1963 AMC 12 真题（P版）

1

以下哪一点不在函数 \(y=\dfrac{x}{x+1}\) 的图像上？

A\((0,0)\) B\(\left(-\dfrac12,-1\right)\) C\(\left(\dfrac12,\dfrac13\right)\) D\((-1,1)\) E\((-2,2)\)

English

Which one of the following points is not on the graph of \(y=\dfrac{x}{x+1}?\)

(A) \((0,0)\)(B) \(\left(-\dfrac12,-1\right)\)(C) \(\left(\dfrac12,\dfrac13\right)\)(D) \((-1,1)\)(E) \((-2,2)\)

2

设 \(n=x-y^{x-y}\)。求 \(n\) 在 \(x=2\) 且 \(y=-2\) 时的值。

A\(-14\) B\(0\) C\(1\) D\(18\) E\(256\)

English

Let \(n=x-y^{x-y}.\) Find \(n\) when \(x=2\) and \(y=-2.\)

(A) \(-14\)(B) \(0\)(C) \(1\)(D) \(18\)(E) \(256\)

3

若 \(x+1\) 的倒数是 \(x-1\)，则 \(x\) 等于：

A\(0\) B\(1\) C\(-1\) D\(\pm1\) E以上都不是

English

If the reciprocal of \(x+1\) is \(x-1,\) then \(x\) equals:

(A) \(0\)(B) \(1\)(C) \(-1\)(D) \(\pm1\)(E) none of these

4

当 \(k\) 取何值时，方程 \(y=x^2\) 与 \(y=3x+k\) 有两个相同的解？

A\(\dfrac49\) B\(-\dfrac49\) C\(\dfrac94\) D\(-\dfrac94\) E\(\pm\dfrac94\)

English

For what value(s) of \(k\) does the pair of equations \(y=x^2\) and \(y=3x+k\) have two identical solutions?

(A) \(\dfrac49\)(B) \(-\dfrac49\)(C) \(\dfrac94\)(D) \(-\dfrac94\)(E) \(\pm\dfrac94\)

5

若 \(x\) 和 \(\log_{10}x\) 都是实数，并且 \(\log_{10}x\lt0\)，则：

A\(x\lt0\) B\(-1\lt x\lt1\) C\(0\lt x\leq1\) D\(-1\lt x\lt0\) E\(0\lt x\lt1\)

English

If \(x\) and \(\log_{10}x\) are real numbers and \(\log_{10}x\lt0,\) then:

(A) \(x\lt0\)(B) \(-1\lt x\lt1\)(C) \(0\lt x\leq1\)(D) \(-1\lt x\lt0\)(E) \(0\lt x\lt1\)

6

三角形 \(BAD\) 在 \(B\) 点为直角。线段 \(AD\) 上有一点 \(C\)，满足 \(AC=CD\) 且 \(AB=BC\)。角 \(DAB\) 的度数是：

A\(67\dfrac12\) B\(60\) C\(45\) D\(30\) E\(22\dfrac12\)

English

Triangle \(BAD\) is right-angled at \(B.\) On \(AD\) there is a point \(C\) for which \(AC=CD\) and \(AB=BC.\) The magnitude of angle \(DAB,\) in degrees, is:

(A) \(67\dfrac12\)(B) \(60\)(C) \(45\)(D) \(30\)(E) \(22\dfrac12\)

7

给出以下四个方程：

\[ \begin{aligned} (1)\quad&3y-2x=12,\\ (2)\quad&-2x-3y=10,\\ (3)\quad&3y+2x=12,\\ (4)\quad&2y+3x=10 \end{aligned}\text{。}\]

其中表示两条垂直直线的一组是：

A\((1)\) 和 \((4)\) B\((1)\) 和 \((3)\) C\((1)\) 和 \((2)\) D\((2)\) 和 \((4)\) E\((2)\) 和 \((3)\)

English

Given the four equations:

\[ \begin{aligned} (1)\quad&3y-2x=12,\\ (2)\quad&-2x-3y=10,\\ (3)\quad&3y+2x=12,\\ (4)\quad&2y+3x=10. \end{aligned} \]

The pair representing perpendicular lines is:

(A) \((1)\) and \((4)\)(B) \((1)\) and \((3)\)(C) \((1)\) and \((2)\)(D) \((2)\) and \((4)\)(E) \((2)\) and \((3)\)

8

最小正整数 \(x\) 是多少，能使 \(1260x=N^3\) 成立？其中 \(N\) 是整数。

A\(1050\) B\(1260\) C\(1260^2\) D\(7350\) E\(44{,}100\)

English

The smallest positive integer \(x\) for which \(1260x=N^3,\) where \(N\) is an integer, is:

(A) \(1050\)(B) \(1260\)(C) \(1260^2\)(D) \(7350\)(E) \(44{,}100\)

9

在 \(\left(a-\dfrac1{\sqrt a}\right)^7\) 的展开式中，\(a^{-\frac{1}{2}}\) 的系数是：

A\(-7\) B\(7\) C\(-21\) D\(21\) E\(35\)

English

In the expansion of \(\left(a-\dfrac1{\sqrt a}\right)^7\) the coefficient of \(a^{-\frac{1}{2}}\) is:

(A) \(-7\)(B) \(7\)(C) \(-21\)(D) \(21\)(E) \(35\)

10

在正方形内部取一点 \(P\)，该正方形的边长为 \(a\)，所取点到两个相邻顶点的距离以及到这两个顶点对边的距离都相等。若 \(d\) 表示这个公共距离，则 \(d\) 等于：

A\(\dfrac{3a}{5}\) B\(\dfrac{5a}{8}\) C\(\dfrac{3a}{8}\) D\(\dfrac{a\sqrt2}{2}\) E\(\dfrac a2\)

English

Point \(P\) is taken interior to a square with side-length \(a\) and such that it is equally distant from two consecutive vertices and from the side opposite these vertices. If \(d\) represents the common distance, then \(d\) equals:

(A) \(\dfrac{3a}{5}\)(B) \(\dfrac{5a}{8}\)(C) \(\dfrac{3a}{8}\)(D) \(\dfrac{a\sqrt2}{2}\)(E) \(\dfrac a2\)

11

一组 \(50\) 个数的算术平均数是 \(38\)。若从中去掉 \(45\) 和 \(55\) 这两个数，剩余各数的算术平均数是：

A\(38.5\) B\(37.5\) C\(37\) D\(36.5\) E\(36\)

English

The arithmetic mean of a set of \(50\) numbers is \(38.\) If two numbers of the set, namely \(45\) and \(55,\) are discarded, the arithmetic mean of the remaining set of numbers is:

(A) \(38.5\)(B) \(37.5\)(C) \(37\)(D) \(36.5\)(E) \(36\)

12

平行四边形 \(PQRS\) 的三个顶点是 \(P(-3,-2)\)、\(Q(1,-5)\)、\(R(9,1)\)，其中 \(P\) 与 \(R\) 是对角顶点。顶点 \(S\) 的两个坐标之和是：

A\(13\) B\(12\) C\(11\) D\(10\) E\(9\)

English

Three vertices of parallelogram \(PQRS\) are \(P(-3,-2),\) \(Q(1,-5),\) \(R(9,1)\) with \(P\) and \(R\) diagonally opposite. The sum of the coordinates of vertex \(S\) is:

(A) \(13\)(B) \(12\)(C) \(11\)(D) \(10\)(E) \(9\)

13

若 \(2^a+2^b=3^c+3^d\)，则整数 \(a\)、\(b\)、\(c\)、\(d\) 中最多有几个可以是负数？

A\(4\) B\(3\) C\(2\) D\(1\) E\(0\)

English

If \(2^a+2^b=3^c+3^d,\) the number of integers \(a,\) \(b,\) \(c,\) \(d\) which can possibly be negative is, at most:

(A) \(4\)(B) \(3\)(C) \(2\)(D) \(1\)(E) \(0\)

14

给定方程 \(x^2+kx+6=0\) 和 \(x^2-kx+6=0\)。如果适当排列两方程的根后，第二个方程的每个根都比第一个方程的对应根大 \(5\)，那么 \(k\) 等于：

A\(5\) B\(-5\) C\(7\) D\(-7\) E以上都不是

English

Given the equations \(x^2+kx+6=0\) and \(x^2-kx+6=0.\) If, when the roots of the equations are suitably listed, each root of the second equation is \(5\) more than the corresponding root of the first equation, then \(k\) equals:

(A) \(5\)(B) \(-5\)(C) \(7\)(D) \(-7\)(E) none of these

15

一个圆内切于等边三角形，一个正方形内接于该圆。三角形面积与正方形面积之比是：

A\(\sqrt3:1\) B\(\sqrt3:\sqrt2\) C\(3\sqrt3:2\) D\(3:\sqrt2\) E\(3:2\sqrt2\)

English

A circle is inscribed in an equilateral triangle, and a square is inscribed in the circle. The ratio of the area of the triangle to the area of the square is:

(A) \(\sqrt3:1\)(B) \(\sqrt3:\sqrt2\)(C) \(3\sqrt3:2\)(D) \(3:\sqrt2\)(E) \(3:2\sqrt2\)

16

三个非零数 \(a\)、\(b\)、\(c\) 成等差数列。把 \(a\) 增加 \(1\) 或把 \(c\) 增加 \(2\)，所得三个数都成等比数列。则 \(b\) 等于：

A\(16\) B\(14\) C\(12\) D\(10\) E\(8\)

English

Three numbers \(a,\) \(b,\) \(c,\) none zero, form an arithmetic progression. Increasing \(a\) by \(1\) or increasing \(c\) by \(2\) results in a geometric progression. Then \(b\) equals:

(A) \(16\)(B) \(14\)(C) \(12\)(D) \(10\)(E) \(8\)

17

表达式

\[ \frac{\dfrac{a}{a+y}+\dfrac{y}{a-y}} {\dfrac{y}{a+y}-\dfrac{a}{a-y}}\text{，}\]

其中 \(a\) 为实数且 \(a\ne0\)，对于下列哪些取值，表达式的值为 \(-1\)？

A除两个实数值外的所有 \(y\) B仅两个实数值的 \(y\) C所有实数值的 \(y\) D仅一个实数值的 \(y\) E没有实数值的 \(y\)

English

The expression

\[ \frac{\dfrac{a}{a+y}+\dfrac{y}{a-y}} {\dfrac{y}{a+y}-\dfrac{a}{a-y}}, \]

\(a\) real, \(a\ne0,\) has the value \(-1\) for:

(A) all but two real values of \(y\)(B) only two real values of \(y\)(C) all real values of \(y\)(D) only one real value of \(y\)(E) no real values of \(y\)

18

弦 \(EF\) 是弦 \(BC\) 的垂直平分线，并与其交于 \(M\)。在 \(B\) 与 \(M\) 之间取一点 \(U\)，延长 \(EU\) 与圆交于 \(A\)。对于任意如此选取的 \(U\)，三角形 \(EUM\) 与下列哪个三角形相似？

A\(EFA\) B\(EFC\) C\(ABM\) D\(ABU\) E\(FMC\)

English

Chord \(EF\) is the perpendicular bisector of chord \(BC,\) intersecting it in \(M.\) Between \(B\) and \(M\) point \(U\) is taken, and \(EU\) extended meets the circle in \(A.\) Then, for any selection of \(U,\) as described, triangle \(EUM\) is similar to triangle:

(A) \(EFA\)(B) \(EFC\)(C) \(ABM\)(D) \(ABU\)(E) \(FMC\)

19

清点 \(n\) 个彩球，其中一些是红球，另一些是黑球。最先清点的 \(50\) 个球中有 \(49\) 个红球；此后每清点 \(8\) 个球，就有 \(7\) 个红球。如果清点的所有球中至少有 \(90\%\) 是红球，那么 \(n\) 的最大值是：

A\(225\) B\(210\) C\(200\) D\(180\) E\(175\)

English

In counting \(n\) colored balls, some red and some black, it was found that \(49\) of the first \(50\) counted were red. Thereafter, \(7\) out of every \(8\) counted were red. If, in all, \(90\%\) or more of the balls counted were red, the maximum value of \(n\) is:

(A) \(225\)(B) \(210\)(C) \(200\)(D) \(180\)(E) \(175\)

20

\(R\) 点与 \(S\) 点相距 \(76\) 英里。两人分别从这两点同时出发，相向而行。从 \(R\) 点出发的人始终以每小时 \(4\dfrac12\) 英里的速度行走；从 \(S\) 点出发的人第一小时以每小时 \(3\dfrac14\) 英里的速度行走，第二小时以每小时 \(3\dfrac34\) 英里的速度行走，之后各小时的速度依次成等差数列。若两人在整数小时后相遇，且相遇点到 \(R\) 点比到 \(S\) 点近 \(x\) 英里，则 \(x\) 是：

A\(10\) B\(8\) C\(6\) D\(4\) E\(2\)

English

Two men at points \(R\) and \(S,\) \(76\) miles apart, set out at the same time to walk towards each other. The man at \(R\) walks uniformly at \(4\dfrac12\) miles per hour; the man at \(S\) walks at \(3\dfrac14\) miles per hour for the first hour, at \(3\dfrac34\) miles per hour for the second hour, and so on, in arithmetic progression. If the men meet \(x\) miles nearer \(R\) than \(S\) in an integral number of hours, then \(x\) is:

(A) \(10\)(B) \(8\)(C) \(6\)(D) \(4\)(E) \(2\)

21

表达式 \(x^2-y^2-z^2+2yz+x+y-z\) 具有：

A没有系数和指数均为整数的一次因式 B因式 \(-x+y+z\) C因式 \(x-y-z+1\) D因式 \(x+y-z+1\) E因式 \(x-y+z+1\)

English

The expression \(x^2-y^2-z^2+2yz+x+y-z\) has:

(A) no linear factor with integer coefficients and integer exponents(B) the factor \(-x+y+z\)(C) the factor \(x-y-z+1\)(D) the factor \(x+y-z+1\)(E) the factor \(x-y+z+1\)

22

锐角三角形 \(ABC\) 内接于以 \(O\) 为圆心的圆，且 \(\overset{\frown}{AB}=120^\circ\)、\(\overset{\frown}{BC}=72^\circ\)。取一点 \(E\) 于劣弧 \(AC\) 上，使 \(OE\) 垂直于 \(AC\)。角 \(OBE\) 与角 \(BAC\) 的度数之比是：

A\(\dfrac5{18}\) B\(\dfrac29\) C\(\dfrac14\) D\(\dfrac13\) E\(\dfrac49\)

English

Acute-angled triangle \(ABC\) is inscribed in a circle with center at \(O;\) \(\overset{\frown}{AB}=120^\circ\) and \(\overset{\frown}{BC}=72^\circ.\) A point \(E\) is taken in minor arc \(AC\) such that \(OE\) is perpendicular to \(AC.\) Then the ratio of the magnitudes of angles \(OBE\) and \(BAC\) is:

(A) \(\dfrac5{18}\)(B) \(\dfrac29\)(C) \(\dfrac14\)(D) \(\dfrac13\)(E) \(\dfrac49\)

23

\(A\) 先给 \(B\) 与 \(B\) 原有钱数相同的美分，再给 \(C\) 与 \(C\) 原有钱数相同的美分。接着，\(B\) 同样分别给 \(A\) 和 \(C\) 与他们当时各自钱数相同的美分。最后，\(C\) 也同样分别给 \(A\) 和 \(B\) 与他们当时各自钱数相同的美分。如果最终每人都有 \(16\) 美分，那么 \(A\) 最初有多少美分？

A\(24\) B\(26\) C\(28\) D\(30\) E\(32\)

English

\(A\) gives \(B\) as many cents as \(B\) has and \(C\) as many cents as \(C\) has. Similarly, \(B\) then gives \(A\) and \(C\) as many cents as each then has. \(C\), similarly, then gives \(A\) and \(B\) as many cents as each then has. If each finally has \(16\) cents, with how many cents does \(A\) start?

(A) \(24\)(B) \(26\)(C) \(28\)(D) \(30\)(E) \(32\)

24

考虑形如 \(x^2+bx+c=0\) 的方程。若系数 \(b\) 和 \(c\) 都从整数集合 \(\{1,2,3,4,5,6\}\) 中选取，其中有多少个方程具有实根？

A\(20\) B\(19\) C\(18\) D\(17\) E\(16\)

English

Consider equations of the form \(x^2+bx+c=0.\) How many such equations have real roots and have coefficients \(b\) and \(c\) selected from the set of integers \(\{1,2,3,4,5,6\}?\)

(A) \(20\)(B) \(19\)(C) \(18\)(D) \(17\)(E) \(16\)

25

在正方形 \(ABCD\) 的边 \(AD\) 上取一点 \(F\)。过 \(C\) 作 \(CF\) 的垂线，与 \(AB\) 的延长线交于 \(E\)。\(ABCD\) 的面积为 \(256\) 平方英寸，三角形 \(CEF\) 的面积为 \(200\) 平方英寸。则 \(BE\) 的长度为多少英寸？

A\(12\) B\(14\) C\(15\) D\(16\) E\(20\)

English

Point \(F\) is taken in side \(AD\) of square \(ABCD.\) At \(C\) a perpendicular is drawn to \(CF,\) meeting \(AB\) extended at \(E.\) The area of \(ABCD\) is \(256\) square inches and the area of triangle \(CEF\) is \(200\) square inches. Then the number of inches in \(BE\) is:

(A) \(12\)(B) \(14\)(C) \(15\)(D) \(16\)(E) \(20\)

26

形式 I。考虑以下命题：

\[ \begin{aligned} (1)\quad&p\land\neg q\land r,\\ (2)\quad&\neg p\land\neg q\land r,\\ (3)\quad&p\land\neg q\land\neg r,\\ (4)\quad&\neg p\land q\land r \end{aligned}\text{，}\]

其中 \(p\)、\(q\)、\(r\) 是命题。以上有多少个能推出 \((p\to q)\to r\) 为真？

形式 II。考虑以下命题：\((1)\) \(p\) 和 \(r\) 为真且 \(q\) 为假；\((2)\) \(r\) 为真且 \(p\) 和 \(q\) 为假；\((3)\) \(p\) 为真且 \(q\) 和 \(r\) 为假；\((4)\) \(q\) 和 \(r\) 为真且 \(p\) 为假。以上有多少个能推出命题“\(r\) 可由‘\(p\) 蕴含 \(q\)’推出”为真？

A\(0\) B\(1\) C\(2\) D\(3\) E\(4\)

English

Form I. Consider the statements

\[ \begin{aligned} (1)\quad&p\land\neg q\land r,\\ (2)\quad&\neg p\land\neg q\land r,\\ (3)\quad&p\land\neg q\land\neg r,\\ (4)\quad&\neg p\land q\land r, \end{aligned} \]

where \(p,\) \(q,\) \(r\) are propositions. How many of these imply the truth of \((p\to q)\to r?\)

Form II. Consider the statements \((1)\) \(p\) and \(r\) are true and \(q\) is false, \((2)\) \(r\) is true and \(p\) and \(q\) are false, \((3)\) \(p\) is true and \(q\) and \(r\) are false, \((4)\) \(q\) and \(r\) are true and \(p\) is false. How many of these imply the truth of the statement “\(r\) is implied by the statement that \(p\) implies \(q\)”?

(A) \(0\)(B) \(1\)(C) \(2\)(D) \(3\)(E) \(4\)

27

在平面上画六条直线，其中任意两条都不平行，任意三条都不共点。它们把平面分成的区域数是：

A\(16\) B\(20\) C\(22\) D\(24\) E\(26\)

English

Six straight lines are drawn in a plane with no two parallel and no three concurrent. The number of regions into which they divide the plane is:

(A) \(16\)(B) \(20\)(C) \(22\)(D) \(24\)(E) \(26\)

28

已知方程 \(3x^2-4x+k=0\) 有实根。使该方程两根之积最大的 \(k\) 值是：

A\(\dfrac{16}{9}\) B\(\dfrac{16}{3}\) C\(\dfrac49\) D\(\dfrac43\) E\(-\dfrac43\)

English

Given the equation \(3x^2-4x+k=0\) with real roots. The value of \(k\) for which the product of the roots of the equation is a maximum is:

(A) \(\dfrac{16}{9}\)(B) \(\dfrac{16}{3}\)(C) \(\dfrac49\)(D) \(\dfrac43\)(E) \(-\dfrac43\)

29

一个竖直向上发射的质点在 \(t\) 秒末到达 \(s\) 英尺的高度，其中 \(s=160t-16t^2\)。它能达到的最大高度是：

A\(800\) B\(640\) C\(400\) D\(320\) E\(160\)

English

A particle projected vertically upward reaches, at the end of \(t\) seconds, an elevation of \(s\) feet where \(s=160t-16t^2.\) The highest elevation is:

(A) \(800\)(B) \(640\)(C) \(400\)(D) \(320\)(E) \(160\)

30

设

\[ F=\log\frac{1+x}{1-x}\text{。}\]

求新函数 \(G\)：把 \(x\) 在 \(F\) 中的每一次出现都替换为

\[ \frac{3x+x^3}{1+3x^2}\text{，}\]

并化简。化简后的 \(G\) 等于：

A\(-F\) B\(F\) C\(3F\) D\(F^3\) E\(F^3-F\)

English

Let

\[ F=\log\frac{1+x}{1-x}. \]

Form a new function \(G\) by replacing each \(x\) in \(F\) by

\[ \frac{3x+x^3}{1+3x^2}, \]

and simplify. The simplified expression \(G\) is equal to:

(A) \(-F\)(B) \(F\)(C) \(3F\)(D) \(F^3\)(E) \(F^3-F\)

31

方程 \(2x+3y=763\) 的正整数解共有：

A\(255\) B\(254\) C\(128\) D\(127\) E\(0\)

English

The number of solutions in positive integers of \(2x+3y=763\) is:

(A) \(255\)(B) \(254\)(C) \(128\)(D) \(127\)(E) \(0\)

32

矩形 \(R\) 的边长为 \(a\) 和 \(b\)，其中 \(a\lt b\)。现要得到一个边长为 \(x\) 和 \(y\) 的矩形，其中 \(x\lt a\)、\(y\lt a\)，使它的周长是 \(R\) 的三分之一，面积也是 \(R\) 的三分之一。这样的不同矩形共有：

A\(0\) B\(1\) C\(2\) D\(4\) E无穷多个

English

The dimensions of a rectangle \(R\) are \(a\) and \(b,\) \(a\lt b.\) It is required to obtain a rectangle with dimensions \(x\) and \(y,\) \(x\lt a,\) \(y\lt a,\) so that its perimeter is one-third that of \(R,\) and its area is one-third that of \(R.\) The number of such (different) rectangles is:

(A) \(0\)(B) \(1\)(C) \(2\)(D) \(4\)(E) infinitely many

33

已知直线 \(y=\dfrac34x+6\)，另有直线 \(L\) 与它平行且相距 \(4\) 个单位。下列哪一个可能是 \(L\) 的方程？

A\(y=\dfrac34x+1\) B\(y=\dfrac34x\) C\(y=\dfrac34x-\dfrac23\) D\(y=\dfrac34x-1\) E\(y=\dfrac34x+2\)

English

Given the line \(y=\dfrac34x+6\) and a line \(L\) parallel to the given line and \(4\) units from it. A possible equation for \(L\) is:

(A) \(y=\dfrac34x+1\)(B) \(y=\dfrac34x\)(C) \(y=\dfrac34x-\dfrac23\)(D) \(y=\dfrac34x-1\)(E) \(y=\dfrac34x+2\)

34

在三角形 \(ABC\) 中，边 \(a=\sqrt3\)，边 \(b=\sqrt3\)，且边 \(c\gt3\)。设 \(x\) 是满足“边 \(c\) 所对角的度数必大于 \(x\)”的最大数，则 \(x\) 等于：

A\(150\) B\(120\) C\(105\) D\(90\) E\(60\)

English

In triangle \(ABC,\) side \(a=\sqrt3,\) side \(b=\sqrt3,\) and side \(c\gt3.\) Let \(x\) be the largest number such that the magnitude, in degrees, of the angle opposite side \(c\) exceeds \(x.\) Then \(x\) equals:

(A) \(150\)(B) \(120\)(C) \(105\)(D) \(90\)(E) \(60\)

35

一个三角形的三边长都是整数，面积也是整数。其中一边长为 \(21\)，周长为 \(48\)。最短边的长度是：

A\(8\) B\(10\) C\(12\) D\(14\) E\(16\)

English

The lengths of the sides of a triangle are integers, and its area is also an integer. One side is \(21\) and the perimeter is \(48.\) The shortest side is:

(A) \(8\)(B) \(10\)(C) \(12\)(D) \(14\)(E) \(16\)

36

某人起初有 \(64\) 美分，共下注 \(6\) 次，赢三次、输三次，输赢的次序随机，每次获胜和失败的概率相同。每次下注金额都是下注时剩余钱数的一半，则最终结果是：

A损失 \(27\) 美分 B获利 \(27\) 美分 C损失 \(37\) 美分 D既没有获利也没有损失 E获利或损失取决于输赢的次序

English

A person starting with \(64\) cents and making \(6\) bets, wins three times and loses three times, the wins and losses occurring in random order. The chance for a win is equal to the chance for a loss. If each wager is for half the money remaining at the time of the bet, then the final result is:

(A) a loss of \(27\)¢(B) a gain of \(27\)¢(C) a loss of \(37\)¢(D) neither a gain nor a loss(E) a gain or a loss depending upon the order in which the wins and losses occur

37

直线上的点 \(P_1\)、\(P_2\)、\(\ldots\)、\(P_7\) 按此顺序排列，但间距不一定相等。在该直线上任取一点 \(P\)，令 \(s\) 为以下无向线段长度之和：

\[ PP_1,\ PP_2,\ \ldots,\ PP_7\text{。}\]

则 \(s\) 取得最小值的充要条件是点 \(P\) 位于：

A\(P_1\) 与 \(P_7\) 的中点 B\(P_2\) 与 \(P_6\) 的中点 C\(P_3\) 与 \(P_5\) 的中点 D\(P_4\) 处 E\(P_1\) 处

English

Given points \(P_1,\) \(P_2,\) \(\ldots,\) \(P_7\) on a straight line, in the order stated (not necessarily evenly spaced). Let \(P\) be an arbitrarily selected point on the line and let \(s\) be the sum of the undirected lengths

\[ PP_1,\ PP_2,\ \ldots,\ PP_7. \]

Then \(s\) is smallest if and only if the point \(P\) is:

(A) midway between \(P_1\) and \(P_7\)(B) midway between \(P_2\) and \(P_6\)(C) midway between \(P_3\) and \(P_5\)(D) at \(P_4\)(E) at \(P_1\)

38

点 \(F\) 位于平行四边形 \(ABCD\) 的边 \(AD\) 的延长线上。\(BF\) 与对角线 \(AC\) 交于 \(E\)，与边 \(DC\) 交于 \(G\)。若 \(EF=32\) 且 \(GF=24\)，则 \(BE\) 等于：

A\(4\) B\(8\) C\(10\) D\(12\) E\(16\)

English

Point \(F\) is taken on the extension of side \(AD\) of parallelogram \(ABCD.\) \(BF\) intersects diagonal \(AC\) at \(E\) and side \(DC\) at \(G.\) If \(EF=32\) and \(GF=24,\) then \(BE\) equals:

(A) \(4\)(B) \(8\)(C) \(10\)(D) \(12\)(E) \(16\)

39

在三角形 \(ABC\) 中，作线段 \(CE\) 和 \(AD\)，使 \(\dfrac{CD}{DB}=\dfrac31\) 且 \(\dfrac{AE}{EB}=\dfrac32\)。

设 \(r=\dfrac{CP}{PE}\)，其中 \(P\) 是 \(CE\) 与 \(AD\) 的交点。则 \(r\) 等于：

A\(3\) B\(\dfrac32\) C\(4\) D\(5\) E\(\dfrac52\)

English

In triangle \(ABC\) lines \(CE\) and \(AD\) are drawn so that \(\dfrac{CD}{DB}=\dfrac31\) and \(\dfrac{AE}{EB}=\dfrac32.\)

Let \(r=\dfrac{CP}{PE},\) where \(P\) is the intersection point of \(CE\) and \(AD.\) Then \(r\) equals:

(A) \(3\)(B) \(\dfrac32\)(C) \(4\)(D) \(5\)(E) \(\dfrac52\)

40

若数 \(x\) 满足方程 \(\sqrt[3]{x+9}-\sqrt[3]{x-9}=3\)，则 \(x^2\) 介于：

A\(55\) 与 \(65\) 之间 B\(65\) 与 \(75\) 之间 C\(75\) 与 \(85\) 之间 D\(85\) 与 \(95\) 之间 E\(95\) 与 \(105\) 之间

English

If \(x\) is a number satisfying the equation \(\sqrt[3]{x+9}-\sqrt[3]{x-9}=3,\) then \(x^2\) is between:

(A) \(55\) and \(65\)(B) \(65\) and \(75\)(C) \(75\) and \(85\)(D) \(85\) and \(95\)(E) \(95\) and \(105\)