answer is D .

25 答案 A

要证明“元音字母必对应偶数”这个规则是错的，玛丽需要找到一张一面是元音字母、另一面是奇数的卡片。没有卡片显示元音字母，因为 \(P\) 和 \(Q\) 是辅音。

唯一显示奇数的卡片是 \(3\)。翻开它后露出元音字母，就会反驳简的说法。

所以正确答案是 A。

English

To show the rule “a vowel forces an even number” is false, Mary needs a card with a vowel on one side and an odd number on the other. No card shows a vowel, since \(P\) and \(Q\) are consonants.

The only card showing an odd number is \(3.\) Turning it over reveals a vowel, which contradicts Jane’s claim.

Thus, the correct answer is A .