1975 AMC 12 真题（P版）

1

下列表达式 \[ \frac{1}{2-\dfrac{1}{2-\dfrac{1}{2-\frac12}}} \] 的值是

A\(\frac34\) B\(\frac45\) C\(\frac56\) D\(\frac67\) E\(\frac65\)

English

The value of \[ \frac{1}{2-\dfrac{1}{2-\dfrac{1}{2-\frac12}}} \] is

(A) \(\frac34\)(B) \(\frac45\)(C) \(\frac56\)(D) \(\frac67\)(E) \(\frac65\)

2

对于哪些实数 \(m\)，联立方程 \[ \begin{aligned} y&=mx+3,\\ y&=(2m-1)x+4 \end{aligned} \] 至少有一组实数解 \((x,y)\)？

A所有 \(m\) B所有 \(m\ne0\) C所有 \(m\ne\frac12\) D所有 \(m\ne1\) E不存在这样的 \(m\)

English

For which real values of \(m\) are the simultaneous equations \[ \begin{aligned} y&=mx+3,\\ y&=(2m-1)x+4 \end{aligned} \] satisfied by at least one pair of real numbers \((x,y)?\)

(A) all \(m\)(B) all \(m\ne0\)(C) all \(m\ne\frac12\)(D) all \(m\ne1\)(E) no values of \(m\)

3

对于所有实数 \(a\)、\(b\)、\(c\)、\(x\)、\(y\)、\(z\)，若它们满足 \(x\lt a\)、\(y\lt b\)、\(z\lt c\)，则下列哪些不等式恒成立？

I. \(xy+yz+zx\lt ab+bc+ca\)

II. \(x^2+y^2+z^2\lt a^2+b^2+c^2\)

III. \(xyz\lt abc\)

A没有一个恒成立。 B只有 \(\mathrm{I}\) C只有 \(\mathrm{II}\) D只有 \(\mathrm{III}\) E全部恒成立。

English

Which of the following inequalities are satisfied for all real numbers \(a,\) \(b,\) \(c,\) \(x,\) \(y,\) \(z\) which satisfy \(x\lt a,\) \(y\lt b,\) and \(z\lt c?\)

I. \(xy+yz+zx\lt ab+bc+ca\)

II. \(x^2+y^2+z^2\lt a^2+b^2+c^2\)

III. \(xyz\lt abc\)

(A) None are satisfied.(B) \(\mathrm{I}\) only(C) \(\mathrm{II}\) only(D) \(\mathrm{III}\) only(E) All are satisfied.

4

若第一个正方形的一条边等于第二个正方形的一条对角线，则第一个正方形与第二个正方形的面积之比是多少？

A\(2\) B\(\sqrt2\) C\(\frac12\) D\(2\sqrt2\) E\(4\)

English

If the side of one square is the diagonal of a second square, what is the ratio of the area of the first square to the area of the second?

(A) \(2\)(B) \(\sqrt2\)(C) \(\frac12\)(D) \(2\sqrt2\)(E) \(4\)

5

将多项式 \((x+y)^9\) 按 \(x\) 的降幂展开。在 \(x=p\)、\(y=q\) 处，展开式的第二项与第三项取值相等，其中 \(p\)、\(q\) 均为正数且和为一。\(p\) 的值是多少？

A\(\frac15\) B\(\frac45\) C\(\frac14\) D\(\frac34\) E\(\frac89\)

English

The polynomial \((x+y)^9\) is expanded in decreasing powers of \(x.\) The second and third terms have equal values when evaluated at \(x=p\) and \(y=q,\) where \(p\) and \(q\) are positive numbers whose sum is one. What is the value of \(p?\)

(A) \(\frac15\)(B) \(\frac45\)(C) \(\frac14\)(D) \(\frac34\)(E) \(\frac89\)

6

前八十个正偶数之和减去前八十个正奇数之和等于

A\(0\) B\(20\) C\(40\) D\(60\) E\(80\)

English

The sum of the first eighty positive odd integers subtracted from the sum of the first eighty positive even integers is

(A) \(0\)(B) \(20\)(C) \(40\)(D) \(60\)(E) \(80\)

7

对于哪些非零实数 \(x\)，\(\frac{|x-|x||}{x}\) 是正整数？

A仅当 \(x\) 为负数时 B仅当 \(x\) 为正数时 C仅当 \(x\) 为偶整数时 D对所有非零实数 \(x\) E不存在这样的非零实数 \(x\)

English

For which nonzero real numbers \(x\) is \(\frac{|x-|x||}{x}\) a positive integer?

(A) for negative \(x\) only(B) for positive \(x\) only(C) only for \(x\) an even integer(D) for all nonzero real numbers \(x\)(E) for no nonzero real numbers \(x\)

8

若命题“这家商店的所有衬衫都在打折”为假，则下列哪些命题一定为真？

I. 这家商店的所有衬衫都按原价出售。

II. 这家商店里至少有一件衬衫不打折。

III. 这家商店里没有衬衫在打折。

IV. 这家商店的衬衫并非全都在打折。

注：命题 I 原文为：这家商店的所有衬衫都不打折。

A只有 \(\mathrm{II}\) B只有 \(\mathrm{IV}\) C只有 \(\mathrm{I}\) 和 \(\mathrm{III}\) D只有 \(\mathrm{II}\) 和 \(\mathrm{IV}\) E只有 \(\mathrm{I}\)、\(\mathrm{II}\) 和 \(\mathrm{IV}\)

English

If the statement “All shirts in this store are on sale.” is false, then which of the following statements must be true?

I. All shirts in this store are at non-sale prices.

II. There is some shirt in this store not on sale.

III. No shirt in this store is on sale.

IV. Not all shirts in this store are on sale.

Note: Originally, statement I read: All shirts in this store are not on sale.

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

9

已知 \(a_1\)、\(a_2\)、\(\ldots\) 与 \(b_1\)、\(b_2\)、\(\ldots\) 都是等差数列，且 \(a_1=25\)、\(b_1=75\)、\(a_{100}+b_{100}=100\)。求数列 \(a_1+b_1\)、\(a_2+b_2\)、\(\ldots\) 的前一百项之和。

A\(0\) B\(100\) C\(10{,}000\) D\(505{,}000\) E所给信息不足以求解

English

Let \(a_1,\) \(a_2,\) \(\ldots\) and \(b_1,\) \(b_2,\) \(\ldots\) be arithmetic progressions such that \(a_1=25,\) \(b_1=75,\) and \(a_{100}+b_{100}=100.\) Find the sum of the first one hundred terms of the progression \(a_1+b_1,\) \(a_2+b_2,\) \(\ldots.\)

(A) \(0\)(B) \(100\)(C) \(10{,}000\)(D) \(505{,}000\)(E) not enough information given to solve the problem

10

设 \(n\) 是正整数，则 \(\left(10^{4n^2+8}+1\right)^2\) 的十进制各位数字之和为

A\(4\) B\(4n\) C\(2+2n\) D\(4n^2\) E\(n^2+n+2\)

English

The sum of the digits in base ten of \(\left(10^{4n^2+8}+1\right)^2,\) where \(n\) is a positive integer, is

(A) \(4\)(B) \(4n\)(C) \(2+2n\)(D) \(4n^2\)(E) \(n^2+n+2\)

11

设 \(P\) 是圆 \(K\) 内异于圆 \(K\) 圆心的一点。作圆 \(K\) 中所有经过 \(P\) 的弦，并取这些弦的中点。这些中点的轨迹是

A去掉一点的圆 B若 \(P\) 到圆 \(K\) 圆心的距离小于圆 \(K\) 半径的一半，则为一个圆；否则为小于 \(360^\circ\) 的一段圆弧 C去掉一点的半圆 D一个半圆 E一个圆

English

Let \(P\) be an interior point of circle \(K\) other than the center of \(K.\) Form all chords of \(K\) which pass through \(P,\) and determine their midpoints. The locus of these midpoints is

(A) a circle with one point deleted(B) a circle if the distance from \(P\) to the center of \(K\) is less than one half the radius of \(K;\) otherwise a circular arc of less than \(360^\circ\)(C) a semicircle with one point deleted(D) a semicircle(E) a circle

12

若 \(a\ne b\)、\(a^3-b^3=19x^3\)、\(a-b=x\)，则下列哪个结论正确？

A\(a=3x\) B\(a=3x\) 或 \(a=-2x\) C\(a=-3x\) 或 \(a=2x\) D\(a=3x\) 或 \(a=2x\) E\(a=2x\)

English

If \(a\ne b,\) \(a^3-b^3=19x^3,\) and \(a-b=x,\) which of the following conclusions is correct?

(A) \(a=3x\)(B) \(a=3x\) or \(a=-2x\)(C) \(a=-3x\) or \(a=2x\)(D) \(a=3x\) or \(a=2x\)(E) \(a=2x\)

13

方程 \(x^6-3x^5-6x^3-x+8=0\) 有

A没有实根 B恰有两个不同的负根 C恰有一个负根 D没有负根，但至少有一个正根 E以上都不对

English

The equation \(x^6-3x^5-6x^3-x+8=0\) has

(A) no real roots(B) exactly two distinct negative roots(C) exactly one negative root(D) no negative roots, but at least one positive root(E) none of these

14

若当“乙量”等于“丙量”且“丁量加丁量”等于“丙量”\(\cdot\)“丁量”时，“甲量”等于“丁量”，那么当“乙量”等于“丁量”、“丁量加丁量”等于“丁量”\(\cdot\)“丁量”且“丙量”等于二时，“乙量”\(\cdot\)“甲量”等于什么（甲量、乙量、丙量和丁量都是取正值的变量）？

A乙量 \(\cdot\) 丙量 \(\cdot\) 丁量 B乙量 C丙量 D丁量 E丁量加丁量

English

If the whatsis is so when the whosis is is and the so and so is is \(\cdot\) so, what is the whosis \(\cdot\) whatsis when the whosis is so, the so and so is so \(\cdot\) so, and the is is two (whatsis, whosis, is and so are variables taking positive values)?

(A) whosis \(\cdot\) is \(\cdot\) so(B) whosis(C) is(D) so(E) so and so

15

数列 \(1\)、\(3\)、\(2\)、\(\ldots\) 中，从第三项起，每一项都等于紧邻它的前一项减去再前一项。该数列前一百项之和为

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

English

In the sequence of numbers \(1,\) \(3,\) \(2,\) \(\ldots,\) each term after the first two is equal to the term preceding it minus the term preceding that. The sum of the first one hundred terms of the sequence is

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

16

一个无穷等比级数的首项为正整数，公比为正整数的倒数，且级数之和为 \(3\)，则其前两项之和为

A\(\frac13\) B\(\frac23\) C\(\frac83\) D\(2\) E\(\frac92\)

English

If the first term of an infinite geometric series is a positive integer, the common ratio is the reciprocal of a positive integer, and the sum of the series is \(3,\) then the sum of the first two terms of the series is

(A) \(\frac13\)(B) \(\frac23\)(C) \(\frac83\)(D) \(2\)(E) \(\frac92\)

17

一名男子上下班可以乘火车或公交车。如果早晨乘火车上班，他下午就乘公交车回家；如果下午乘火车回家，他早晨就乘公交车上班。在总共 \(x\) 个工作日中，他早晨乘公交车上班 \(8\) 次，下午乘公交车回家 \(15\) 次，并且共乘火车通勤（早晨或下午）\(9\) 次。求 \(x\)。

A\(19\) B\(18\) C\(17\) D\(16\) E所给信息不足以求解

English

A man can commute either by train or by bus. If he goes to work on the train in the morning, he comes home on the bus in the afternoon; and if he comes home in the afternoon on the train, he took the bus in the morning. During a total of \(x\) working days, the man took the bus to work in the morning \(8\) times, came home by bus in the afternoon \(15\) times, and commuted by train (either morning or afternoon) \(9\) times. Find \(x.\)

(A) \(19\)(B) \(18\)(C) \(17\)(D) \(16\)(E) not enough information given to solve the problem

18

从所有十进制三位正整数中等可能地随机选取一个正整数 \(N\)。\(\log_2N\) 为整数的概率是

A\(0\) B\(\frac3{899}\) C\(\frac1{225}\) D\(\frac1{300}\) E\(\frac1{450}\)

English

A positive integer \(N\) with three digits in its base ten representation is chosen at random, with each three-digit number having an equal chance of being chosen. The probability that \(\log_2N\) is an integer is

(A) \(0\)(B) \(\frac3{899}\)(C) \(\frac1{225}\)(D) \(\frac1{300}\)(E) \(\frac1{450}\)

19

哪些正数 \(x\) 满足方程 \((\log_3x)(\log_x5)=\log_35\)？

A只有 \(3\) 和 \(5\) B只有 \(3\)、\(5\) 和 \(15\) C只有形如 \(5^n\cdot3^m\) 的数，其中 \(n\) 和 \(m\) 为正整数 D所有满足 \(x\ne1\) 的正数 E以上都不对

English

Which positive numbers \(x\) satisfy the equation \((\log_3x)(\log_x5)=\log_35?\)

(A) \(3\) and \(5\) only(B) \(3,\) \(5\) and \(15\) only(C) only numbers of the form \(5^n\cdot3^m,\) where \(n\) and \(m\) are positive integers(D) all positive \(x\ne1\)(E) none of these

20

如图，在三角形 \(ABC\) 中，\(AB=4\)、\(AC=8\)。若 \(M\) 是 \(BC\) 的中点且 \(AM=3\)，则 \(BC\) 的长度是多少？

A\(2\sqrt{26}\) B\(2\sqrt{31}\) C\(9\) D\(4+2\sqrt{13}\) E所给信息不足以求解

English

In the adjoining figure triangle \(ABC\) is such that \(AB=4\) and \(AC=8.\) If \(M\) is the midpoint of \(BC\) and \(AM=3,\) what is the length of \(BC?\)

(A) \(2\sqrt{26}\)(B) \(2\sqrt{31}\)(C) \(9\)(D) \(4+2\sqrt{13}\)(E) not enough information given to solve the problem

21

设 \(f(x)\) 对所有实数 \(x\) 都有定义；\(f(x)\gt0\) 对所有 \(x\) 成立；且 \(f(a)f(b)=f(a+b)\) 对所有 \(a\) 和 \(b\) 成立。下列哪些命题为真？

I. \(f(0)=1\)

II. \(f(-a)=\frac{1}{f(a)}\) 对所有 \(a\) 成立

III. \(f(a)=\sqrt[3]{f(3a)}\) 对所有 \(a\) 成立

IV. \(f(b)\gt f(a)\)，只要 \(b\gt a\)

A只有 \(\mathrm{III}\) 和 \(\mathrm{IV}\) B只有 \(\mathrm{I}\)、\(\mathrm{III}\) 和 \(\mathrm{IV}\) C只有 \(\mathrm{I}\)、\(\mathrm{II}\) 和 \(\mathrm{IV}\) D只有 \(\mathrm{I}\)、\(\mathrm{II}\) 和 \(\mathrm{III}\) E全部为真。

English

Suppose \(f(x)\) is defined for all real numbers \(x;\) \(f(x)\gt0\) for all \(x;\) and \(f(a)f(b)=f(a+b)\) for all \(a\) and \(b.\) Which of the following statements are true?

I. \(f(0)=1\)

II. \(f(-a)=\frac{1}{f(a)}\) for all \(a\)

III. \(f(a)=\sqrt[3]{f(3a)}\) for all \(a\)

IV. \(f(b)\gt f(a)\) if \(b\gt a\)

(A) \(\mathrm{III}\) and \(\mathrm{IV}\) only(B) \(\mathrm{I},\) \(\mathrm{III}\) and \(\mathrm{IV}\) only(C) \(\mathrm{I},\) \(\mathrm{II}\) and \(\mathrm{IV}\) only(D) \(\mathrm{I},\) \(\mathrm{II}\) and \(\mathrm{III}\) only(E) All are true.

22

若 \(p\) 和 \(q\) 都是质数，且 \(x^2-px+q=0\) 有两个不同的正整数根，则下列哪些命题为真？

I. 两根之差为奇数。

II. 至少有一个根是质数。

III. \(p^2-q\) 是质数。

IV. \(p+q\) 是质数。

A只有 \(\mathrm{I}\) B只有 \(\mathrm{II}\) C只有 \(\mathrm{II}\) 和 \(\mathrm{III}\) D只有 \(\mathrm{I}\)、\(\mathrm{II}\) 和 \(\mathrm{IV}\) E全部为真。

English

If \(p\) and \(q\) are primes and \(x^2-px+q=0\) has distinct positive integral roots, then which of the following statements are true?

I. The difference of the roots is odd.

II. At least one root is prime.

III. \(p^2-q\) is prime.

IV. \(p+q\) is prime.

(A) \(\mathrm{I}\) only(B) \(\mathrm{II}\) only(C) \(\mathrm{II}\) and \(\mathrm{III}\) only(D) \(\mathrm{I},\) \(\mathrm{II}\) and \(\mathrm{IV}\) only(E) All are true.

23

如图，\(AB\) 和 \(BC\) 是正方形 \(ABCD\) 的相邻两边；\(M\) 是 \(AB\) 的中点；\(N\) 是 \(BC\) 的中点；\(AN\) 与 \(CM\) 交于 \(O\)。四边形 \(AOCD\) 与正方形 \(ABCD\) 的面积之比为

A\(\frac56\) B\(\frac34\) C\(\frac23\) D\(\frac{\sqrt3}{2}\) E\(\frac{\sqrt3-1}{2}\)

English

In the adjoining figure \(AB\) and \(BC\) are adjacent sides of square \(ABCD;\) \(M\) is the midpoint of \(AB;\) \(N\) is the midpoint of \(BC;\) and \(AN\) and \(CM\) intersect at \(O.\) The ratio of the area of \(AOCD\) to the area of \(ABCD\) is

(A) \(\frac56\)(B) \(\frac34\)(C) \(\frac23\)(D) \(\frac{\sqrt3}{2}\)(E) \(\frac{\sqrt3-1}{2}\)

24

在三角形 \(ABC\) 中，\(\angle C=\theta\)、\(\angle B=2\theta\)，其中 \(0^\circ\lt\theta\lt60^\circ\)。以 \(A\) 为圆心、\(AB\) 为半径的圆与 \(AC\) 交于 \(D\)，并与 \(BC\)（必要时延长）交于 \(B\) 和 \(E\)（\(E\) 可能与 \(B\) 重合）。则等式 \(EC=AD\)

A对任何 \(\theta\) 都不成立 B仅当 \(\theta=45^\circ\) 时成立 C仅当 \(0^\circ\lt\theta\le45^\circ\) 时成立 D仅当 \(45^\circ\le\theta\lt60^\circ\) 时成立 E对所有 \(\theta\) 都成立，其中 \(0^\circ\lt\theta\lt60^\circ\)

English

In triangle \(ABC,\) \(\angle C=\theta\) and \(\angle B=2\theta,\) where \(0^\circ\lt\theta\lt60^\circ.\) The circle with center \(A\) and radius \(AB\) intersects \(AC\) at \(D\) and intersects \(BC,\) extended if necessary, at \(B\) and at \(E\) (\(E\) may coincide with \(B\)). Then \(EC=AD\)

(A) for no values of \(\theta\)(B) only if \(\theta=45^\circ\)(C) only if \(0^\circ\lt\theta\le45^\circ\)(D) only if \(45^\circ\le\theta\lt60^\circ\)(E) for all \(\theta\) such that \(0^\circ\lt\theta\lt60^\circ\)

25

一名女子、她的兄弟、她的儿子和她的女儿都是国际象棋棋手（所有亲属关系均为血缘关系）。水平最差者的双胞胎手足也是这四人之一，且与水平最佳者性别相反。水平最差者与水平最佳者年龄相同。谁的水平最差？

A这名女子 B她的儿子 C她的兄弟 D她的女儿 E不存在符合所给信息的答案。

English

A woman, her brother, her son and her daughter are chess players (all relations by birth). The worst player’s twin (who is one of the four players) and the best player are of opposite sex. The worst player and the best player are the same age. Who is the worst player?

(A) the woman(B) her son(C) her brother(D) her daughter(E) No solution is consistent with the given information.

26

在锐角三角形 \(ABC\) 中，\(\angle A\) 的角平分线与边 \(BC\) 交于 \(D\)。以 \(B\) 为圆心、\(BD\) 为半径的圆与边 \(AB\) 交于 \(M\)；以 \(C\) 为圆心、\(CD\) 为半径的圆与边 \(AC\) 交于 \(N\)。则恒有

A\(\begin{aligned}\angle CND+\angle BMD\\{}-\angle DAC=120^\circ\end{aligned}\) B\(AMDN\) 是梯形 C\(BC\) 平行于 \(MN\) D\(AM-AN=\frac{3(DB-DC)}2\) E\(AB-AC=\frac{3(DB-DC)}2\)

English

In acute triangle \(ABC\) the bisector of \(\angle A\) meets side \(BC\) at \(D.\) The circle with center \(B\) and radius \(BD\) intersects side \(AB\) at \(M;\) and the circle with center \(C\) and radius \(CD\) intersects side \(AC\) at \(N.\) Then it is always true that

(A) \(\begin{aligned}\angle CND+\angle BMD\\{}-\angle DAC=120^\circ\end{aligned}\)(B) \(AMDN\) is a trapezoid(C) \(BC\) is parallel to \(MN\)(D) \(AM-AN=\frac{3(DB-DC)}2\)(E) \(AB-AC=\frac{3(DB-DC)}2\)

27

若 \(p\)、\(q\)、\(r\) 是方程 \(x^3-x^2+x-2=0\) 的三个不同根，则 \(p^3+q^3+r^3\) 等于

A\(-1\) B\(1\) C\(3\) D\(5\) E以上都不对

English

If \(p,\) \(q,\) and \(r\) are distinct roots of \(x^3-x^2+x-2=0,\) then \(p^3+q^3+r^3\) equals

(A) \(-1\)(B) \(1\)(C) \(3\)(D) \(5\)(E) none of these

28

如图，在三角形 \(ABC\) 中，\(M\) 是边 \(BC\) 的中点，\(AB=12\)、\(AC=16\)。点 \(E\) 和 \(F\) 分别取在 \(AC\) 和 \(AB\) 上，直线 \(EF\) 与 \(AM\) 交于 \(G\)。若 \(AE=2AF\)，则 \(\frac{EG}{GF}\) 等于

A\(\frac32\) B\(\frac43\) C\(\frac54\) D\(\frac65\) E所给信息不足以求解

English

In triangle \(ABC\) shown in the adjoining figure, \(M\) is the midpoint of side \(BC,\) \(AB=12,\) and \(AC=16.\) Points \(E\) and \(F\) are taken on \(AC\) and \(AB,\) respectively, and lines \(EF\) and \(AM\) intersect at \(G.\) If \(AE=2AF,\) then \(\frac{EG}{GF}\) equals

(A) \(\frac32\)(B) \(\frac43\)(C) \(\frac54\)(D) \(\frac65\)(E) not enough information given to solve the problem

29

大于 \((\sqrt3+\sqrt2)^6\) 的最小整数是多少？

A\(972\) B\(971\) C\(970\) D\(969\) E\(968\)

English

What is the smallest integer larger than \((\sqrt3+\sqrt2)^6?\)

(A) \(972\)(B) \(971\)(C) \(970\)(D) \(969\)(E) \(968\)

30

令 \(x=\cos36^\circ-\cos72^\circ\)。则 \(x\) 等于

A\(\frac13\) B\(\frac12\) C\(3-\sqrt6\) D\(2\sqrt3-3\) E以上都不对

English

Let \(x=\cos36^\circ-\cos72^\circ.\) Then \(x\) equals

(A) \(\frac13\)(B) \(\frac12\)(C) \(3-\sqrt6\)(D) \(2\sqrt3-3\)(E) none of these