1993 年 AMC 8 真题（P版）

1 答案 C

逐项检查：\((-4)(-9) = 36\)，\((-3)(-12) = 36\)，\(\dfrac12 \times (-72) = -36\)，\((1)(36) = 36\)，且 \(\dfrac32 \times 24 = 36\)。

只有 \(\dfrac12 \times (-72) = -36\) 不等于 \(36\)。

所以正确答案是 C。

English

Checking each pair: \((-4)(-9) = 36,\) \((-3)(-12) = 36,\) \(\dfrac12 \times (-72) = -36,\) \((1)(36) = 36,\) and \(\dfrac32 \times 24 = 36.\)

Only \(\dfrac12 \times (-72) = -36\) fails to equal \(36.\)

Thus, the correct answer is C .

2 答案 C

因为 \(49 = 7 \times 7\)，且 \(84 = 7 \times 12\)，所以分数可约为 \(\dfrac{7}{12}\)。

分子和分母之和为 \(7 + 12 = 19\)。

所以正确答案是 C。

English

Since \(49 = 7 \times 7\) and \(84 = 7 \times 12,\) the fraction reduces to \(\dfrac{7}{12}.\)

The sum of numerator and denominator is \(7 + 12 = 19.\)

Thus, the correct answer is C .

3 答案 B

分解得：\(39 = 3 \times 13\)，\(51 = 3 \times 17\)，\(77 = 7 \times 11\)，\(91 = 7 \times 13\)，\(121 = 11 \times 11\)。

这些最大质因数中最大的是 \(17\)，它是 \(51\) 的因数。

所以正确答案是 B。

English

Factoring: \(39 = 3 \times 13,\) \(51 = 3 \times 17,\) \(77 = 7 \times 11,\) \(91 = 7 \times 13,\) and \(121 = 11 \times 11.\)

The largest prime factor among these is \(17,\) which is a factor of \(51.\)

Thus, the correct answer is B .

4 答案 E

重新分组为 \((1000 \times 10)\) \(\times 0.1993\) \(\times 1993\) \(= 10000\) \(\times 0.1993\) \(\times 1993\)。

因为 \(10000 \times 0.1993 = 1993\)，所以原乘积是 \(1993 \times 1993 = (1993)^2\)。

所以正确答案是 E。

English

Regroup as \((1000 \times 10)\) \(\times 0.1993\) \(\times 1993\) \(= 10000\) \(\times 0.1993\) \(\times 1993.\)

Since \(10000 \times 0.1993 = 1993,\) the product is \(1993 \times 1993 = (1993)^2.\)

Thus, the correct answer is E .

5 答案 C

两个涂色区域各占圆的四分之一，未涂色区域占一半。所以三个数量的比例是 \(\tfrac14 : \tfrac14 : \tfrac12\)，即 \(1 : 1 : 2\)。

相应条形图必须有两个涂色条高度相等，白色条高度正好是它们的两倍。只有一个条形图符合这些条件。

所以正确答案是 C。

English

The two shaded regions are each one quarter of the circle, and the unshaded region is one half. So the three quantities are in the ratio \(\tfrac14 : \tfrac14 : \tfrac12,\) or \(1 : 1 : 2.\)

A matching bar graph must have the two shaded bars equal in height and the unshaded bar exactly twice as tall. Only one bar graph has two equal shaded bars with the white bar double their height.

Thus, the correct answer is C .

6 答案 B

供 \(15\) 个儿童食用需要 \(15 \div 5 = 3\) 罐汤，剩下 \(5 - 3 = 2\) 罐。

这 \(2\) 罐可以供 \(2 \times 3 = 6\) 个成人食用。

所以正确答案是 B。

English

Feeding \(15\) children uses \(15 \div 5 = 3\) cans, leaving \(5 - 3 = 2\) cans.

Those \(2\) cans feed \(2 \times 3 = 6\) adults.

Thus, the correct answer is B .

7 答案 A

三个相同项相加，\(3^3 + 3^3 + 3^3 = 3 \times 3^3 = 3^4 = 81\)。

所以正确答案是 A。

English

Adding three equal terms, \(3^3 + 3^3 + 3^3 = 3 \times 3^3 = 3^4 = 81.\)

Thus, the correct answer is A .

8 答案 D

她每两天服用半片，所以一片药可用 \(4\) 天。\(60\) 片药可用 \(60 \times 4 = 240\) 天。

按每月约 \(30\) 天计算，约为 \(240 \div 30 = 8\) 个月。

所以正确答案是 D。

English

She takes half a pill every two days, so one pill lasts \(4\) days. Then \(60\) pills last \(60 \times 4 = 240\) days.

At about \(30\) days per month, that is roughly \(240 \div 30 = 8\) months.

Thus, the correct answer is D .

9 答案 D

从表中读出 \(2 * 4 = 3\)，且 \(1 * 3 = 3\)。

因此 \((2 * 4) * (1 * 3) = 3 * 3 = 4\)。

所以正确答案是 D。

English

From the table, \(2 * 4 = 3\) and \(1 * 3 = 3.\)

Then \((2 * 4) * (1 * 3) = 3 * 3 = 4.\)

Thus, the correct answer is D .

10 答案 B

月度变化为：一月从 \(\$2.50 \to \$2.00\)，下降 \(\$0.50\)；二月从 \(\$2.00 \to \$4.00\)，价格上升；三月从 \(\$4.00 \to \$1.50\)，下降 \(\$2.50\)；四月从 \(\$1.50 \to \$4.50\)，价格上升；五月从 \(\$4.50 \to \$3.00\)，下降 \(\$1.50\)；六月从 \(\$3.00 \to \$1.00\)，下降 \(\$2.00\)。

最大下降是 \(\$2.50\)，发生在三月。

所以正确答案是 B。

English

The price changes month to month are: January \(\$2.50 \to \$2.00\) (drop \(\$0.50\)), February \(\$2.00 \to \$4.00\) (rise), March \(\$4.00 \to \$1.50\) (drop \(\$2.50\)), April \(\$1.50 \to \$4.50\) (rise), May \(\$4.50 \to \$3.00\) (drop \(\$1.50\)), and June \(\$3.00 \to \$1.00\) (drop \(\$2.00\)).

The largest drop is \(\$2.50,\) which occurred during March.

Thus, the correct answer is B .

11 答案 C

因为有 \(81\) 名学生，中位数是从低到高数的第 \(41\) 个分数。

从左往右累加柱高，累计人数为 \(1\)、\(3\)、\(7\)、\(12\)、\(18\)、\(28\)、\(42\)、\(\ldots\)。累计数第一次超过 \(41\) 时位于标号 \(70\) 的区间，其中包含第 \(29\) 到第 \(42\) 个分数。所以第 \(41\) 个分数在标号 \(70\) 的区间中。

所以正确答案是 C。

English

Since \(81\) students took the test, the median is the \(41\)st score counting up from the lowest.

Adding the bar heights from the left gives running totals \(1,\) \(3,\) \(7,\) \(12,\) \(18,\) \(28,\) \(42,\) \(\ldots\) The total first passes \(41\) at the interval labeled \(70,\) which contains the \(29\)th through \(42\)nd scores. So the \(41\)st score lies in the interval labeled \(70.\)

Thus, the correct answer is C .

12 答案 E

六种排列分别给出：\(5 \times 4 + 6 - 3 = 23\)，\(5 \times 4 - 6 + 3 = 17\)，\(5 + 4 \times 6 - 3 = 26\)，\(5 - 4 \times 6 + 3 = -16\)，\(5 + 4 - 6 \times 3 = -9\)，以及 \(5 - 4 + 6 \times 3 = 19\)。

选项中唯一可能的值是 \(19\)。

所以正确答案是 E。

English

The six arrangements give \(5 \times 4 + 6 - 3 = 23,\) \(5 \times 4 - 6 + 3 = 17,\) \(5 + 4 \times 6 - 3 = 26,\) \(5 - 4 \times 6 + 3 = -16,\) \(5 + 4 - 6 \times 3 = -9,\) and \(5 - 4 + 6 \times 3 = 19.\)

The only value among the choices is \(19.\)

Thus, the correct answer is E .

13 答案 D

整个标牌面积为 \(5 \times 15 = 75\) 平方单位。数涂色单位方格可得 \(H = 11\)、\(E = 11\)、\(L = 7\)、\(P = 10\)，涂色总面积为 \(11 + 11 + 7 + 10 = 39\)。

未涂色面积是 \(75 - 39 = 36\)。

所以正确答案是 D。

English

The full sign has area \(5 \times 15 = 75\) square units. Counting the shaded unit squares in each letter gives \(H = 11,\) \(E = 11,\) \(L = 7,\) and \(P = 10,\) for a shaded total of \(11 + 11 + 7 + 10 = 39.\)

The unshaded area is \(75 - 39 = 36.\)

Thus, the correct answer is D .

14 答案 C

填表使每行每列都有 \(1\)、\(2\)、\(3\)：第一行变为 \(1\)、\(3\)、\(2\)，第二行为 \(3\)、\(2\)、\(A\)，第三行为 \(2\)、\(1\)、\(B\)。第二行迫使 \(A = 1\)，最后一列 \(2\)、\(1\)、\(B\) 迫使 \(B = 3\)。

所以 \(A + B = 1 + 3 = 4\)。

所以正确答案是 C。

English

Filling the grid so each row and column has \(1,\) \(2,\) \(3,\) the top row becomes \(1,\) \(3,\) \(2,\) the middle row \(3,\) \(2,\) \(A,\) and the bottom row \(2,\) \(1,\) \(B.\) The middle row forces \(A = 1,\) and the last column \(2,\) \(1,\) \(B\) forces \(B = 3.\)

So \(A + B = 1 + 3 = 4.\)

Thus, the correct answer is C .

15 答案 A

四个数的和是 \(4 \times 85 = 340\)，所以其余三个数的和是 \(340 - 97 = 243\)。

它们的平均数是 \(243 \div 3 = 81\)。

所以正确答案是 A。

English

The four numbers sum to \(4 \times 85 = 340,\) so the remaining three sum to \(340 - 97 = 243.\)

Their mean is \(243 \div 3 = 81.\)

Thus, the correct answer is A .

16 答案 C

从里面开始，\(2 + \dfrac13 = \dfrac73\)，所以 \(\dfrac{1}{\frac{7}{3}} = \dfrac37\)。

接着 \(1 + \dfrac37 = \dfrac{10}{7}\)，整个表达式为 \(\dfrac{1}{\frac{10}{7}} = \dfrac{7}{10}\)。

所以正确答案是 C。

English

Starting inside, \(2 + \dfrac13 = \dfrac73,\) so \(\dfrac{1}{\frac{7}{3}} = \dfrac37.\)

Then \(1 + \dfrac37 = \dfrac{10}{7},\) and the whole expression is \(\dfrac{1}{\frac{10}{7}} = \dfrac{7}{10}.\)

Thus, the correct answer is C .

17 答案 B

盒子内部表面正好是切去四角后纸板的一面。原纸板面积为 \(20 \times 30 = 600\)，每个被切去的角面积为 \(5 \times 5 = 25\)。

所以内部表面积是 \(600 - 4 \times 25 = 500\)。

所以正确答案是 B。

English

The interior surface is exactly one face of the cardboard after the corners are removed. The sheet has area \(20 \times 30 = 600,\) and each removed corner has area \(5 \times 5 = 25.\)

So the interior surface area is \(600 - 4 \times 25 = 500.\)

Thus, the correct answer is B .

18 答案 A

长方形 \(ACDE\) 面积为 \(32 \times 20 = 640\)。三角形 \(BCD\) 面积为 \(\dfrac{16 \times 20}{2} = 160\)，三角形 \(DEF\) 面积为 \(\dfrac{10 \times 32}{2} = 160\)。

剩下的四边形 \(ABDF\) 面积为 \(640 - (160 + 160) = 320\)。

所以正确答案是 A。

English

Rectangle \(ACDE\) has area \(32 \times 20 = 640.\) Triangle \(BCD\) has area \(\dfrac{16 \times 20}{2} = 160,\) and triangle \(DEF\) has area \(\dfrac{10 \times 32}{2} = 160.\)

The remaining region \(ABDF\) has area \(640 - (160 + 160) = 320.\)

Thus, the correct answer is A .

19 答案 A

第一个和中的每个数都比第二个和中对应的数大 \(1800\)，并且共有 \(93\) 对。

所以差为 \(93 \times 1800 = 167{,}400\)。

所以正确答案是 A。

English

Each number in the first sum is exactly \(1800\) more than the matching number in the second sum, and there are \(93\) such pairs.

So the difference is \(93 \times 1800 = 167{,}400.\)

Thus, the correct answer is A .

20 答案 D

从 \(10^{93}\)（\(1\) 后面跟 \(93\) 个零）减去 \(93\)，结果为 \(91\) 个九后接 \(07\)。

数字和为 \(91 \times 9 + 0 + 7 = 819 + 7 = 826\)。

所以正确答案是 D。

English

Subtracting \(93\) from \(10^{93}\) (a \(1\) followed by \(93\) zeros) gives a number that is \(91\) nines followed by \(07.\)

The digit sum is \(91 \times 9 + 0 + 7 = 819 + 7 = 826.\)

Thus, the correct answer is D .

21 答案 D

新长是原长的 \(1.2\) 倍，新宽是原宽的 \(1.5\) 倍，所以新面积是原面积的 \(1.2 \times 1.5 = 1.8\) 倍。

这表示面积增加了 \(80\%\)。

所以正确答案是 D。

English

The new length is \(1.2\) times the old and the new width is \(1.5\) times the old, so the new area is \(1.2 \times 1.5 = 1.8\) times the old area.

That is an increase of \(80\%.\)

Thus, the correct answer is D .

22 答案 D

编号 \(1\) 到 \(99\) 时，个位出现十个 \(2\)，十位上也出现十个，共用掉二十个 \(2\)。页码 \(100\) 和 \(101\) 一个也不用到。

剩下两个 \(2\) 用在页码 \(102\) 和 \(112\)。之后 \(113\) 到 \(119\) 不需要 \(2\)，但 \(120\) 需要另一个 \(2\)。所以最多能编号到 \(119\)。

所以正确答案是 D。

English

Numbering \(1\) through \(99\) uses ten \(2\)’s in the units place and ten in the tens place, a total of twenty \(2\)’s. Pages \(100\) and \(101\) use none.

The remaining two \(2\)’s are used on pages \(102\) and \(112.\) After that, pages \(113\) through \(119\) need no \(2,\) but \(120\) would require another \(2,\) so he can number up to \(119.\)

Thus, the correct answer is D .

23 答案 C

因为 \(P\) 领先 \(Q\) 和 \(R\)，也在 \(T\) 之前，而 \(Q\) 又领先 \(S\)，所以 \(P\) 第一，因此不可能第三。

条件还给出完赛先后链：\(P\)、\(T\)、\(Q\)、\(S\) 依次在前。所以 \(P\)、\(T\) 和 \(Q\) 都在 \(S\) 前，\(S\) 最好也只能第四，不可能第三。

\(Q\)、\(R\)、\(T\) 都可能第三。例如，\(P\)、\(T\)、\(Q\)、\(R\)、\(S\) 使 \(Q\) 第三；\(P\)、\(R\)、\(T\)、\(Q\)、\(S\) 使 \(T\) 第三；\(P\)、\(T\)、\(R\)、\(Q\)、\(S\) 使 \(R\) 第三。因此只有 \(P\) 和 \(S\) 不可能第三。

所以正确答案是 C。

English

Since \(P\) beats \(Q\) and \(R,\) finishes ahead of \(T,\) and \(Q\) beats \(S,\) runner \(P\) finishes first and so cannot be third.

The clues give the chain \(P\) before \(T\) before \(Q\) before \(S.\) So \(P,\) \(T,\) and \(Q\) all finish ahead of \(S,\) meaning \(S\) is no better than fourth and cannot be third either.

Each of \(Q,\) \(R,\) \(T\) can finish third: for example \(P,\) \(T,\) \(Q,\) \(R,\) \(S\) puts \(Q\) third; \(P,\) \(R,\) \(T,\) \(Q,\) \(S\) puts \(T\) third; and \(P,\) \(T,\) \(R,\) \(Q,\) \(S\) puts \(R\) third. So only \(P\) and \(S\) cannot be third.

Thus, the correct answer is C .

24 答案 C

每一行都以一个完全平方数结尾，所以含 \(142\) 的那一行以 \(144 = 12^2\) 结尾，上一行以 \(121 = 11^2\) 结尾。

因为各行右端对齐，\(121\) 在 \(143\) 的正上方，所以 \(120\) 在 \(142\) 的正上方。

所以正确答案是 C。

English

Each row ends at a perfect square, so the row containing \(142\) ends at \(144 = 12^2,\) and the row above it ends at \(121 = 11^2.\)

Since the rows are aligned at their right edges, \(121\) sits directly above \(143,\) and therefore \(120\) sits directly above \(142.\)

Thus, the correct answer is C .

25 答案 E

如图所示，将卡片旋转 \(45^\circ\)，并把中心放在四个棋盘小格相交的角点上。因为卡片对角线 \(\sqrt{1.5^2 + 1.5^2} = \sqrt{4.5} \approx 2.1\) 大于 \(2\)，卡片的四个角都会越过网格线伸入下一格。

卡片覆盖中央的 \(2 \times 2\) 共 \(4\) 个小正方形，并且在四边各伸入另外 \(2\) 个小正方形，总共 \(4 + 4 \times 2 = 12\) 个。

这也是最大可能值。卡片边长只有 \(1.5\) 英寸，所以水平和竖直方向跨度都最多约 \(2.1\) 英寸，因此它位于一个 \(4 \times 4\)、共含 \(16\) 格的方格块内。只有卡片的四个尖角能到达该方格块边缘，它不可能到达这个方格块的四个角格，所以最多覆盖 \(12\) 个。既然 \(12\) 可以达到，最大值就是 \(12\)，属于“\(12\) 或更多”。

所以正确答案是 E。

English

Tilt the card \(45^\circ\) and center it on a corner where four grid squares meet, as shown. Because the card’s diagonal, \(\sqrt{1.5^2 + 1.5^2} = \sqrt{4.5} \approx 2.1,\) is longer than \(2,\) each of the four corners of the card reaches past a grid line into the next square.

The card covers the central \(2 \times 2\) block of \(4\) squares and pokes into \(2\) more squares on each of its four sides, giving \(4 + 4 \times 2 = 12\) squares.

This is also the most possible. The card is only \(1.5\) inches wide, so its overall width and height are each at most \(2.1\) inches; it therefore lies within a \(4 \times 4\) block of \(16\) squares. Its four pointed corners are the only parts that reach the edge of that block, so it can never reach the four corner squares of the block, leaving at most \(12.\) Since \(12\) is achievable, the maximum is \(12,\) which falls in the range “\(12\) or more.”

Thus, the correct answer is E .