1966 AMC 12 真题（P版）

1

已知 \(3x-4\) 与 \(y+15\) 之比为常数，且当 \(x=2\) 时 \(y=3\)。那么，当 \(y=12\) 时，\(x\) 等于：

A\(\dfrac18\) B\(\dfrac87\) C\(\dfrac73\) D\(\dfrac72\) E\(8\)

English

Given that the ratio of \(3x-4\) to \(y+15\) is constant, and \(y=3\) when \(x=2,\) then, when \(y=12,\) \(x\) equals:

(A) \(\dfrac18\)(B) \(\dfrac87\)(C) \(\dfrac73\)(D) \(\dfrac72\)(E) \(8\)

2

若三角形的底边增加 \(10\%\)，而这条底边上的高减少 \(10\%\)，则面积的变化为：

A增加 \(1\%\) B增加 \(\dfrac12\%\) C\(0\%\) D减少 \(\dfrac12\%\) E减少 \(1\%\)

English

When the base of a triangle is increased \(10\%\) and the altitude to this base is decreased \(10\%,\) the change in area is:

(A) \(1\%\) increase(B) \(\dfrac12\%\) increase(C) \(0\%\)(D) \(\dfrac12\%\) decrease(E) \(1\%\) decrease

3

若两个数的算术平均数为 \(6\)，几何平均数为 \(10\)，则以这两个数为根的一个方程是：

A\(x^2+12x+100=0\) B\(x^2+6x+100=0\) C\(x^2-12x-10=0\) D\(x^2-12x+100=0\) E\(x^2-6x+100=0\)

English

If the arithmetic mean of two numbers is \(6\) and their geometric mean is \(10,\) then an equation with the given two numbers as roots is:

(A) \(x^2+12x+100=0\)(B) \(x^2+6x+100=0\)(C) \(x^2-12x-10=0\)(D) \(x^2-12x+100=0\)(E) \(x^2-6x+100=0\)

4

圆 \(\mathrm{I}\) 外接于一个给定的正方形，圆 \(\mathrm{II}\) 内切于该正方形。若 \(r\) 是圆 \(\mathrm{I}\) 与圆 \(\mathrm{II}\) 的面积之比，则 \(r\) 等于：

A\(\sqrt2\) B\(2\) C\(\sqrt3\) D\(2\sqrt2\) E\(2\sqrt3\)

English

Circle \(\mathrm{I}\) is circumscribed about a given square and circle \(\mathrm{II}\) is inscribed in the given square. If \(r\) is the ratio of the area of circle \(\mathrm{I}\) to that of circle \(\mathrm{II},\) then \(r\) equals:

(A) \(\sqrt2\)(B) \(2\)(C) \(\sqrt3\)(D) \(2\sqrt2\)(E) \(2\sqrt3\)

5

满足方程的 \(x\) 值的个数

\[ \frac{2x^2-10x}{x^2-5x}=x-3 \]

为：

A零 B一 C二 D三 E大于 \(3\) 的整数

English

The number of values of \(x\) satisfying the equation

\[ \frac{2x^2-10x}{x^2-5x}=x-3 \]

is:

(A) zero(B) one(C) two(D) three(E) an integer greater than \(3\)

6

\(AB\) 是一个圆的直径，圆心为 \(O\)。点 \(C\) 在圆上，且角 \(BOC\) 为 \(60^\circ\)。若圆的直径为 \(5\) 英寸，则弦 \(AC\) 的长度（单位：英寸）为：

A\(3\) B\(\dfrac{5\sqrt2}{2}\) C\(\dfrac{5\sqrt3}{2}\) D\(3\sqrt3\) E以上均不是

English

\(AB\) is a diameter of a circle centered at \(O.\) \(C\) is a point on the circle such that angle \(BOC\) is \(60^\circ.\) If the diameter of the circle is \(5\) inches, the length of chord \(AC,\) expressed in inches, is:

(A) \(3\)(B) \(\dfrac{5\sqrt2}{2}\)(C) \(\dfrac{5\sqrt3}{2}\)(D) \(3\sqrt3\)(E) none of these

7

设

\[ \frac{35x-29}{x^2-3x+2} =\frac{N_1}{x-1}+\frac{N_2}{x-2} \]

是关于 \(x\) 的恒等式。则 \(N_1N_2\) 的数值为：

A\(-246\) B\(-210\) C\(-29\) D\(210\) E\(246\)

English

Let

\[ \frac{35x-29}{x^2-3x+2} =\frac{N_1}{x-1}+\frac{N_2}{x-2} \]

be an identity in \(x.\) The numerical value of \(N_1N_2\) is:

(A) \(-246\)(B) \(-210\)(C) \(-29\)(D) \(210\)(E) \(246\)

8

两个相交圆的公共弦长为 \(16\) 英尺。若两个半径分别为 \(10\) 英尺和 \(17\) 英尺，则两圆心之间的距离可能为（单位：英尺）：

A\(27\) B\(21\) C\(\sqrt{389}\) D\(15\) E无法确定

English

The length of the common chord of two intersecting circles is \(16\) feet. If the radii are \(10\) feet and \(17\) feet, a possible value for the distance between the centers of the circles, expressed in feet, is:

(A) \(27\)(B) \(21\)(C) \(\sqrt{389}\)(D) \(15\)(E) undetermined

9

若 \(x=(\log_8 2)^{(\log_2 8)}\)，则 \(\log_3 x\) 等于：

A\(-3\) B\(-\dfrac13\) C\(\dfrac13\) D\(3\) E\(9\)

English

If \(x=(\log_8 2)^{(\log_2 8)},\) then \(\log_3 x\) equals:

(A) \(-3\)(B) \(-\dfrac13\)(C) \(\dfrac13\)(D) \(3\)(E) \(9\)

10

若两个数的和为 \(1\)，积为 \(1\)，则它们的立方和为（其中 \(i=\sqrt{-1}\)）：

A\(2\) B\(-2-\dfrac{3\sqrt3i}{4}\) C\(0\) D\(-\dfrac{3\sqrt3i}{4}\) E\(-2\)

English

If the sum of two numbers is \(1\) and their product is \(1,\) then the sum of their cubes is, where \(i=\sqrt{-1}:\)

(A) \(2\)(B) \(-2-\dfrac{3\sqrt3i}{4}\)(C) \(0\)(D) \(-\dfrac{3\sqrt3i}{4}\)(E) \(-2\)

11

三角形 \(BAC\) 的三边之比为 \(2:3:4\)。\(BD\) 是作到最短边 \(AC\) 的角平分线，将其分成线段 \(AD\) 与 \(CD\)。若 \(AC\) 的长度为 \(10\)，则 \(AC\) 上较长线段的长度为：

A\(3\dfrac12\) B\(5\) C\(5\dfrac57\) D\(6\) E\(7\dfrac12\)

English

The sides of triangle \(BAC\) are in the ratio \(2:3:4.\) \(BD\) is the angle-bisector drawn to the shortest side \(AC,\) dividing it into segments \(AD\) and \(CD.\) If the length of \(AC\) is \(10,\) then the length of the longer segment of \(AC\) is:

(A) \(3\dfrac12\)(B) \(5\)(C) \(5\dfrac57\)(D) \(6\)(E) \(7\dfrac12\)

12

满足方程的实数 \(x\) 的个数

\[ (2^{6x+3})(4^{3x+6})=8^{4x+5} \]

为：

A\(0\) B\(1\) C\(2\) D\(3\) E大于 \(3\)

English

The number of real values of \(x\) that satisfy the equation

\[ (2^{6x+3})(4^{3x+6})=8^{4x+5} \]

is:

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

13

在 \(xy\) 平面中满足 \(x+y\leq5\) 的点中，坐标均为正有理数的点共有：

A\(9\) B\(10\) C\(14\) D\(15\) E无限多个

English

The number of points with positive rational coordinates selected from the set of points in the \(xy\)-plane such that \(x+y\leq5,\) is:

(A) \(9\)(B) \(10\)(C) \(14\)(D) \(15\)(E) infinite

14

矩形 \(ABCD\) 的长为 \(5\) 英寸，宽为 \(3\) 英寸。点 \(E\) 与 \(F\) 将对角线 \(AC\) 三等分。三角形 \(BEF\) 的面积（单位：平方英寸）为：

A\(\dfrac32\) B\(\dfrac53\) C\(\dfrac52\) D\(\dfrac13\sqrt{34}\) E\(\dfrac13\sqrt{68}\)

English

The length of rectangle \(ABCD\) is \(5\) inches and its width is \(3\) inches. Diagonal \(AC\) is divided into three equal segments by points \(E\) and \(F.\) The area of triangle \(BEF,\) expressed in square inches, is:

(A) \(\dfrac32\)(B) \(\dfrac53\)(C) \(\dfrac52\)(D) \(\dfrac13\sqrt{34}\)(E) \(\dfrac13\sqrt{68}\)

15

若 \(x-y\gt x\) 且 \(x+y\lt y\)，则：

A\(y\lt x\) B\(x\lt y\) C\(x\lt y\lt0\) D\(x\lt0\)，且 \(y\lt0\) E\(x\lt0\)，且 \(y\gt0\)

English

If \(x-y\gt x\) and \(x+y\lt y,\) then:

(A) \(y\lt x\)(B) \(x\lt y\)(C) \(x\lt y\lt0\)(D) \(x\lt0,\) and \(y\lt0\)(E) \(x\lt0,\) and \(y\gt0\)

16

若

\[ \begin{gathered} \dfrac{4^x}{2^{x+y}}=8,\\ \dfrac{9^{x+y}}{3^{5y}}=243 \end{gathered}\text{，}\]

其中 \(x\) 与 \(y\) 为实数，则 \(xy\) 等于：

A\(1\dfrac34\) B\(4\) C\(6\) D\(12\) E\(-4\)

English

If

\[ \begin{gathered} \dfrac{4^x}{2^{x+y}}=8,\\ \dfrac{9^{x+y}}{3^{5y}}=243, \end{gathered} \]

\(x\) and \(y\) are real numbers, then \(xy\) equals:

(A) \(1\dfrac34\)(B) \(4\)(C) \(6\)(D) \(12\)(E) \(-4\)

17

曲线 \(x^2+4y^2=1\) 与 \(4x^2+y^2=4\) 共有的不同交点个数为：

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

English

The number of distinct points common to the curves \(x^2+4y^2=1\) and \(4x^2+y^2=4\) is:

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

18

某等差数列的首项为 \(2\)，末项为 \(29\)，所有项之和为 \(155\)。其公差为：

A\(3\) B\(2\) C\(\dfrac{27}{19}\) D\(\dfrac{13}{9}\) E\(\dfrac{23}{38}\)

English

In a given arithmetic sequence the first term is \(2,\) the last term is \(29,\) and the sum of all the terms is \(155.\) The common difference is:

(A) \(3\)(B) \(2\)(C) \(\dfrac{27}{19}\)(D) \(\dfrac{13}{9}\)(E) \(\dfrac{23}{38}\)

19

设 \(s_1\) 为等差数列 \(8\)、\(12\)、\(\ldots\) 的前 \(n\) 项之和，\(s_2\) 为等差数列 \(17\)、\(19\)、\(\ldots\) 的前 \(n\) 项之和。则使 \(s_1=s_2\) 成立的值共有：

A无 \(n\) 值 B一个 \(n\) 值 C两个 \(n\) 值 D四个 \(n\) 值 E多于四个 \(n\) 值

English

Let \(s_1\) be the sum of the first \(n\) terms of the arithmetic sequence \(8,\) \(12,\) \(\ldots\) and let \(s_2\) be the sum of the first \(n\) terms of the arithmetic sequence \(17,\) \(19,\) \(\ldots.\) Then \(s_1=s_2\) for:

(A) no value of \(n\)(B) one value of \(n\)(C) two values of \(n\)(D) four values of \(n\)(E) more than four values of \(n\)

20

若命题“\(a=0\)”为真，则命题“对于实数 \(a\) 和 \(b\)，若 \(a=0\)，则 \(ab=0\)”的否定为：

A若 \(a\ne0\)，则 \(ab\ne0\) B若 \(a\ne0\)，则 \(ab=0\) C若 \(a=0\)，则 \(ab\ne0\) D若 \(ab\ne0\)，则 \(a\ne0\) E若 \(ab=0\)，则 \(a\ne0\)

English

If the proposition “\(a=0\)” is true, the negation of the proposition “For real values of \(a\) and \(b,\) if \(a=0,\) then \(ab=0\)” is:

(A) If \(a\ne0,\) then \(ab\ne0\)(B) If \(a\ne0,\) then \(ab=0\)(C) If \(a=0,\) then \(ab\ne0\)(D) If \(ab\ne0,\) then \(a\ne0\)(E) If \(ab=0,\) then \(a\ne0\)

21

按以下方法作一个“\(n\) 角星”：将一个凸多边形的各边依次编号为 \(1\)、\(2\)、\(\ldots\)、\(k\)、\(\ldots\)、\(n\)，其中 \(n\geq5\)；对于 \(k\) 的全部 \(n\) 个取值，边 \(k\) 与边 \(k+2\) 均不平行，并约定边 \(n+1\) 和边 \(n+2\) 分别与边 \(1\) 和边 \(2\) 相同；将编号为 \(k\) 和 \(k+2\) 的 \(n\) 对边延长至相交。（图示为 \(n=5\) 的情形。）

设 \(S\) 为该星形 \(n\) 个尖角的内角度数之和，则 \(S\) 等于：

A\(180\) B\(360\) C\(180(n+2)\) D\(180(n-2)\) E\(180(n-4)\)

English

An “\(n\)-pointed star” is formed as follows: the sides of a convex polygon are numbered consecutively \(1,\) \(2,\) \(\ldots,\) \(k,\) \(\ldots,\) \(n,\) \(n\geq5;\) for all \(n\) values of \(k,\) sides \(k\) and \(k+2\) are non-parallel, sides \(n+1\) and \(n+2\) being respectively identical with sides \(1\) and \(2;\) prolong the \(n\) pairs of sides numbered \(k\) and \(k+2\) until they meet. (A figure is shown for the case \(n=5.\))

Let \(S\) be the degree-sum of the interior angles at the \(n\) points of the star; then \(S\) equals:

(A) \(180\)(B) \(360\)(C) \(180(n+2)\)(D) \(180(n-2)\)(E) \(180(n-4)\)

22

考虑下列等式：

\[ \text{(I) }\sqrt{a^2+b^2}=0 \] \[ \text{(II) }\sqrt{a^2+b^2}=ab \] \[ \text{(III) }\sqrt{a^2+b^2}=a+b \] \[ \text{(IV) }\sqrt{a^2+b^2}=a-b \]

其中允许 \(a\) 与 \(b\) 为实数或复数。除 \(a=0\) 且 \(b=0\) 外仍存在解的等式是：

A\((\mathrm{I})\)，\((\mathrm{II})\)， \((\mathrm{III})\)，\((\mathrm{IV})\) B仅 \((\mathrm{II})\)，\((\mathrm{III})\)，\((\mathrm{IV})\) C仅 \((\mathrm{I})\)，\((\mathrm{III})\)，\((\mathrm{IV})\) D仅 \((\mathrm{III})\)，\((\mathrm{IV})\) E仅 \((\mathrm{I})\)

English

Consider the statements:

\[ \text{(I) }\sqrt{a^2+b^2}=0 \] \[ \text{(II) }\sqrt{a^2+b^2}=ab \] \[ \text{(III) }\sqrt{a^2+b^2}=a+b \] \[ \text{(IV) }\sqrt{a^2+b^2}=a-b \]

where we allow \(a\) and \(b\) to be real or complex numbers. Those statements for which there exist solutions other than \(a=0\) and \(b=0,\) are:

(A) \((\mathrm{I}),\) \((\mathrm{II}),\) \((\mathrm{III}),\) \((\mathrm{IV})\)(B) \((\mathrm{II}),\) \((\mathrm{III}),\) \((\mathrm{IV})\) only(C) \((\mathrm{I}),\) \((\mathrm{III}),\) \((\mathrm{IV})\) only(D) \((\mathrm{III}),\) \((\mathrm{IV})\) only(E) \((\mathrm{I})\) only

23

若 \(x\) 为实数且 \(4y^2+4xy+x+6=0\)，则使 \(y\) 为实数的全部 \(x\) 值为：

A\(x\leq-2\) 或 \(x\geq3\) B\(x\leq2\) 或 \(x\geq3\) C\(x\leq-3\) 或 \(x\geq2\) D\(-3\leq x\leq2\) E\(-2\leq x\leq3\)

English

If \(x\) is real and \(4y^2+4xy+x+6=0,\) then the complete set of values of \(x\) for which \(y\) is real, is:

(A) \(x\leq-2\) or \(x\geq3\)(B) \(x\leq2\) or \(x\geq3\)(C) \(x\leq-3\) or \(x\geq2\)(D) \(-3\leq x\leq2\)(E) \(-2\leq x\leq3\)

24

若 \(\log_MN=\log_NM\)、\(M\ne N\)、\(MN\gt0\)，且 \(M\ne1\)、\(N\ne1\)，则 \(MN\) 等于：

A\(\dfrac12\) B\(1\) C\(2\) D\(10\) E大于 \(2\) 且小于 \(10\) 的数

English

If \(\log_MN=\log_NM,\) \(M\ne N,\) \(MN\gt0,\) and \(M\ne1,\) \(N\ne1,\) then \(MN\) equals:

(A) \(\dfrac12\)(B) \(1\)(C) \(2\)(D) \(10\)(E) a number greater than \(2\) and less than \(10\)

25

若当 \(n=1\)、\(2\)、\(\ldots\) 时，\(F(n+1)=\dfrac{2F(n)+1}{2}\)，且 \(F(1)=2\)，则 \(F(101)\) 等于：

A\(49\) B\(50\) C\(51\) D\(52\) E\(53\)

English

If \(F(n+1)=\dfrac{2F(n)+1}{2}\) for \(n=1,\) \(2,\) \(\ldots,\) and \(F(1)=2,\) then \(F(101)\) equals:

(A) \(49\)(B) \(50\)(C) \(51\)(D) \(52\)(E) \(53\)

26

设 \(m\) 为正整数，且直线 \(13x+11y=700\) 与 \(y=mx-1\) 交于一个坐标均为整数的点。则 \(m\) 可以是：

A仅 \(4\) B仅 \(5\) C仅 \(6\) D仅 \(7\) E整数 \(4\)、\(5\)、\(6\)、\(7\) 之一以及另一个正整数

English

Let \(m\) be a positive integer and let the lines \(13x+11y=700\) and \(y=mx-1\) intersect in a point whose coordinates are integers. Then \(m\) can be:

(A) \(4\) only(B) \(5\) only(C) \(6\) only(D) \(7\) only(E) one of the integers \(4,\) \(5,\) \(6,\) \(7\) and one other positive integer

27

一个人按通常速度顺流划行 \(15\) 英里，所用时间比返程少五小时。若他将通常划船速度加倍，则顺流时间只比逆流时间少一小时。水流速度（单位：英里/小时）为：

A\(2\) B\(\dfrac52\) C\(3\) D\(\dfrac72\) E\(4\)

English

At his usual rate a man rows \(15\) miles downstream in five hours less time than it takes him to return. If he doubles his usual rate, the time downstream is only one hour less than the time upstream. In miles per hour, the rate of the stream’s current is:

(A) \(2\)(B) \(\dfrac52\)(C) \(3\)(D) \(\dfrac72\)(E) \(4\)

28

在一条直线上依次取五个点 \(O\)、\(A\)、\(B\)、\(C\)、\(D\)，且距离 \(OA=a\)、\(OB=b\)、\(OC=c\)、\(OD=d\)。点 \(P\) 位于 \(B\) 与 \(C\) 之间，并满足 \(AP:PD=BP:PC\)。则 \(OP\) 等于：

A\(\dfrac{b^2-bc}{a-b+c-d}\) B\(\dfrac{ac-bd}{a-b+c-d}\) C\(-\dfrac{bd+ac}{a-b+c-d}\) D\(\dfrac{bc+ad}{a+b+c+d}\) E\(\dfrac{ac-bd}{a+b+c+d}\)

English

Five points \(O,\) \(A,\) \(B,\) \(C,\) \(D\) are taken in order on a straight line with distances \(OA=a,\) \(OB=b,\) \(OC=c,\) and \(OD=d.\) \(P\) is a point on the line between \(B\) and \(C\) and such that \(AP:PD=BP:PC.\) Then \(OP\) equals:

(A) \(\dfrac{b^2-bc}{a-b+c-d}\)(B) \(\dfrac{ac-bd}{a-b+c-d}\)(C) \(-\dfrac{bd+ac}{a-b+c-d}\)(D) \(\dfrac{bc+ad}{a+b+c+d}\)(E) \(\dfrac{ac-bd}{a+b+c+d}\)

29

小于 \(1000\) 且既不能被 \(5\) 整除也不能被 \(7\) 整除的正整数共有：

A\(688\) B\(686\) C\(684\) D\(658\) E\(630\)

English

The number of positive integers less than \(1000\) divisible by neither \(5\) nor \(7\) is:

(A) \(688\)(B) \(686\)(C) \(684\)(D) \(658\)(E) \(630\)

30

若 \(x^4+ax^2+bx+c=0\) 的三个根为 \(1\)、\(2\) 和 \(3\)，则 \(a+c\) 的值为：

A\(35\) B\(24\) C\(-12\) D\(-61\) E\(-63\)

English

If three of the roots of \(x^4+ax^2+bx+c=0\) are \(1,\) \(2,\) and \(3,\) then the value of \(a+c\) is:

(A) \(35\)(B) \(24\)(C) \(-12\)(D) \(-61\)(E) \(-63\)

31

三角形 \(ABC\) 内接于圆心为 \(O'\) 的圆。圆心为 \(O\) 的圆内切于三角形 \(ABC\)。作 \(AO\)，并延长至与大圆交于 \(D\)。则必有：

A\(CD=BD=O'D\) B\(AO=CO=OD\) C\(CD=CO=BD\) D\(CD=OD=BD\) E\(O'B=O'C=OD\)

English

Triangle \(ABC\) is inscribed in a circle with center \(O'.\) A circle with center \(O\) is inscribed in triangle \(ABC.\) \(AO\) is drawn, and extended to intersect the larger circle in \(D.\) Then we must have:

(A) \(CD=BD=O'D\)(B) \(AO=CO=OD\)(C) \(CD=CO=BD\)(D) \(CD=OD=BD\)(E) \(O'B=O'C=OD\)

32

设 \(M\) 为三角形 \(ABC\) 的边 \(AB\) 的中点。点 \(P\) 位于 \(AB\) 上的 \(A\) 与 \(M\) 之间，过该中点作 \(MD\) 平行于 \(PC\)，并与 \(BC\) 交于 \(D\)。若三角形 \(BPD\) 与三角形 \(ABC\) 的面积之比记为 \(r\)，则：

A\(\dfrac12\lt r\lt1\)，取决于 \(P\) 的位置 B\(r=\dfrac12\)，与 \(P\) 的位置无关 C\(\dfrac12\leq r\lt1\)，取决于 \(P\) 的位置 D\(\dfrac13\lt r\lt\dfrac23\)，取决于 \(P\) 的位置 E\(r=\dfrac13\)，与 \(P\) 的位置无关

English

Let \(M\) be the midpoint of side \(AB\) of triangle \(ABC.\) Let \(P\) be a point on \(AB\) between \(A\) and \(M,\) and let \(MD\) be drawn parallel to \(PC\) and intersecting \(BC\) at \(D.\) If the ratio of the area of triangle \(BPD\) to that of triangle \(ABC\) is denoted by \(r,\) then:

(A) \(\dfrac12\lt r\lt1\) depending upon the position of \(P\)(B) \(r=\dfrac12\) independent of the position of \(P\)(C) \(\dfrac12\leq r\lt1\) depending upon the position of \(P\)(D) \(\dfrac13\lt r\lt\dfrac23\) depending upon the position of \(P\)(E) \(r=\dfrac13\) independent of the position of \(P\)

33

若 \(ab\ne0\) 且 \(|a|\ne|b|\)，则满足方程的不同 \(x\) 值的个数

\[ \begin{aligned} \frac{x-a}{b}+\frac{x-b}{a} &=\frac{b}{x-a}\\ &\quad+\frac{a}{x-b} \end{aligned} \]

为：

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

English

If \(ab\ne0\) and \(|a|\ne|b|,\) the number of distinct values of \(x\) satisfying the equation

\[ \begin{aligned} \frac{x-a}{b}+\frac{x-b}{a} &=\frac{b}{x-a}\\ &\quad+\frac{a}{x-b} \end{aligned} \]

is:

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

34

一个周长为 \(11\) 英尺的车轮以每小时 \(r\) 英里的速度行驶。若车轮转一整圈的时间缩短 \(\dfrac14\) 秒，速度 \(r\) 就会增加 \(5\) 英里/小时。则 \(r\) 为：

A\(9\) B\(10\) C\(10\dfrac12\) D\(11\) E\(12\)

English

Let \(r\) be the speed in miles per hour at which a wheel, \(11\) feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by \(\dfrac14\) of a second, the speed \(r\) is increased by \(5\) miles per hour. Then \(r\) is:

(A) \(9\)(B) \(10\)(C) \(10\dfrac12\)(D) \(11\)(E) \(12\)

35

设 \(O\) 为三角形 \(ABC\) 的内点，且 \(s_1=OA+OB+OC\)。若 \(s_2=AB+BC+CA\)，则：

A对每个三角形，\(s_1\gt\dfrac12s_2\)，且 \(s_1\leq s_2\) B对每个三角形，\(s_1\geq\dfrac12s_2\)，且 \(s_1\lt s_2\) C对每个三角形，\(s_1\gt\dfrac12s_2\)，且 \(s_1\lt s_2\) D对每个三角形，\(s_1\geq\dfrac12s_2\)，且 \(s_1\leq s_2\) E\((A)\)、\((B)\)、\((C)\)、\((D)\) 均不适用于每个三角形

English

Let \(O\) be an interior point of triangle \(ABC,\) and let \(s_1=OA+OB+OC.\) If \(s_2=AB+BC+CA,\) then:

(A) for every triangle \(s_1\gt\dfrac12s_2,\) and \(s_1\leq s_2\)(B) for every triangle \(s_1\geq\dfrac12s_2,\) and \(s_1\lt s_2\)(C) for every triangle \(s_1\gt\dfrac12s_2,\) and \(s_1\lt s_2\)(D) for every triangle \(s_1\geq\dfrac12s_2,\) and \(s_1\leq s_2\)(E) neither \((A)\) nor \((B)\) nor \((C)\) nor \((D)\) applies to every triangle

36

设

\[ \begin{aligned} (1+x+x^2)^n &=a_0+a_1x\\ &\quad+a_2x^2+\cdots\\ &\quad+a_{2n}x^{2n} \end{aligned} \]

是关于 \(x\) 的恒等式。若令 \(s=a_0+a_2+a_4+\cdots+a_{2n}\)，则 \(s\) 等于：

A\(2^n\) B\(2^n+1\) C\(\dfrac{3^n-1}{2}\) D\(\dfrac{3^n}{2}\) E\(\dfrac{3^n+1}{2}\)

English

Let

\[ \begin{aligned} (1+x+x^2)^n &=a_0+a_1x\\ &\quad+a_2x^2+\cdots\\ &\quad+a_{2n}x^{2n} \end{aligned} \]

be an identity in \(x.\) If we let \(s=a_0+a_2+a_4+\cdots+a_{2n},\) then \(s\) equals:

(A) \(2^n\)(B) \(2^n+1\)(C) \(\dfrac{3^n-1}{2}\)(D) \(\dfrac{3^n}{2}\)(E) \(\dfrac{3^n+1}{2}\)

37

阿尔法、贝塔和伽马三人合作完成一项工作，所需时间比阿尔法单独工作少 \(6\) 小时，比贝塔单独工作少 \(1\) 小时，并且是伽马单独工作所需时间的一半。设阿尔法与贝塔合作完成该工作需要 \(h\) 小时，则 \(h\) 等于：

A\(\dfrac52\) B\(\dfrac32\) C\(\dfrac43\) D\(\dfrac54\) E\(\dfrac34\)

English

Three men, Alpha, Beta, and Gamma, working together, do a job in \(6\) hours less time than Alpha alone, in \(1\) hour less time than Beta alone, and in one-half the time needed by Gamma when working alone. Let \(h\) be the number of hours needed by Alpha and Beta, working together, to do the job. Then \(h\) equals:

(A) \(\dfrac52\)(B) \(\dfrac32\)(C) \(\dfrac43\)(D) \(\dfrac54\)(E) \(\dfrac34\)

38

在三角形 \(ABC\) 中，分别作到边 \(BC\) 与 \(AB\) 的中线 \(AM\) 和 \(CN\)，二者交于点 \(O\)。点 \(P\) 是边 \(AC\) 的中点，且 \(MP\) 与 \(CN\) 交于 \(Q\)。若三角形 \(OMQ\) 的面积为 \(n\)，则三角形 \(ABC\) 的面积为：

A\(16n\) B\(18n\) C\(21n\) D\(24n\) E\(27n\)

English

In triangle \(ABC\) the medians \(AM\) and \(CN\) to sides \(BC\) and \(AB,\) respectively, intersect in point \(O.\) \(P\) is the midpoint of side \(AC,\) and \(MP\) intersects \(CN\) in \(Q.\) If the area of triangle \(OMQ\) is \(n,\) then the area of triangle \(ABC\) is:

(A) \(16n\)(B) \(18n\)(C) \(21n\)(D) \(24n\)(E) \(27n\)

39

在 \(R_1\) 进制中，分数 \(F_1\) 展开为 \(0.373737\ldots\)，分数 \(F_2\) 展开为 \(0.737373\ldots\)。在 \(R_2\) 进制中，分数 \(F_1\) 展开为 \(0.252525\ldots\)，而分数 \(F_2\) 展开为 \(0.525252\ldots\)。将 \(R_1\) 与 \(R_2\) 都写成十进制后，它们的和为：

A\(24\) B\(22\) C\(21\) D\(20\) E\(19\)

English

In base \(R_1\) the expanded fraction \(F_1\) becomes \(0.373737\ldots\) and the expanded fraction \(F_2\) becomes \(0.737373\ldots.\) In base \(R_2\) fraction \(F_1,\) when expanded, becomes \(0.252525\ldots,\) while fraction \(F_2\) becomes \(0.525252\ldots.\) The sum of \(R_1\) and \(R_2,\) each written in base ten, is:

(A) \(24\)(B) \(22\)(C) \(21\)(D) \(20\)(E) \(19\)

40

图中，\(AB\) 是一个圆的直径，圆心为 \(O\)，半径为 \(a\)。作弦 \(AD\) 并延长，与圆在 \(B\) 点的切线交于 \(C\)。在 \(AC\) 上取点 \(E\)，使得 \(AE=DC\)。若 \(E\) 的坐标为 \((x,y)\)，则：

A\(y^2=\dfrac{x^3}{2a-x}\) B\(y^2=\dfrac{x^3}{2a+x}\) C\(y^4=\dfrac{x^2}{2a-x}\) D\(x^2=\dfrac{y^2}{2a-x}\) E\(x^2=\dfrac{y^2}{2a+x}\)

English

In this figure \(AB\) is a diameter of a circle, centered at \(O,\) with radius \(a.\) A chord \(AD\) is drawn and extended to meet the tangent to the circle at \(B,\) in point \(C.\) Point \(E\) is taken on \(AC\) so that \(AE=DC.\) If the coordinates of \(E\) are \((x,y),\) then:

(A) \(y^2=\dfrac{x^3}{2a-x}\)(B) \(y^2=\dfrac{x^3}{2a+x}\)(C) \(y^4=\dfrac{x^2}{2a-x}\)(D) \(x^2=\dfrac{y^2}{2a-x}\)(E) \(x^2=\dfrac{y^2}{2a+x}\)