1960 AMC 12 真题（P版）

1

若 \(2\) 是方程 \(x^3+hx+10=0\) 的一个解（根），则 \(h\) 等于：

A\(10\) B\(9\) C\(2\) D\(-2\) E\(-9\)

English

If \(2\) is a solution (root) of \(x^3+hx+10=0,\) then \(h\) equals:

(A) \(10\)(B) \(9\)(C) \(2\)(D) \(-2\)(E) \(-9\)

2

一只钟从恰好 \(6{:}00\) 开始敲响 \(6\) 下，共用 \(5\) 秒。若每相邻两次敲响的时间间隔相同，那么敲响 \(12\) 下需要多少秒？

A\(9\dfrac13\) B\(10\) C\(11\) D\(14\dfrac23\) E以上都不是

English

It takes \(5\) seconds for a clock to strike \(6\) o’clock beginning at \(6{:}00\) o’clock precisely. If the strikings are uniformly spaced, how long, in seconds, does it take to strike \(12\) o’clock?

(A) \(9\dfrac13\)(B) \(10\)(C) \(11\)(D) \(14\dfrac23\)(E) none of these

3

对一张金额为 \(\$10{,}000\) 的账单，打 \(40\%\) 的折扣与先后打 \(36\%\) 和 \(4\%\) 的两次折扣之间的差额（以美元计）是：

A\(0\) B\(144\) C\(256\) D\(400\) E\(416\)

English

Applied to a bill for \(\$10{,}000\) the difference between a discount of \(40\%\) and two successive discounts of \(36\%\) and \(4\%,\) expressed in dollars, is:

(A) \(0\)(B) \(144\)(C) \(256\)(D) \(400\)(E) \(416\)

4

一个三角形的两个角均为 \(60^\circ\)，它们的夹边长为 \(4\) 英寸。这个三角形的面积（以平方英寸计）是：

A\(8\sqrt3\) B\(8\) C\(4\sqrt3\) D\(4\) E\(2\sqrt3\)

English

Each of two angles of a triangle is \(60^\circ\) and the included side is \(4\) inches. The area of the triangle, in square inches, is:

(A) \(8\sqrt3\)(B) \(8\)(C) \(4\sqrt3\)(D) \(4\)(E) \(2\sqrt3\)

5

图像 \(x^2+y^2=9\) 与 \(y^2=9\) 的不同公共点共有：

A无穷多个 B四个 C两个 D一个 E没有

English

The number of distinct points common to the graphs of \(x^2+y^2=9\) and \(y^2=9\) is:

(A) infinitely many(B) four(C) two(D) one(E) none

6

一个圆的周长为 \(100\) 英寸。该圆内接正方形的边长（以英寸计）是：

A\(\dfrac{25\sqrt2}{\pi}\) B\(\dfrac{50\sqrt2}{\pi}\) C\(\dfrac{100}{\pi}\) D\(\dfrac{100\sqrt2}{\pi}\) E\(50\sqrt2\)

English

The circumference of a circle is \(100\) inches. The side of a square inscribed in this circle, expressed in inches, is:

(A) \(\dfrac{25\sqrt2}{\pi}\)(B) \(\dfrac{50\sqrt2}{\pi}\)(C) \(\dfrac{100}{\pi}\)(D) \(\dfrac{100\sqrt2}{\pi}\)(E) \(50\sqrt2\)

7

圆 I 经过圆 II 的圆心，并与圆 II 相切。圆 I 的面积为 \(4\) 平方英寸。那么圆 II 的面积（以平方英寸计）是：

A\(8\) B\(8\sqrt2\) C\(8\sqrt\pi\) D\(16\) E\(16\sqrt2\)

English

Circle I passes through the center of, and is tangent to, circle II. The area of circle I is \(4\) square inches. Then the area of circle II, in square inches, is:

(A) \(8\)(B) \(8\sqrt2\)(C) \(8\sqrt\pi\)(D) \(16\)(E) \(16\sqrt2\)

8

数 \(2.5252525\ldots\) 可以写成分数。将该分数约成最简分数后，其分子与分母之和是：

A\(7\) B\(29\) C\(141\) D\(349\) E以上都不是

English

The number \(2.5252525\ldots\) can be written as a fraction. When reduced to lowest terms the sum of the numerator and denominator of this fraction is:

(A) \(7\)(B) \(29\)(C) \(141\)(D) \(349\)(E) none of these

9

分式 \[ \frac{a^2+b^2-c^2+2ab}{a^2+c^2-b^2+2ac} \] 在 \(a\)、\(b\) 和 \(c\) 的取值满足适当限制时：

A不能约简 B可约简为 \(-1\) C可约简为一个三项式 D可约简为 \(\dfrac{a-b+c}{a+b-c}\) E可约简为 \(\dfrac{a+b-c}{a-b+c}\)

English

The fraction \[ \frac{a^2+b^2-c^2+2ab}{a^2+c^2-b^2+2ac} \] is (with suitable restrictions on the values of \(a,\) \(b,\) and \(c\)):

(A) irreducible(B) reducible to \(-1\)(C) reducible to a polynomial of three terms(D) reducible to \(\dfrac{a-b+c}{a+b-c}\)(E) reducible to \(\dfrac{a+b-c}{a-b+c}\)

10

给出以下六个陈述：

\((1)\) 所有女性都是好驾驶员。

\((2)\) 有些女性是好驾驶员。

\((3)\) 没有男性是好驾驶员。

\((4)\) 所有男性都是差驾驶员。

\((5)\) 至少有一名男性是差驾驶员。

\((6)\) 所有男性都是好驾驶员。

否定陈述 \((6)\) 的陈述是：

A\((1)\) B\((2)\) C\((3)\) D\((4)\) E\((5)\)

English

Given the following six statements:

\((1)\) All women are good drivers.

\((2)\) Some women are good drivers.

\((3)\) No men are good drivers.

\((4)\) All men are bad drivers.

\((5)\) At least one man is a bad driver.

\((6)\) All men are good drivers.

The statement that negates statement \((6)\) is:

(A) \((1)\)(B) \((2)\)(C) \((3)\)(D) \((4)\)(E) \((5)\)

11

对于某个给定的 \(k\)，方程 \[ x^2-3kx+2k^2-1=0 \] 的两根之积为 \(7\)。这两个根可以描述为：

A正整数 B负整数 C有理数，但不是整数 D无理数 E虚数

English

For a given value of \(k,\) the product of the roots of \[ x^2-3kx+2k^2-1=0 \] is \(7.\) The roots may be characterized as:

(A) integral and positive(B) integral and negative(C) rational, but not integral(D) irrational(E) imaginary

12

在同一平面内，所有半径为给定值 \(a\)，且经过一个定点的圆，其圆心的轨迹是：

A一个点 B一条直线 C两条直线 D一个圆 E两个圆

English

The locus of the centers of all circles of given radius \(a,\) in the same plane, passing through a fixed point, is:

(A) a point(B) a straight line(C) two straight lines(D) a circle(E) two circles

13

由 \(y=3x+2\)、\(y=-3x+2\) 和 \(y=-2\) 围成的多边形是：

A等边三角形 B等腰三角形 C直角三角形 D一个三角形和一个梯形 E四边形

English

The polygon(s) formed by \(y=3x+2,\) \(y=-3x+2,\) and \(y=-2,\) is (are):

(A) an equilateral triangle(B) an isosceles triangle(C) a right triangle(D) a triangle and a trapezoid(E) a quadrilateral

14

若 \(a\) 和 \(b\) 是实数，则方程 \(3x-5+a=bx+1\) 有唯一解 \(x\) 的条件是［符号 \(a\ne0\) 表示 \(a\) 不等于零］：

A对所有 \(a\) 和 \(b\) B当 \(a\ne2b\) 时 C当 \(a\ne6\) 时 D当 \(b\ne0\) 时 E当 \(b\ne3\) 时

English

If \(a\) and \(b\) are real numbers, the equation \(3x-5+a=bx+1\) has a unique solution \(x\) [the symbol \(a\ne0\) means that \(a\) is different from zero]:

(A) for all \(a\) and \(b\)(B) if \(a\ne2b\)(C) if \(a\ne6\)(D) if \(b\ne0\)(E) if \(b\ne3\)

15

三角形 I 是等边三角形，边长为 \(A\)、周长为 \(P\)、面积为 \(K\)、外接圆半径为 \(R\)。三角形 II 也是等边三角形，边长为 \(a\)、周长为 \(p\)、面积为 \(k\)、外接圆半径为 \(r\)。若 \(A\) 不同于 \(a\)，则：

A\(P:p=R:r\) 仅在某些情况下成立 B\(P:p=R:r\) 总是成立 C\(P:p=K:k\) 仅在某些情况下成立 D\(R:r=K:k\) 总是成立 E\(R:r=K:k\) 仅在某些情况下成立

English

Triangle I is equilateral with side \(A,\) perimeter \(P,\) area \(K,\) and circumradius \(R\) (radius of the circumscribed circle). Triangle II is equilateral with side \(a,\) perimeter \(p,\) area \(k,\) and circumradius \(r.\) If \(A\) is different from \(a,\) then:

(A) \(P:p=R:r\) only sometimes(B) \(P:p=R:r\) always(C) \(P:p=K:k\) only sometimes(D) \(R:r=K:k\) always(E) \(R:r=K:k\) only sometimes

16

在以 \(5\) 为底的计数系统中，数数方式如下：\(1\)、\(2\)、\(3\)、\(4\)、\(10\)、\(11\)、\(12\)、\(13\)、\(14\)、\(20\)、\(\ldots\)。在十进位系统中表示为 \(69\) 的数，用 \(5\) 进制表示时，其数位是：

A两个连续的数字 B两个不连续的数字 C三个连续的数字 D三个不连续的数字 E四个数字

English

In the numeration system with base \(5,\) counting is as follows: \(1,\) \(2,\) \(3,\) \(4,\) \(10,\) \(11,\) \(12,\) \(13,\) \(14,\) \(20,\) \(\ldots\) The number whose description in the decimal system is \(69,\) when described in the base \(5\) system, is a number with:

(A) two consecutive digits(B) two non-consecutive digits(C) three consecutive digits(D) three non-consecutive digits(E) four digits

17

对某一群体，公式 \(N=8\cdot10^8\cdot x^{-\frac{3}{2}}\) 给出了收入超过 \(x\) 美元的人数。收入最高的 \(800\) 人中，最低收入至少为多少美元？

A\(10^4\) B\(10^6\) C\(10^8\) D\(10^{12}\) E\(10^{16}\)

English

The formula \(N=8\cdot10^8\cdot x^{-\frac{3}{2}}\) gives, for a certain group, the number of individuals whose income exceeds \(x\) dollars. The lowest income, in dollars, of the wealthiest \(800\) individuals is at least:

(A) \(10^4\)(B) \(10^6\)(C) \(10^8\)(D) \(10^{12}\)(E) \(10^{16}\)

18

方程组 \(3^{x+y}=81\) 和 \(81^{x-y}=3\)：

A没有公共解 B解为 \(x=2\)、\(y=2\) C解为 \(x=2\dfrac12\)、\(y=1\dfrac12\) D有由一正一负两个整数组成的公共解 E以上都不是

English

The pair of equations \(3^{x+y}=81\) and \(81^{x-y}=3\) has:

(A) no common solution(B) the solution \(x=2,\) \(y=2\)(C) the solution \(x=2\dfrac12,\) \(y=1\dfrac12\)(D) a common solution in positive and negative integers(E) none of these

19

考虑方程 I：\(x+y+z=46\)，其中 \(x\)、\(y\) 和 \(z\) 为正整数；以及方程 II：\(x+y+z+w=46\)，其中 \(x\)、\(y\)、\(z\) 和 \(w\) 为正整数。则：

A\(\mathrm{I}\) 可由连续整数满足 B\(\mathrm{I}\) 可由连续偶数满足 C\(\mathrm{II}\) 可由连续整数满足 D\(\mathrm{II}\) 可由连续偶数满足 E\(\mathrm{II}\) 可由连续奇数满足

English

Consider equation I: \(x+y+z=46,\) where \(x,\) \(y,\) and \(z\) are positive integers, and equation II: \(x+y+z+w=46,\) where \(x,\) \(y,\) \(z,\) and \(w\) are positive integers. Then:

(A) \(\mathrm{I}\) can be solved in consecutive integers(B) \(\mathrm{I}\) can be solved in consecutive even integers(C) \(\mathrm{II}\) can be solved in consecutive integers(D) \(\mathrm{II}\) can be solved in consecutive even integers(E) \(\mathrm{II}\) can be solved in consecutive odd integers

20

在 \[ \left(\frac{x^2}{2}-\frac2x\right)^8 \] 的展开式中，\(x^7\) 的系数是：

A\(56\) B\(-56\) C\(14\) D\(-14\) E\(0\)

English

The coefficient of \(x^7\) in the expansion of \[ \left(\frac{x^2}{2}-\frac2x\right)^8 \] is:

(A) \(56\)(B) \(-56\)(C) \(14\)(D) \(-14\)(E) \(0\)

21

正方形 I 的对角线长为 \(a+b\)。面积为正方形 I 两倍的正方形 II，其周长是：

A\((a+b)^2\) B\(\sqrt2(a+b)^2\) C\(2(a+b)\) D\(\sqrt8(a+b)\) E\(4(a+b)\)

English

The diagonal of square I is \(a+b.\) The perimeter of square II with twice the area of I is:

(A) \((a+b)^2\)(B) \(\sqrt2(a+b)^2\)(C) \(2(a+b)\)(D) \(\sqrt8(a+b)\)(E) \(4(a+b)\)

22

在等式 \((x+m)^2-(x+n)^2=(m-n)^2\) 中，\(m\) 和 \(n\) 是不相等的非零常数。若 \(x=am+bn\) 满足该等式，则：

A\(a=0\)，且 \(b\) 有唯一的非零值 B\(a=0\)，且 \(b\) 有两个非零值 C\(b=0\)，且 \(a\) 有唯一的非零值 D\(b=0\)，且 \(a\) 有两个非零值 E\(a\) 和 \(b\) 各有唯一的非零值

English

The equality \((x+m)^2-(x+n)^2=(m-n)^2,\) where \(m\) and \(n\) are unequal non-zero constants, is satisfied by \(x=am+bn\) where:

(A) \(a=0,\) \(b\) has a unique non-zero value(B) \(a=0,\) \(b\) has two non-zero values(C) \(b=0,\) \(a\) has a unique non-zero value(D) \(b=0,\) \(a\) has two non-zero values(E) \(a\) and \(b\) each have a unique non-zero value

23

一个圆柱形盒子的半径 \(R\) 为 \(8\) 英寸，高 \(H\) 为 \(3\) 英寸。体积为 \(V=\pi R^2H\)。将 \(R\) 增加 \(x\) 英寸与将 \(H\) 增加 \(x\) 英寸时，体积要增加同一个固定的正数。满足这一条件的是：

A\(x\) 没有实数取值 B\(x\) 有一个整数取值 C\(x\) 有一个有理数但非整数的取值 D\(x\) 有一个无理数取值 E\(x\) 有两个实数取值

English

The radius \(R\) of a cylindrical box is \(8\) inches, the height \(H\) is \(3\) inches. The volume \(V=\pi R^2H\) is to be increased by the same fixed positive amount when \(R\) is increased by \(x\) inches as when \(H\) is increased by \(x\) inches. This condition is satisfied by:

(A) no real value of \(x\)(B) one integral value of \(x\)(C) one rational, but not integral, value of \(x\)(D) one irrational value of \(x\)(E) two real values of \(x\)

24

若 \(\log_{2x}216=x\)，其中 \(x\) 为实数，则 \(x\) 是：

A既不是完全平方数也不是完全立方数的整数 B既不是完全平方数也不是完全立方数的非整数有理数 C无理数 D完全平方数 E完全立方数

English

If \(\log_{2x}216=x,\) where \(x\) is real, then \(x\) is:

(A) a non-square, non-cube integer(B) a non-square, non-cube, non-integral rational number(C) an irrational number(D) a perfect square(E) a perfect cube

25

设 \(m\) 和 \(n\) 为任意两个奇数，且 \(n\) 小于 \(m\)。能整除所有可能的 \(m^2-n^2\) 的最大整数是：

A\(2\) B\(4\) C\(6\) D\(8\) E\(16\)

English

Let \(m\) and \(n\) be any two odd numbers, with \(n\) less than \(m.\) The largest integer which divides all possible numbers of the form \(m^2-n^2\) is:

(A) \(2\)(B) \(4\)(C) \(6\)(D) \(8\)(E) \(16\)

26

求满足下列不等式的所有 \(x\) 值：\[ \left|\frac{5-x}{3}\right|\lt2 \] ［符号 \(|a|\) 表示：若 \(a\) 为正，则取 \(+a\)；若 \(a\) 为负，则取 \(-a\)；若 \(a\) 为零，则取 \(0\)。记号 \(1\lt a\lt2\) 表示 \(a\) 可以取 \(1\) 与 \(2\) 之间的任意值，但不包括 \(1\) 和 \(2\)。］

A\(1\lt x\lt11\) B\(-1\lt x\lt11\) C\(x\lt11\) D\(x\gt11\) E\(|x|\lt6\)

English

Find the set of \(x\)-values satisfying the inequality \[ \left|\frac{5-x}{3}\right|\lt2. \] [The symbol \(|a|\) means \(+a\) if \(a\) is positive, \(-a\) if \(a\) is negative, \(0\) if \(a\) is zero. The notation \(1\lt a\lt2\) means that \(a\) can have any value between \(1\) and \(2,\) excluding \(1\) and \(2.\)]

(A) \(1\lt x\lt11\)(B) \(-1\lt x\lt11\)(C) \(x\lt11\)(D) \(x\gt11\)(E) \(|x|\lt6\)

27

设多边形 \(P\) 的内角和为 \(S\)，且每个内角都是同一顶点处外角的 \(7\dfrac12\) 倍。则：

A\(S=2660^\circ\)，且 \(P\) 可能是正多边形 B\(S=2660^\circ\)，且 \(P\) 不是正多边形 C\(S=2700^\circ\)，且 \(P\) 是正多边形 D\(S=2700^\circ\)，且 \(P\) 不是正多边形 E\(S=2700^\circ\)，且 \(P\) 可能是正多边形，也可能不是

English

Let \(S\) be the sum of the interior angles of a polygon \(P\) for which each interior angle is \(7\dfrac12\) times the exterior angle at the same vertex. Then:

(A) \(S=2660^\circ\) and \(P\) may be regular(B) \(S=2660^\circ\) and \(P\) is not regular(C) \(S=2700^\circ\) and \(P\) is regular(D) \(S=2700^\circ\) and \(P\) is not regular(E) \(S=2700^\circ\) and \(P\) may or may not be regular

28

方程 \[ x-\frac7{x-3}=3-\frac7{x-3} \] 的根的情况是：

A有无穷多个整数根 B无根 C有一个整数根 D有两个相等的整数根 E有两个相等的非整数根

English

The equation \[ x-\frac7{x-3}=3-\frac7{x-3} \] has:

(A) infinitely many integral roots(B) no root(C) one integral root(D) two equal integral roots(E) two equal non-integral roots

29

\(A\) 的钱的五倍加上 \(B\) 的钱多于 \(\$51.00\)。\(A\) 的钱的三倍减去 \(B\) 的钱等于 \(\$21.00\)。若 \(a\) 表示 \(A\) 的钱（美元），\(b\) 表示 \(B\) 的钱（美元），则：

A\(a\gt9\)，\(b\gt6\) B\(a\gt9\)，\(b\lt6\) C\(a\gt9\)，\(b=6\) D\(a\gt9\)，但无法给 \(b\) 确定界限 E\(2a=3b\)

English

Five times \(A\)’s money added to \(B\)’s money is more than \(\$51.00.\) Three times \(A\)’s money minus \(B\)’s money is \(\$21.00.\) If \(a\) represents \(A\)’s money in dollars and \(b\) represents \(B\)’s money in dollars, then:

(A) \(a\gt9,\) \(b\gt6\)(B) \(a\gt9,\) \(b\lt6\)(C) \(a\gt9,\) \(b=6\)(D) \(a\gt9,\) but we can put no bounds on \(b\)(E) \(2a=3b\)

30

给定直线 \(3x+5y=15\)，考察该直线上到两坐标轴距离相等的点。这样的点存在于：

A任何象限都没有 B仅第 \(\mathrm{I}\) 象限 C仅第 \(\mathrm{I}\)、\(\mathrm{II}\) 象限 D仅第 \(\mathrm{I}\)、\(\mathrm{II}\)、\(\mathrm{III}\) 象限 E每个象限

English

Given the line \(3x+5y=15\) and a point on this line equidistant from the coordinate axes. Such a point exists in:

(A) none of the quadrants(B) quadrant \(\mathrm{I}\) only(C) quadrants \(\mathrm{I},\) \(\mathrm{II}\) only(D) quadrants \(\mathrm{I},\) \(\mathrm{II},\) \(\mathrm{III}\) only(E) each of the quadrants

31

要使 \(x^2+2x+5\) 成为 \(x^4+px^2+q\) 的因式，\(p\) 和 \(q\) 的值必须依次为：

A\(-2\)，\(5\) B\(5\)，\(25\) C\(10\)，\(20\) D\(6\)，\(25\) E\(14\)，\(25\)

English

For \(x^2+2x+5\) to be a factor of \(x^4+px^2+q,\) the values of \(p\) and \(q\) must be, respectively:

(A) \(-2,\) \(5\)(B) \(5,\) \(25\)(C) \(10,\) \(20\)(D) \(6,\) \(25\)(E) \(14,\) \(25\)

32

在图中，圆心为 \(O\)。\(\overline{AB}\perp\overline{BC}\)，\(ADOE\) 是一条直线，\(\overline{AP}=\overline{AD}\)，且 \(\overline{AB}\) 的长度是半径的两倍。则：

A\(\overline{AP}^{\,2}=\overline{PB}\cdot\overline{AB}\) B\(\overline{AP}\cdot\overline{DO}=\overline{PB}\cdot\overline{AD}\) C\(\overline{AB}^{\,2}=\overline{AD}\cdot\overline{DE}\) D\(\overline{AB}\cdot\overline{AD}=\overline{OB}\cdot\overline{AO}\) E以上都不是

English

In this figure the center of the circle is \(O.\) \(\overline{AB}\perp\overline{BC},\) \(ADOE\) is a straight line, \(\overline{AP}=\overline{AD},\) and \(\overline{AB}\) has a length twice the radius. Then:

(A) \(\overline{AP}^{\,2}=\overline{PB}\cdot\overline{AB}\)(B) \(\overline{AP}\cdot\overline{DO}=\overline{PB}\cdot\overline{AD}\)(C) \(\overline{AB}^{\,2}=\overline{AD}\cdot\overline{DE}\)(D) \(\overline{AB}\cdot\overline{AD}=\overline{OB}\cdot\overline{AO}\)(E) none of these

33

给定一个含 \(58\) 项的数列；每一项都形如 \(P+n\)，其中 \(P\) 表示所有不超过 \(61\) 的质数的乘积 \(2\cdot3\cdot5\cdots61\)，而 \(n\) 依次取值 \(2\)、\(3\)、\(4\)、\(\ldots\)、\(59\)。设该数列中质数的个数为 \(N\)。则 \(N\) 是：

A\(0\) B\(16\) C\(17\) D\(57\) E\(58\)

English

You are given a sequence of \(58\) terms; each term has the form \(P+n\) where \(P\) stands for the product \(2\cdot3\cdot5\cdots61\) of all prime numbers less than or equal to \(61,\) and \(n\) takes, successively, the values \(2,\) \(3,\) \(4,\) \(\ldots,\) \(59.\) Let \(N\) be the number of primes appearing in this sequence. Then \(N\) is:

(A) \(0\)(B) \(16\)(C) \(17\)(D) \(57\)(E) \(58\)

34

两名游泳者分别位于一个长 \(90\) 英尺的泳池两端，同时开始沿泳池长度方向游泳，一人的速度为每秒 \(3\) 英尺，另一人的速度为每秒 \(2\) 英尺。他们来回游了 \(12\) 分钟。假设转身不耗费时间，求他们相互经过的次数。

A\(24\) B\(21\) C\(20\) D\(19\) E\(18\)

English

Two swimmers, at opposite ends of a \(90\)-foot pool, start to swim the length of the pool, one at the rate of \(3\) feet per second, the other at \(2\) feet per second. They swim back and forth for \(12\) minutes. Allowing no loss of time at the turns, find the number of times they pass each other.

(A) \(24\)(B) \(21\)(C) \(20\)(D) \(19\)(E) \(18\)

35

从周长为 \(10\) 个单位的圆外一点 \(P\) 作圆的切线；又从 \(P\) 作一条割线，将圆分成长度分别为 \(m\) 和 \(n\) 的两段不等弧。已知切线长 \(t\) 是 \(m\) 与 \(n\) 的比例中项。若 \(m\) 和 \(t\) 都是整数，则 \(t\) 可能取值的个数为：

A零个 B一个 C两个 D三个 E无穷多个

English

From point \(P\) outside a circle, with a circumference of \(10\) units, a tangent is drawn. Also from \(P\) a secant is drawn dividing the circle into unequal arcs with lengths \(m\) and \(n.\) It is found that \(t,\) the length of the tangent, is the mean proportional between \(m\) and \(n.\) If \(m\) and \(t\) are integers, then \(t\) may have the following number of values:

(A) zero(B) one(C) two(D) three(E) infinitely many

36

在同一个首项为 \(a\)、公差为 \(d\) 的等差数列中，设前 \(n\)、\(2n\)、\(3n\) 项的和依次为 \(s_1\)、\(s_2\)、\(s_3\)。令 \(R=s_3-s_2-s_1\)。则 \(R\) 取决于：

A\(a\) 和 \(d\) B\(d\) 和 \(n\) C\(a\) 和 \(n\) D\(a\)、\(d\) 和 \(n\) E不取决于 \(a\)、\(d\) 或 \(n\) 中的任何一个

English

Let \(s_1,\) \(s_2,\) \(s_3\) be the respective sums of \(n,\) \(2n,\) \(3n\) terms of the same arithmetic progression with \(a\) as the first term and \(d\) as the common difference. Let \(R=s_3-s_2-s_1.\) Then \(R\) is dependent on:

(A) \(a\) and \(d\)(B) \(d\) and \(n\)(C) \(a\) and \(n\)(D) \(a,\) \(d,\) and \(n\)(E) neither \(a\) nor \(d\) nor \(n\)

37

一个三角形的底长为 \(b\)，高为 \(h\)。在该三角形内接一个高为 \(x\) 的矩形，且矩形的底边在三角形的底边上。该矩形的面积是：

A\(\dfrac{bx}{h}(h-x)\) B\(\dfrac{hx}{b}(b-x)\) C\(\dfrac{bx}{h}(h-2x)\) D\(x(b-x)\) E\(x(h-x)\)

English

The base of a triangle is of length \(b,\) and the altitude is of length \(h.\) A rectangle of height \(x\) is inscribed in the triangle with the base of the rectangle in the base of the triangle. The area of the rectangle is:

(A) \(\dfrac{bx}{h}(h-x)\)(B) \(\dfrac{hx}{b}(b-x)\)(C) \(\dfrac{bx}{h}(h-2x)\)(D) \(x(b-x)\)(E) \(x(h-x)\)

38

在图中，\(\overline{AB}\) 和 \(\overline{AC}\) 是等腰三角形 \(ABC\) 的两条等长边，其中内接等边三角形 \(DEF\)。记角 \(BFD\) 为 \(a\)、角 \(ADE\) 为 \(b\)、角 \(FEC\) 为 \(c\)。则：

A\(b=\dfrac{a+c}{2}\) B\(b=\dfrac{a-c}{2}\) C\(a=\dfrac{b-c}{2}\) D\(a=\dfrac{b+c}{2}\) E以上都不是

English

In this diagram \(\overline{AB}\) and \(\overline{AC}\) are the equal sides of an isosceles triangle \(ABC,\) in which is inscribed equilateral triangle \(DEF.\) Designate angle \(BFD\) by \(a,\) angle \(ADE\) by \(b,\) and angle \(FEC\) by \(c.\) Then:

(A) \(b=\dfrac{a+c}{2}\)(B) \(b=\dfrac{a-c}{2}\)(C) \(a=\dfrac{b-c}{2}\)(D) \(a=\dfrac{b+c}{2}\)(E) none of these

39

要满足方程 \[ \frac{a+b}{a}=\frac{b}{a+b}\text{，}\] \(a\) 和 \(b\) 必须：

A都是有理数 B都是实数但不是有理数 C都不是实数 D一个是实数，另一个不是实数 E一个是实数而另一个不是，或者两者都不是实数

English

To satisfy the equation \[ \frac{a+b}{a}=\frac{b}{a+b}, \] \(a\) and \(b\) must be:

(A) both rational(B) both real but not rational(C) both not real(D) one real, one not real(E) one real, one not real or both not real

40

给定直角三角形 \(ABC\)，两直角边为 \(\overline{BC}=3\)、\(\overline{AC}=4\)。求从 \(C\) 引向斜边的两条三等分角线中较短一条的长度：

A\(\dfrac{32\sqrt3-24}{13}\) B\(\dfrac{12\sqrt3-9}{13}\) C\(6\sqrt3-8\) D\(\dfrac{5\sqrt{10}}6\) E\(\dfrac{25}{12}\)

English

Given right triangle \(ABC\) with legs \(\overline{BC}=3,\) \(\overline{AC}=4.\) Find the length of the shorter angle trisector from \(C\) to the hypotenuse:

(A) \(\dfrac{32\sqrt3-24}{13}\)(B) \(\dfrac{12\sqrt3-9}{13}\)(C) \(6\sqrt3-8\)(D) \(\dfrac{5\sqrt{10}}6\)(E) \(\dfrac{25}{12}\)