1998 年 AMC 8 真题（P版）

1

当 \(x = 7\) 时，下列哪一个最小？

A\(\dfrac{6}{x}\) B\(\dfrac{6}{x+1}\) C\(\dfrac{6}{x-1}\) D\(\dfrac{x}{6}\) E\(\dfrac{x+1}{6}\)

English

For \(x = 7,\) which of the following is the smallest?

(A) \(\dfrac{6}{x} \)(B) \(\dfrac{6}{x+1} \)(C) \(\dfrac{6}{x-1} \)(D) \(\dfrac{x}{6} \)(E) \(\dfrac{x+1}{6} \)

2

如果 \(~\begin{array}{r|l}a&b \\ \hline c&d\end{array} = a\cdot d-b\cdot c\)，那么 \(~\begin{array}{r|l}3&4 \\ \hline 1&2\end{array}~\) 的值是多少？

A\(-2\) B\(-1\) C\(0\) D\(1\) E\(2\)

English

If \(~\begin{array}{r|l}a&b \\ \hline c&d\end{array} = a\cdot d-b\cdot c,\) what is the value of \(~\begin{array}{r|l}3&4 \\ \hline 1&2\end{array}~?\)

(A) \(-2 \)(B) \(-1 \)(C) \(0 \)(D) \(1 \)(E) \(2 \)

3

下列表达式的值是多少？\[\dfrac{\frac{3}{8}+\frac{7}{8}}{\frac{4}{5}}\]

A\(1\) B\(\dfrac{25}{16}\) C\(2\) D\(\dfrac{43}{20}\) E\(\dfrac{47}{16}\)

English

What is the value of the following expression? \[\dfrac{\frac{3}{8}+\frac{7}{8}}{\frac{4}{5}}\]

(A) \(1 \)(B) \( \dfrac{25}{16} \)(C) \(2 \)(D) \(\dfrac{43}{20} \)(E) \(\dfrac{47}{16} \)

4

这个图形中有多少个三角形？（有些三角形可能与其他三角形重叠。）

A\(9\) B\(8\) C\(7\) D\(6\) E\(5\)

English

How many triangles are in this figure? (Some triangles may overlap other triangles.)

(A) \(9 \)(B) \(8 \)(C) \(7 \)(D) \(6 \)(E) \(5 \)

5

下列哪个数最大？

A\(9.12344\) B\(9.123\overline{4}\) C\(9.12\overline{34}\) D\(9.1\overline{234}\) E\(9.\overline{1234}\)

English

Which of the following numbers is largest?

(A) \(9.12344 \)(B) \(9.123\overline{4} \)(C) \(9.12\overline{34} \)(D) \(9.1\overline{234} \)(E) \(9.\overline{1234} \)

6

点在水平方向和竖直方向上都相隔一个单位。这个多边形围成的面积是多少平方单位？

A\(5\) B\(6\) C\(7\) D\(8\) E\(9\)

English

Dots are spaced one unit apart, horizontally and vertically. The number of square units enclosed by the polygon is

(A) \(5 \)(B) \(6 \)(C) \(7 \)(D) \(8 \)(E) \(9 \)

7

\(100\times 19.98\times 1.998\times 1000=\)

A\((1.998)^2\) B\((19.98)^2\) C\((199.8)^2\) D\((1998)^2\) E\((19980)^2\)

English

\(100\times 19.98\times 1.998\times 1000=\)

(A) \((1.998)^2 \)(B) \((19.98)^2 \)(C) \((199.8)^2 \)(D) \((1998)^2 \)(E) \((19980)^2 \)

8

一个儿童浅水池中有 \(200\) 加仑水。如果水以每天 \(0.5\) 加仑的速度蒸发，且没有其他水加入或取出，那么 \(30\) 天后池中还有多少加仑水？

A\(140\) B\(170\) C\(185\) D\(198.5\) E\(199.85\)

English

A child’s wading pool contains \(200\) gallons of water. If water evaporates at the rate of \(0.5\) gallons per day and no other water is added or removed, how many gallons of water will be in the pool after \(30\) days?

(A) \(140 \)(B) \(170 \)(C) \(185 \)(D) \(198.5 \)(E) \(199.85 \)

9

促销时，店主把一条 \(\$10\) 围巾降价 \(20\%\)。后来价格再次降低，这次降低为折后价的一半。现在价格是

A\(\$2.00\) B\(\$3.75\) C\(\$4.00\) D\(\$4.90\) E\(\$6.40\)

English

For a sale, a store owner reduces the price of a \(\$10\) scarf by \(20\%.\) Later the price is lowered again, this time by one-half the reduced price. The price is now

(A) \( \$2.00 \)(B) \(\$3.75 \)(C) \(\$4.00 \)(D) \(\$4.90 \)(E) \(\$6.40 \)

10

字母 \(W\)、\(X\)、\(Y\) 和 \(Z\) 分别表示集合 \(\{1,2,3,4\}\) 中不同的整数，但顺序不一定如此。它们满足 \[\dfrac{W}{X}-\dfrac{Y}{Z}=1\] \(W\) 与 \(Y\) 的和是多少？

A\(3\) B\(4\) C\(5\) D\(6\) E\(7\)

English

Each of the letters \(W,\) \(X,\) \(Y,\) and \(Z\) represents a different integer in the set \(\{1,2,3,4\},\) but not necessarily in that order. They satisfy \[\dfrac{W}{X}-\dfrac{Y}{Z}=1\] What is the sum of \(W\) and \(Y?\)

(A) \(3 \)(B) \(4 \)(C) \(5 \)(D) \(6 \)(E) \(7 \)

11

Harry 有 \(3\) 个姐妹和 \(5\) 个兄弟。他的姐妹 Harriet 有 \(S\) 个姐妹和 \(B\) 个兄弟。\(S\) 与 \(B\) 的乘积是多少？

A\(8\) B\(10\) C\(12\) D\(15\) E\(18\)

English

Harry has \(3\) sisters and \(5\) brothers. His sister Harriet has \(S\) sisters and \(B\) brothers. What is the product of \(S\) and \(B?\)

(A) \(8 \)(B) \(10 \)(C) \(12 \)(D) \(15 \)(E) \(18 \)

12

下列表达式的值是多少？\[ \begin{aligned} &2\left(1-\dfrac{1}{2}\right)+3\left(1-\dfrac{1}{3}\right) \\ &\quad {}+4\left(1-\dfrac{1}{4}\right)+\cdots \\ &\quad {}+10\left(1-\dfrac{1}{10}\right) \end{aligned} \]

A\(45\) B\(49\) C\(50\) D\(54\) E\(55\)

English

What is the value of the following expression? \[ \begin{aligned} &2\left(1-\dfrac{1}{2}\right)+3\left(1-\dfrac{1}{3}\right) \\ &\quad {}+4\left(1-\dfrac{1}{4}\right)+\cdots \\ &\quad {}+10\left(1-\dfrac{1}{10}\right) \end{aligned} \]

(A) \(45 \)(B) \(49 \)(C) \(50 \)(D) \(54 \)(E) \(55 \)

13

阴影正方形的面积与大正方形面积之比是多少？（图按比例绘制。）

A\(\dfrac{1}{6}\) B\(\dfrac{1}{7}\) C\(\dfrac{1}{8}\) D\(\dfrac{1}{12}\) E\(\dfrac{1}{16}\)

English

What is the ratio of the area of the shaded square to the area of the large square? (The figure is drawn to scale.)

(A) \(\dfrac{1}{6} \)(B) \(\dfrac{1}{7} \)(C) \(\dfrac{1}{8} \)(D) \(\dfrac{1}{12} \)(E) \(\dfrac{1}{16} \)

14

在 Annville Junior High School，数学俱乐部中 \(30\%\) 的学生也参加科学俱乐部，科学俱乐部中 \(80\%\) 的学生也参加数学俱乐部。科学俱乐部有 \(15\) 名学生。数学俱乐部有多少名学生？

A\(12\) B\(15\) C\(30\) D\(36\) E\(40\)

English

At Annville Junior High School, \(30\%\) of the students in the Math Club are in the Science Club, and \(80\%\) of the students in the Science Club are in the Math Club. There are \(15\) students in the Science Club. How many students are in the Math Club?

(A) \(12 \)(B) \(15 \)(C) \(30 \)(D) \(36 \)(E) \(40 \)

15

第 \(15\)，\(16\) 和 \(17\) 题都参考以下内容：

不要让群岛太拥挤

在 Irenic Sea 的正中央，有美丽的 Nisos Isles。\(1998\) 年这些岛上的人口只有 \(200\)，但人口每 \(25\) 年增长为原来的三倍。Queen Irene 规定，岛上每位居民至少必须有 \(1.5\) 平方英里的土地。Nisos Isles 的总面积是 \(24{,}900\) 平方英里。

估计 \(2050\) 年 Nisos 的人口。

A\(600\) B\(800\) C\(1000\) D\(2000\) E\(3000\)

English

Problems \(15,\) \(16,\) and \(17\) all refer to the following:

Don’t Crowd The Isles

In the very center of the Irenic Sea lie the beautiful Nisos Isles. In \(1998\) the number of people on these islands is only \(200,\) but the population triples every \(25\) years. Queen Irene has decreed that there must be at least \(1.5\) square miles for every person living in the Isles. The total area of the Nisos Isles is \(24{,}900\) square miles.

Estimate the population of Nisos in the year \(2050.\)

(A) \(600 \)(B) \(800 \)(C) \(1000 \)(D) \(2000 \)(E) \(3000 \)

16

估计 Nisos 人口约为 \(6000\) 的年份。

A\(2050\) B\(2075\) C\(2100\) D\(2125\) E\(2150\)

English

Estimate the year in which the population of Nisos will be approximately \(6000.\)

(A) \(2050 \)(B) \(2075 \)(C) \(2100 \)(D) \(2125 \)(E) \(2150 \)

17

从 \(1998\) 年起，大约多少年后，Nisos 的人口会达到 Queen Irene 宣称这些岛屿可以承载的人口数量？

A\(50\) 年 B\(75\) 年 C\(100\) 年 D\(125\) 年 E\(150\) 年

English

In how many years, approximately, from \(1998\) will the population of Nisos be as much as Queen Irene has proclaimed that the islands can support?

(A) \(50\) years(B) \(75\) years(C) \(100\) years(D) \(125\) years(E) \(150\) years

18

如下图所示，一张长方形纸先从下往上折，再从左往右折，最后在 \(X\) 处打一个孔。展开后纸是什么样子？

A B C D E

English

As indicated by the diagram below, a rectangular piece of paper is folded bottom to top, then left to right, and finally, a hole is punched at \(X.\) What does the paper look like when unfolded?

(A) (B) (C) (D) (E)

19

Tamika 从集合 \(\{8,9,10\}\) 中随机选择两个不同的数并相加。Carlos 从集合 \(\{3,5,6\}\) 中随机选择两个不同的数并相乘。Tamika 的结果大于 Carlos 的结果的概率是多少？

A\(\dfrac{4}{9}\) B\(\dfrac{5}{9}\) C\(\dfrac{1}{2}\) D\(\dfrac{1}{3}\) E\(\dfrac{2}{3}\)

English

Tamika selects two different numbers at random from the set \(\{8,9,10\}\) and adds them. Carlos takes two different numbers at random from the set \(\{3,5,6\}\) and multiplies them. What is the probability that Tamika’s result is greater than Carlos’ result?

(A) \(\dfrac{4}{9} \)(B) \(\dfrac{5}{9} \)(C) \(\dfrac{1}{2} \)(D) \(\dfrac{1}{3} \)(E) \(\dfrac{2}{3} \)

20

设 \(PQRS\) 是一张正方形纸。把 \(P\) 折到 \(R\)，再把 \(Q\) 折到 \(S\)。所得图形的面积为 \(9\) 平方英寸。求正方形 \(PQRS\) 的周长。

A\(9\) B\(16\) C\(18\) D\(24\) E\(36\)

English

Let \(PQRS\) be a square piece of paper. \(P\) is folded onto \(R\) and then \(Q\) is folded onto \(S.\) The area of the resulting figure is \(9\) square inches. Find the perimeter of square \(PQRS.\)

(A) \(9 \)(B) \(16 \)(C) \(18 \)(D) \(24 \)(E) \(36 \)

21

一个 \(4\times 4\times 4\) 的立方体盒子内装有 \(64\) 个完全填满盒子的相同小立方体。有多少个小立方体接触盒子的侧面或底面？

A\(48\) B\(52\) C\(60\) D\(64\) E\(80\)

English

A \(4\times 4\times 4\) cubical box contains \(64\) identical small cubes that exactly fill the box. How many of these small cubes touch a side or the bottom of the box?

(A) \(48 \)(B) \(52 \)(C) \(60 \)(D) \(64 \)(E) \(80 \)

22

Terri 按三条规则生成一个正整数序列。她从一个正整数开始，然后对结果应用合适的规则，并持续这样做。

规则 \(1\)：如果整数小于 \(10\)，则乘以 \(9\)。

规则 \(2\)：如果整数是偶数且大于 \(9\)，则除以 \(2\)。

规则 \(3\)：如果整数是奇数且大于 \(9\)，则减去 \(5\)。

例如，样例序列为 \(23\)，\(18\)，\(9\)，\(81\)，\(76\)，\(\ldots\)。

求以 \(98\)、\(49\)、\(\ldots\) 开头的这个序列的第 \(98\) 项。

A\(6\) B\(11\) C\(22\) D\(27\) E\(54\)

English

Terri produces a sequence of positive integers by following three rules. She starts with a positive integer, then applies the appropriate rule to the result, and continues in this fashion.

Rule \(1:\) If the integer is less than \(10,\) multiply it by \(9.\)

Rule \(2:\) If the integer is even and greater than \(9,\) divide it by \(2.\)

Rule \(3:\) If the integer is odd and greater than \(9,\) subtract \(5\) from it.

For example, consider the sample sequence \(23,\) \(18,\) \(9,\) \(81,\) \(76,\) \(\ldots.\)

Find the \(98\)th term of the sequence that begins \(98,\) \(49,\) \(\ldots.\)

(A) \(6 \)(B) \(11 \)(C) \(22 \)(D) \(27 \)(E) \(54 \)

23

如果图中的模式继续，第八个三角形内部有多少比例会被涂色？

A\(\dfrac{3}{8}\) B\(\dfrac{5}{27}\) C\(\dfrac{7}{16}\) D\(\dfrac{9}{16}\) E\(\dfrac{11}{45}\)

English

If the pattern in the diagram continues, what fraction of the interior would be shaded in the eighth triangle?

(A) \(\dfrac{3}{8} \)(B) \(\dfrac{5}{27} \)(C) \(\dfrac{7}{16} \)(D) \(\dfrac{9}{16} \)(E) \(\dfrac{11}{45} \)

24

一个长方形板有 \(8\) 列小方格，从左上角开始编号，第一行从左到右编号为 \(1\) 到 \(8\)，第二行为 \(9\) 到 \(16\)，依此类推。一个学生涂色方格 \(1\)，然后跳过一个方格涂色方格 \(3\)，跳过两个方格涂色方格 \(6\)，跳过 \(3\) 个方格涂色方格 \(10\)，并继续这样做，直到每一列至少有一个涂色方格。

第一次达到这个结果时，被涂色方格的编号是多少？

A\(36\) B\(64\) C\(78\) D\(91\) E\(120\)

English

A rectangular board of \(8\) columns has squares numbered beginning in the upper left corner and moving left to right so row one is numbered \(1\) through \(8,\) row two is \(9\) through \(16,\) and so on. A student shades square \(1,\) then skips one square and shades square \(3,\) skips two squares and shades square \(6,\) skips \(3\) squares and shades square \(10,\) and continues in this way until there is at least one shaded square in each column.

What is the number of the shaded square that first achieves this result?

(A) \(36 \)(B) \(64 \)(C) \(78 \)(D) \(91 \)(E) \(120 \)

25

三位慷慨的朋友各有一些现金，并按如下方式重新分配：Ami 给 Jan 和 Toy 足够的钱，使他们各自的钱数加倍。然后 Jan 给 Ami 和 Toy 足够的钱，使他们各自的钱数加倍。最后 Toy 给 Ami 和 Jan 足够的钱，使他们各自的钱数加倍。如果 Toy 开始时有 \(\$36\)，结束时也有 \(\$36\)，那么三位朋友一共有多少钱？

A\(\$108\) B\(\$180\) C\(\$216\) D\(\$252\) E\(\$288\)

English

Three generous friends, each with some cash, redistribute their money as follows: Ami gives enough money to Jan and Toy to double the amount that each has. Jan then gives enough to Ami and Toy to double their amounts. Finally, Toy gives Ami and Jan enough to double their amounts. If Toy has \(\$36\) when they begin and \(\$36\) when they end, what is the total amount that all three friends have?

(A) \(\$108 \)(B) \(\$180 \)(C) \(\$216 \)(D) \(\$252 \)(E) \(\$288 \)