1988 AMC 12 真题（P版）

1

\(\sqrt8+\sqrt{18}=\)

A\(\sqrt{20}\) B\(2(\sqrt2+\sqrt3)\) C\(7\) D\(5\sqrt2\) E\(2\sqrt{13}\)

English

\(\sqrt8+\sqrt{18}=\)

(A) \(\sqrt{20}\)(B) \(2(\sqrt2+\sqrt3)\)(C) \(7\)(D) \(5\sqrt2\)(E) \(2\sqrt{13}\)

2

三角形 \(ABC\) 与 \(XYZ\) 相似，其中 \(A\) 对应 \(X\)，\(B\) 对应 \(Y\)。若 \(AB=3\)、\(BC=4\)、\(XY=5\)，则 \(YZ\) 为

A\(3\frac34\) B\(6\) C\(6\frac14\) D\(6\frac23\) E\(8\)

English

Triangles \(ABC\) and \(XYZ\) are similar, with \(A\) corresponding to \(X\) and \(B\) to \(Y.\) If \(AB=3,\) \(BC=4,\) and \(XY=5,\) then \(YZ\) is

(A) \(3\frac34\)(B) \(6\)(C) \(6\frac14\)(D) \(6\frac23\)(E) \(8\)

3

四条长 \(10\)、宽 \(1\) 的矩形纸条平放在桌面上，并按图示方式两两垂直交叠。桌面上被覆盖的面积是多少？

A\(36\) B\(40\) C\(44\) D\(98\) E\(100\)

English

Four rectangular paper strips of length \(10\) and width \(1\) are put flat on a table and overlap perpendicularly as shown. How much area of the table is covered?

(A) \(36\)(B) \(40\)(C) \(44\)(D) \(98\)(E) \(100\)

4

直线 \(\frac{x}{3}+\frac{y}{2}=1\) 的斜率为

A\(-\frac32\) B\(-\frac23\) C\(\frac13\) D\(\frac23\) E\(\frac32\)

English

The slope of the line \(\frac{x}{3}+\frac{y}{2}=1\) is

(A) \(-\frac32\)(B) \(-\frac23\)(C) \(\frac13\)(D) \(\frac23\)(E) \(\frac32\)

5

若 \(b\) 和 \(c\) 是常数，且 \[(x+2)(x+b)=x^2+cx+6\text{，}\]则 \(c\) 为

A\(-5\) B\(-3\) C\(-1\) D\(3\) E\(5\)

English

If \(b\) and \(c\) are constants and \[(x+2)(x+b)=x^2+cx+6,\] then \(c\) is

(A) \(-5\)(B) \(-3\)(C) \(-1\)(D) \(3\)(E) \(5\)

6

一个图形是等角平行四边形的充要条件是它为

A矩形 B正多边形 C菱形 D正方形 E梯形

English

A figure is an equiangular parallelogram if and only if it is a

(A) rectangle(B) regular polygon(C) rhombus(D) square(E) trapezoid

7

估算通过一条通信信道发送 \(60\) 个数据块所需的时间。每个数据块由 \(512\) 个“数据片”组成，信道每秒可传输 \(120\) 个数据片。

A\(0.04\) 秒 B\(0.4\) 秒 C\(4\) 秒 D\(4\) 分钟 E\(4\) 小时

English

Estimate the time it takes to send \(60\) blocks of data over a communications channel if each block consists of \(512\) “chunks” and the channel can transmit \(120\) chunks per second.

(A) \(0.04\) seconds(B) \(0.4\) seconds(C) \(4\) seconds(D) \(4\) minutes(E) \(4\) hours

8

若 \(\frac ba=2\)，且 \(\frac cb=3\)，则 \(a+b\) 与 \(b+c\) 之比是多少？

A\(\frac13\) B\(\frac38\) C\(\frac35\) D\(\frac23\) E\(\frac34\)

English

If \(\frac ba=2\) and \(\frac cb=3,\) what is the ratio of \(a+b\) to \(b+c?\)

(A) \(\frac13\)(B) \(\frac38\)(C) \(\frac35\)(D) \(\frac23\)(E) \(\frac34\)

9

一张 \(8'\times10'\) 的桌子位于一间正方形房间的角落，如下面图 \(1\) 所示。主人希望把桌子移到图 \(2\) 所示的位置。房间边长为 \(S\) 英尺。若移动时不能倾斜或拆卸桌子，能完成所需移动的最小整数 \(S\) 是多少？

A\(11\) B\(12\) C\(13\) D\(14\) E\(15\)

English

An \(8'\times10'\) table sits in the corner of a square room, as in Figure \(1\) below. The owners desire to move the table to the position shown in Figure \(2.\) The side of the room is \(S\) feet. What is the smallest integer value of \(S\) for which the table can be moved as desired without tilting it or taking it apart?

(A) \(11\)(B) \(12\)(C) \(13\)(D) \(14\)(E) \(15\)

10

在一次实验中，测得某科学常数 \(C\) 为 \(2.43865\)，误差至多为 \(\pm0.00312\)。实验者希望公布一个每一位都可靠的 \(C\) 值。也就是说，无论 \(C\) 的实际值是多少，把 \(C\) 舍入到所公布的位数时，都必须得到这个公布值。实验者能公布的最精确 \(C\) 值是

A\(2\) B\(2.4\) C\(2.43\) D\(2.44\) E\(2.439\)

English

In an experiment, a scientific constant \(C\) is determined to be \(2.43865\) with an error of at most \(\pm0.00312.\) The experimenter wishes to announce a value for \(C\) in which every digit is significant. That is, whatever \(C\) is, the announced value must be the correct result when \(C\) is rounded to that number of digits. The most accurate value the experimenter can announce for \(C\) is

(A) \(2\)(B) \(2.4\)(C) \(2.43\)(D) \(2.44\)(E) \(2.439\)

11

下图每条水平线上的五个大点表示城市 \(A\)、\(B\)、\(C\)、\(D\)、\(E\) 在所标年份的人口。从 \(1970\) 年到 \(1980\) 年，哪个城市的人口增长百分比最大？

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

English

On each horizontal line in the figure below, the five large dots indicate the populations of cities \(A,\) \(B,\) \(C,\) \(D,\) and \(E\) in the year indicated. Which city had the greatest percentage increase in population from \(1970\) to \(1980?\)

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

12

将整数 \(1\) 到 \(9\) 分别写在九张纸条上，再把它们全部放入一顶帽子中。杰克随机抽出一张纸条后放回。然后吉尔再随机抽一张。杰克与吉尔所抽整数之和的个位数字最有可能是哪一个？

A\(0\) B\(1\) C\(8\) D\(9\) E每个数字的可能性都相同

English

Each integer \(1\) through \(9\) is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of these slips at random and puts it back. Then Jill picks a slip at random. Which digit is most likely to be the units digit of the sum of Jack’s integer and Jill’s integer?

(A) \(0\)(B) \(1\)(C) \(8\)(D) \(9\)(E) each digit is equally likely

13

若 \(\sin x=3\cos x\)，则 \(\sin x\cos x\) 是多少？

A\(\frac16\) B\(\frac15\) C\(\frac29\) D\(\frac14\) E\(\frac3{10}\)

English

If \(\sin x=3\cos x,\) then what is \(\sin x\cos x?\)

(A) \(\frac16\)(B) \(\frac15\)(C) \(\frac29\)(D) \(\frac14\)(E) \(\frac3{10}\)

14

对任意实数 \(a\) 和正整数 \(k\)，定义 \[ \binom ak= \frac{\begin{gathered} a(a-1)(a-2)\cdots\\ {}\cdot(a-(k-1)) \end{gathered}} {\begin{gathered} k(k-1)(k-2)\cdots\\ {}\cdot(2)(1) \end{gathered}}\text{。}\]求 \[\binom{-\frac12}{100}\div\binom{\frac12}{100}\ \text{？}\]

A\(-199\) B\(-197\) C\(-1\) D\(197\) E\(199\)

English

For any real number \(a\) and positive integer \(k,\) define \[ \binom ak= \frac{\begin{gathered} a(a-1)(a-2)\cdots\\ {}\cdot(a-(k-1)) \end{gathered}} {\begin{gathered} k(k-1)(k-2)\cdots\\ {}\cdot(2)(1) \end{gathered}}. \] What is \[\binom{-\frac12}{100}\div\binom{\frac12}{100}\ ?\]

(A) \(-199\)(B) \(-197\)(C) \(-1\)(D) \(197\)(E) \(199\)

15

若 \(a\) 和 \(b\) 是整数，且 \(x^2-x-1\) 是 \(ax^3+bx^2+1\) 的因式，则 \(b\) 为

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

English

If \(a\) and \(b\) are integers such that \(x^2-x-1\) is a factor of \(ax^3+bx^2+1,\) then \(b\) is

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

16

\(ABC\) 与 \(A'B'C'\) 是边互相平行且中心相同的等边三角形，如图所示。边 \(BC\) 与边 \(B'C'\) 之间的距离是 \(\triangle ABC\) 的高的 \(\frac16\)。\(\triangle A'B'C'\) 的面积与 \(\triangle ABC\) 的面积之比为

A\(\frac1{36}\) B\(\frac16\) C\(\frac14\) D\(\frac{\sqrt3}{4}\) E\(\frac{9+8\sqrt3}{36}\)

English

\(ABC\) and \(A'B'C'\) are equilateral triangles with parallel sides and the same center, as in the figure. The distance between side \(BC\) and side \(B'C'\) is \(\frac16\) the altitude of \(\triangle ABC.\) The ratio of the area of \(\triangle A'B'C'\) to the area of \(\triangle ABC\) is

(A) \(\frac1{36}\)(B) \(\frac16\)(C) \(\frac14\)(D) \(\frac{\sqrt3}{4}\)(E) \(\frac{9+8\sqrt3}{36}\)

17

若 \(\lvert x\rvert+x+y=10\)，且 \(x+\lvert y\rvert-y=12\)，求 \(x+y\)。

A\(-2\) B\(2\) C\(\frac{18}{5}\) D\(\frac{22}{3}\) E\(22\)

English

If \(\lvert x\rvert+x+y=10\) and \(x+\lvert y\rvert-y=12,\) find \(x+y.\)

(A) \(-2\)(B) \(2\)(C) \(\frac{18}{5}\)(D) \(\frac{22}{3}\)(E) \(22\)

18

一项职业保龄球锦标赛结束时，排名前 \(5\) 的选手进行附加赛。首先，编号 \(5\) 与编号 \(4\) 比赛。负者获得第 \(5\) 名奖金，胜者再与编号 \(3\) 比赛。这场比赛的负者获得第 \(4\) 名奖金，胜者再与编号 \(2\) 比赛。这场比赛的负者获得第 \(3\) 名奖金，胜者再与编号 \(1\) 比赛。最后一场的胜者获得第 \(1\) 名奖金，负者获得第 \(2\) 名奖金。编号 \(1\) 到编号 \(5\) 的选手共有多少种获奖名次顺序？

A\(10\) B\(16\) C\(24\) D\(120\) E以上都不是

English

At the end of a professional bowling tournament, the top \(5\) bowlers have a play-off. First #\(5\) bowls #\(4.\) The loser receives \(5\)th prize and the winner bowls #\(3\) in another game. The loser of this game receives \(4\)th prize and the winner bowls #\(2.\) The loser of this game receives \(3\)rd prize and the winner bowls #\(1.\) The winner of this game gets \(1\)st prize and the loser gets \(2\)nd prize. In how many orders can bowlers #\(1\) through #\(5\) receive the prizes?

(A) \(10\)(B) \(16\)(C) \(24\)(D) \(120\)(E) none of these

19

化简 \[ \frac{\begin{gathered} bx(a^2x^2+2a^2y^2+b^2y^2)\\ {}+ay(a^2x^2+2b^2x^2+b^2y^2) \end{gathered}}{bx+ay}\text{。}\]

A\(a^2x^2+b^2y^2\) B\((ax+by)^2\) C\((ax+by)(bx+ay)\) D\(2(a^2x^2+b^2y^2)\) E\((bx+ay)^2\)

English

Simplify \[ \frac{\begin{gathered} bx(a^2x^2+2a^2y^2+b^2y^2)\\ {}+ay(a^2x^2+2b^2x^2+b^2y^2) \end{gathered}}{bx+ay}. \]

(A) \(a^2x^2+b^2y^2\)(B) \((ax+by)^2\)(C) \((ax+by)(bx+ay)\)(D) \(2(a^2x^2+b^2y^2)\)(E) \((bx+ay)^2\)

20

在相邻的两幅图中，第一幅把一个边长为 \(2\) 的正方形分成四块，其中 \(E\) 和 \(F\) 是一对相对边的中点，且 \(AG\) 垂直于 \(BF\)。然后可将这四块重新拼成第二幅图所示的矩形。该矩形高与底边之比 \(\frac{XY}{YZ}\) 为

A\(4\) B\(1+2\sqrt3\) C\(2\sqrt5\) D\(\frac{8+4\sqrt3}{3}\) E\(5\)

English

In one of the adjoining figures a square of side \(2\) is dissected into four pieces so that \(E\) and \(F\) are the midpoints of opposite sides and \(AG\) is perpendicular to \(BF.\) These four pieces can then be reassembled into a rectangle as shown in the second figure. The ratio of height to base, \(\frac{XY}{YZ},\) in this rectangle is

(A) \(4\)(B) \(1+2\sqrt3\)(C) \(2\sqrt5\)(D) \(\frac{8+4\sqrt3}{3}\)(E) \(5\)

21

复数 \(z\) 满足 \(z+\lvert z\rvert=2+8i\)。\(\lvert z\rvert^2\) 等于多少？注：若 \(z=a+bi\)，则 \(\lvert z\rvert=\sqrt{a^2+b^2}\)。

A\(68\) B\(100\) C\(169\) D\(208\) E\(289\)

English

The complex number \(z\) satisfies \(z+\lvert z\rvert=2+8i.\) What is \(\lvert z\rvert^2?\) Note: if \(z=a+bi,\) then \(\lvert z\rvert=\sqrt{a^2+b^2}.\)

(A) \(68\)(B) \(100\)(C) \(169\)(D) \(208\)(E) \(289\)

22

共有多少个整数 \(x\)，使得以 \(10\)、\(24\) 和 \(x\) 为边长的三角形的所有内角均为锐角？

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

English

For how many integers \(x\) does a triangle with side lengths \(10,\) \(24,\) and \(x\) have all its angles acute?

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

23

四面体 \(ABCD\) 的六条棱长分别为 \(7\)、\(13\)、\(18\)、\(27\)、\(36\) 和 \(41\) 个单位。若棱 \(AB\) 的长度为 \(41\)，则棱 \(CD\) 的长度为

A\(7\) B\(13\) C\(18\) D\(27\) E\(36\)

English

The six edges of tetrahedron \(ABCD\) measure \(7,\) \(13,\) \(18,\) \(27,\) \(36,\) and \(41\) units. If the length of edge \(AB\) is \(41,\) then the length of edge \(CD\) is

(A) \(7\)(B) \(13\)(C) \(18\)(D) \(27\)(E) \(36\)

24

一个等腰梯形外切于一个圆。梯形的长底为 \(16\)，其中一个底角为 \(\arcsin(0.8)\)。求梯形的面积。

A\(72\) B\(75\) C\(80\) D\(90\) E不能唯一确定

English

An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is \(16,\) and one of the base angles is \(\arcsin(0.8).\) Find the area of the trapezoid.

(A) \(72\)(B) \(75\)(C) \(80\)(D) \(90\)(E) not uniquely determined

25

\(X\)、\(Y\) 和 \(Z\) 是两两不相交的人员集合。集合 \(X\)、\(Y\)、\(Z\)、\(X\cup Y\)、\(X\cup Z\) 和 \(Y\cup Z\) 中人员的平均年龄如下表所示。

求集合 \(X\cup Y\cup Z\) 中人员的平均年龄。

A\(33\) B\(33.5\) C\(33.66\) D\(33.833\) E\(34\)

English

\(X,\) \(Y,\) and \(Z\) are pairwise disjoint sets of people. The average ages of people in the sets \(X,\) \(Y,\) \(Z,\) \(X\cup Y,\) \(X\cup Z,\) and \(Y\cup Z\) are given in the table below.

Find the average age of the people in the set \(X\cup Y\cup Z.\)

(A) \(33\)(B) \(33.5\)(C) \(33.66\)(D) \(33.833\)(E) \(34\)

26

设 \(p\) 和 \(q\) 是满足下式的正数： \[ \begin{aligned} \log_9(p)&=\log_{12}(q)\\ &=\log_{16}(p+q) \end{aligned} \] 求 \(\frac{q}{p}\) 的值。

A\(\frac43\) B\(\frac12(1+\sqrt3)\) C\(\frac85\) D\(\frac12(1+\sqrt5)\) E\(\frac{16}{9}\)

English

Suppose that \(p\) and \(q\) are positive numbers for which \[ \begin{aligned} \log_9(p)&=\log_{12}(q)\\ &=\log_{16}(p+q). \end{aligned} \] What is the value of \(\frac{q}{p}?\)

(A) \(\frac43\)(B) \(\frac12(1+\sqrt3)\)(C) \(\frac85\)(D) \(\frac12(1+\sqrt5)\)(E) \(\frac{16}{9}\)

27

图中，\(AB\perp BC\)、\(BC\perp CD\)，且 \(BC\) 与以 \(O\) 为圆心、\(AD\) 为直径的圆相切。在以下哪一种情况下，\(ABCD\) 的面积为整数？

A\(AB=3\)，\(CD=1\) B\(AB=5\)，\(CD=2\) C\(AB=7\)，\(CD=3\) D\(AB=9\)，\(CD=4\) E\(AB=11\)，\(CD=5\)

English

In the figure, \(AB\perp BC,\) \(BC\perp CD,\) and \(BC\) is tangent to the circle with center \(O\) and diameter \(AD.\) In which one of the following cases is the area of \(ABCD\) an integer?

(A) \(AB=3,\) \(CD=1\)(B) \(AB=5,\) \(CD=2\)(C) \(AB=7,\) \(CD=3\)(D) \(AB=9,\) \(CD=4\)(E) \(AB=11,\) \(CD=5\)

28

一枚不均匀硬币在一次投掷中正面朝上的概率为 \(p\)。设 \(w\) 为独立投掷这枚硬币 \(5\) 次时恰好出现 \(3\) 次正面的概率。若 \(w=\frac{144}{625}\)，则

A\(p\) 必须为 \(\frac{2}{5}\) B\(p\) 必须为 \(\frac{3}{5}\) C\(p\) 必须大于 \(\frac{3}{5}\) D\(p\) 不能唯一确定 E不存在任何 \(p\)，使 \(w=\frac{144}{625}\) 成立

English

An unfair coin has probability \(p\) of coming up heads on a single toss. Let \(w\) be the probability that, in \(5\) independent tosses of this coin, heads come up exactly \(3\) times. If \(w=\frac{144}{625},\) then

(A) \(p\) must be \(\frac{2}{5}\)(B) \(p\) must be \(\frac{3}{5}\)(C) \(p\) must be greater than \(\frac{3}{5}\)(D) \(p\) is not uniquely determined(E) there is no value of \(p\) for which \(w=\frac{144}{625}\)

29

把三位朋友的体重 \((y)\) 对身高 \((x)\) 作图，得到点 \((x_1,y_1)\)、\((x_2,y_2)\)、\((x_3,y_3)\)。若 \[ \begin{aligned} x_1&\lt x_2\lt x_3,\\ x_3-x_2&=x_2-x_1 \end{aligned}\text{，}\]则下列哪一个式子一定是数据最佳拟合直线的斜率？“最佳拟合”是指数据点到直线的竖直距离的平方和小于对任何其他直线所得的平方和。

A\(\frac{y_3-y_1}{x_3-x_1}\) B\(\frac{(y_2-y_1)-(y_3-y_2)}{x_3-x_1}\) C\(\frac{2y_3-y_1-y_2}{2x_3-x_1-x_2}\) D\(\frac{y_2-y_1}{x_2-x_1}+\frac{y_3-y_2}{x_3-x_2}\) E以上均不是

English

You plot weight \((y)\) against height \((x)\) for three of your friends and obtain the points \((x_1,y_1),\) \((x_2,y_2),\) \((x_3,y_3).\) If \[ \begin{aligned} x_1&\lt x_2\lt x_3,\\ x_3-x_2&=x_2-x_1, \end{aligned} \] which of the following is necessarily the slope of the line which best fits the data? “Best fits” means that the sum of the squares of the vertical distances from the data points to the line is smaller than for any other line.

(A) \(\frac{y_3-y_1}{x_3-x_1}\)(B) \(\frac{(y_2-y_1)-(y_3-y_2)}{x_3-x_1}\)(C) \(\frac{2y_3-y_1-y_2}{2x_3-x_1-x_2}\)(D) \(\frac{y_2-y_1}{x_2-x_1}+\frac{y_3-y_2}{x_3-x_2}\)(E) none of these

30

设 \(f(x)=4x-x^2\)。给定 \(x_0\)，考虑由 \(x_n=f(x_{n-1})\)（对所有 \(n\ge1\)）定义的数列。对于多少个实数 \(x_0\)，数列 \(x_0\)、\(x_1\)、\(x_2\)、\(\ldots\) 只取有限多个不同的值？

A\(0\) B\(1\) 或 \(2\) C\(3\)、\(4\)、\(5\) 或 \(6\) D多于 \(6\) 个但只有有限多个 E无限多个

English

Let \(f(x)=4x-x^2.\) Given \(x_0,\) consider the sequence defined by \(x_n=f(x_{n-1})\) for all \(n\ge1.\) For how many real numbers \(x_0\) will the sequence \(x_0,\) \(x_1,\) \(x_2,\) \(\ldots\) take on only a finite number of different values?

(A) \(0\)(B) \(1\) or \(2\)(C) \(3,\) \(4,\) \(5\) or \(6\)(D) more than \(6\) but finitely many(E) infinitely many