答题 · 1992 AMC 12 真题（P版）

1单选

若 \(3(4x+5\pi)=P\)，则 \(6(8x+10\pi)=\)

English（选项同上方）

If \(3(4x+5\pi)=P,\) then \(6(8x+10\pi)=\)

2单选

一个瓮中装有硬币和珠子，每件物品不是银制就是金制。瓮中物品的百分之二十是珠子，硬币的百分之四十是银制的。金制硬币占瓮中全部物品的百分之几？

English（选项同上方）

An urn is filled with coins and beads, all of which are either silver or gold. Twenty percent of the objects in the urn are beads. Forty percent of the coins in the urn are silver. What percent of the objects in the urn are gold coins?

3单选

若 \(m\gt0\)，且点 \((m,3)\) 和 \((1,m)\) 在一条斜率为 \(m\) 的直线上，则 \(m=\)

English（选项同上方）

If \(m\gt0\) and the points \((m,3)\) and \((1,m)\) lie on a line with slope \(m,\) then \(m=\)

4单选

若 \(a\)、\(b\) 和 \(c\) 都是正整数，且 \(a\) 与 \(b\) 都是奇数，则 \(3^a+(b-1)^2c\)

A对任意 \(c\) 都是奇数 B对任意 \(c\) 都是偶数 C当 \(c\) 为偶数时是奇数；当 \(c\) 为奇数时是偶数 D当 \(c\) 为奇数时是奇数；当 \(c\) 为偶数时是偶数 E当 \(c\) 不是 \(3\) 的倍数时是奇数；当 \(c\) 是 \(3\) 的倍数时是偶数

English（选项同上方）

If \(a,\) \(b,\) and \(c\) are positive integers and \(a\) and \(b\) are odd, then \(3^a+(b-1)^2c\) is

5单选

\(6^6+6^6+6^6+6^6+6^6+6^6=\)

English（选项同上方）

\(6^6+6^6+6^6+6^6+6^6+6^6=\)

6单选

若 \(x\gt y\gt0\)，则 \[ \frac{x^y y^x}{y^y x^x}= \]

English（选项同上方）

If \(x\gt y\gt0,\) then \[ \frac{x^y y^x}{y^y x^x}= \]

7单选

\(w\) 与 \(x\) 之比为 \(4:3\)，\(y\) 与 \(z\) 之比为 \(3:2\)，而 \(z\) 与 \(x\) 之比为 \(1:6\)。求 \(w\) 与 \(y\) 之比。

English（选项同上方）

The ratio of \(w\) to \(x\) is \(4:3,\) of \(y\) to \(z\) is \(3:2,\) and of \(z\) to \(x\) is \(1:6.\) What is the ratio of \(w\) to \(y?\)

8单选

一个正方形地面由全等的正方形瓷砖铺成。两条对角线上的瓷砖是黑色的，其余瓷砖是白色的。若共有 \(101\) 块黑色瓷砖，则瓷砖总数为

English（选项同上方）

A square floor is tiled with congruent square tiles. The tiles on the two diagonals of the floor are black. The rest of the tiles are white. If there are \(101\) black tiles, then the total number of tiles is

9单选

五个边长均为 \(2\sqrt3\) 的等边三角形排列在同一直线的一侧，且每个三角形都有一条边在该直线上。沿着这条直线，一个三角形底边的中点是下一个三角形的一个顶点。这五个三角形区域的并集所覆盖的平面区域面积为

English（选项同上方）

Five equilateral triangles, each with side \(2\sqrt3,\) are arranged so they are all on the same side of a line containing one side of each. Along this line, the midpoint of the base of one triangle is a vertex of the next. The area of the region of the plane that is covered by the union of the five triangular regions is

10单选

使方程 \[ kx-12=3k \] 对 \(x\) 有整数解的正整数 \(k\) 的个数为

English（选项同上方）

The number of positive integers \(k\) for which the equation \[ kx-12=3k \] has an integer solution for \(x\) is

11单选

两个同心圆的半径之比为 \(1:3\)。若 \(\overline{AC}\) 是大圆的直径，\(\overline{BC}\) 是大圆的一条弦且与小圆相切，并且 \(AB=12\)，则大圆的半径为

English（选项同上方）

The ratio of the radii of two concentric circles is \(1:3.\) If \(\overline{AC}\) is a diameter of the larger circle, \(\overline{BC}\) is a chord of the larger circle that is tangent to the smaller circle, and \(AB=12,\) then the radius of the larger circle is

12单选

直线 \(x-3y+11=0\) 关于 \(x\) 轴的对称图像为 \(y=mx+b\)。则 \(m+b\) 的值为

English（选项同上方）

Let \(y=mx+b\) be the image when the line \(x-3y+11=0\) is reflected across the \(x\)-axis. The value of \(m+b\) is

13单选

有多少个正整数对 \((a,b)\) 既满足 \(a+b\le100\)，又满足方程 \[ \frac{a+b^{-1}}{a^{-1}+b}=13\text{？}\]

English（选项同上方）

How many pairs of positive integers \((a,b)\) with \(a+b\le100\) satisfy the equation \[ \frac{a+b^{-1}}{a^{-1}+b}=13? \]

14单选

下列哪些方程的图像相同？\[ \begin{aligned} &\text{I. }y=x-2\\ &\text{II. }y=\dfrac{x^2-4}{x+2}\\ &\text{III. }(x+2)y=x^2-4 \end{aligned} \]

A仅 \(\mathrm{I}\) 和 \(\mathrm{II}\) B仅 \(\mathrm{I}\) 和 \(\mathrm{III}\) C仅 \(\mathrm{II}\) 和 \(\mathrm{III}\) D\(\mathrm{I}\)、\(\mathrm{II}\) 和 \(\mathrm{III}\) E都不是，所有方程的图像都不同

English（选项同上方）

Which of the following equations have the same graph? \[ \begin{aligned} &\text{I. }y=x-2\\ &\text{II. }y=\dfrac{x^2-4}{x+2}\\ &\text{III. }(x+2)y=x^2-4 \end{aligned} \]

15单选

令 \(i=\sqrt{-1}\)。对 \(n\ge1\)，按下式定义一个复数数列：\[ \begin{aligned} z_1&=0,\\ z_{n+1}&=z_n^2+i \end{aligned}\text{。}\]在复平面中，\(z_{111}\) 到原点的距离是多少？

English（选项同上方）

Let \(i=\sqrt{-1}.\) For \(n\ge1,\) define a sequence of complex numbers by \[ \begin{aligned} z_1&=0,\\ z_{n+1}&=z_n^2+i. \end{aligned} \] In the complex plane, how far from the origin is \(z_{111}?\)

16单选

若三个互不相同的正数 \(x\)、\(y\) 和 \(z\) 满足 \[ \frac{y}{x-z}=\frac{x+y}{z}=\frac{x}{y} \] 则 \(\frac{x}{y}=\)

English（选项同上方）

If \[ \frac{y}{x-z}=\frac{x+y}{z}=\frac{x}{y} \] for three positive numbers \(x,\) \(y,\) and \(z,\) all different, then \(\frac{x}{y}=\)

17单选

将从 \(19\) 到 \(92\) 的两位整数依次写在一起，形成大整数 \[ N=19202122\ldots909192\text{。}\]若 \(3^k\) 是 \(3\) 的幂中能整除 \(N\) 的最高次者，则 \(k=\)

English（选项同上方）

The two-digit integers from \(19\) to \(92\) are written consecutively to form the large integer \[ N=19202122\ldots909192. \] If \(3^k\) is the highest power of \(3\) that is a factor of \(N,\) then \(k=\)

18单选

递增正整数数列 \(a_1\)、\(a_2\)、\(a_3\)、\(\ldots\) 对每个 \(n\ge1\) 都满足 \[ a_{n+2}=a_n+a_{n+1}\text{。}\]若 \(a_7=120\)，则 \(a_8\) 为

English（选项同上方）

The increasing sequence of positive integers \(a_1,\) \(a_2,\) \(a_3,\) \(\ldots\) has the property that, for every \(n\ge1,\) \[ a_{n+2}=a_n+a_{n+1}. \] If \(a_7=120,\) then \(a_8\) is

19单选

对一个实心正方体的每个顶点，取由该顶点和与它相接的三条棱的中点所确定的四面体。切去这八个四面体后，正方体剩余的部分称为截半立方体。截半立方体与原正方体的体积之比最接近下列哪一项？

English（选项同上方）

For each vertex of a solid cube, consider the tetrahedron determined by the vertex and the midpoints of the three edges that meet at that vertex. The portion of the cube that remains when these eight tetrahedra are cut away is called a cuboctahedron. The ratio of the volume of the cuboctahedron to the volume of the original cube is closest to which of these?

20单选

图中显示了一个“\(n\) 角正星形”的一部分。它是一个简单闭合多边形，其中 \(2n\) 条边全等，角 \(A_1\)、\(A_2\)、\(\ldots\)、\(A_n\) 全等，角 \(B_1\)、\(B_2\)、\(\ldots\)、\(B_n\) 也全等。若 \(A_1\) 处的锐角比 \(B_1\) 处的锐角小 \(10^\circ\)，则 \(n=\)

English（选项同上方）

Part of an “\(n\)-pointed regular star” is shown. It is a simple closed polygon in which all \(2n\) edges are congruent, angles \(A_1,\) \(A_2,\) \(\ldots,\) \(A_n\) are congruent, and angles \(B_1,\) \(B_2,\) \(\ldots,\) \(B_n\) are congruent. If the acute angle at \(A_1\) is \(10^\circ\) less than the acute angle at \(B_1,\) then \(n=\)

21单选

对有限数列 \(A=(a_1,a_2,\ldots,a_n)\)，定义 \(A\) 的切萨罗和为 \[ \frac{S_1+S_2+\cdots+S_n}{n}\text{，}\]其中 \[ \begin{aligned} S_k&=a_1+a_2+a_3+\cdots+a_k\\ &\qquad(1\le k\le n) \end{aligned}\text{。}\]若 \(99\) 项数列 \((a_1,a_2,\ldots,a_{99})\) 的切萨罗和为 \(1000\)，则 \(100\) 项数列 \((1,a_1,a_2,\ldots,a_{99})\) 的切萨罗和是多少？

English（选项同上方）

For a finite sequence \(A=(a_1,a_2,\ldots,a_n)\) of numbers, the Cesàro sum of \(A\) is defined to be \[ \frac{S_1+S_2+\cdots+S_n}{n}, \] where \[ \begin{aligned} S_k&=a_1+a_2+a_3+\cdots+a_k\\ &\qquad(1\le k\le n). \end{aligned} \] If the Cesàro sum of the \(99\)-term sequence \((a_1,a_2,\ldots,a_{99})\) is \(1000,\) what is the Cesàro sum of the \(100\)-term sequence \((1,a_1,a_2,\ldots,a_{99})?\)

22单选

在 \(x\) 轴正半轴 \(\mathrm{X}^+\) 上选取十个点，在 \(y\) 轴正半轴 \(\mathrm{Y}^+\) 上选取五个点。连接 \(\mathrm{X}^+\) 上每个选定点与 \(\mathrm{Y}^+\) 上每个选定点，共画出五十条线段。位于第一象限内部的这些线段交点数最多可能是多少？

English（选项同上方）

Ten points are selected on the positive \(x\)-axis, \(\mathrm{X}^+,\) and five points are selected on the positive \(y\)-axis, \(\mathrm{Y}^+.\) The fifty segments connecting the ten selected points on \(\mathrm{X}^+\) to the five selected points on \(\mathrm{Y}^+\) are drawn. What is the maximum possible number of points of intersection of these fifty segments that could lie in the interior of the first quadrant?

23单选

若 \(S\) 是 \(\{1,2,3,\ldots,50\}\) 的子集，且 \(S\) 中任意两个不同元素之和都不能被 \(7\) 整除，则这样的子集最多能有多少个元素？

English（选项同上方）

What is the size of the largest subset \(S\) of \(\{1,2,3,\ldots,50\}\) such that no pair of distinct elements of \(S\) has a sum divisible by \(7?\)

24单选

设 \(ABCD\) 是面积为 \(10\) 的平行四边形，其中 \(AB=3\)、\(BC=5\)。点 \(E\)、\(F\) 和 \(G\) 分别位于线段 \(\overline{AB}\)、\(\overline{BC}\) 和 \(\overline{AD}\) 上，并且 \(AE=BF=AG=2\)。过 \(G\) 作平行于 \(\overline{EF}\) 的直线，与 \(\overline{CD}\) 交于 \(H\)。四边形 \(EFHG\) 的面积为

English（选项同上方）

Let \(ABCD\) be a parallelogram of area \(10\) with \(AB=3\) and \(BC=5.\) Locate \(E,\) \(F,\) and \(G\) on segments \(\overline{AB},\) \(\overline{BC},\) and \(\overline{AD},\) respectively, with \(AE=BF=AG=2.\) Let the line through \(G\) parallel to \(\overline{EF}\) intersect \(\overline{CD}\) at \(H.\) The area of the quadrilateral \(EFHG\) is

25单选

在三角形 \(ABC\) 中，\(\angle ABC=120^\circ\)、\(AB=3\)，且 \(BC=4\)。过 \(A\) 作 \(\overline{AB}\) 的垂线，过 \(C\) 作 \(\overline{BC}\) 的垂线，两条垂线交于 \(D\)，则 \(CD=\)

English（选项同上方）

In triangle \(ABC,\) \(\angle ABC=120^\circ,\) \(AB=3,\) and \(BC=4.\) If perpendiculars constructed to \(\overline{AB}\) at \(A\) and to \(\overline{BC}\) at \(C\) meet at \(D,\) then \(CD=\)

26单选

半圆弧 \(\overset{\frown}{AB}\) 的圆心为 \(C\)，半径为 \(1\)。点 \(D\) 在 \(\overset{\frown}{AB}\) 上，且 \(CD\perp AB\)。分别延长 \(\overline{BD}\) 和 \(\overline{AD}\) 至 \(E\) 和 \(F\)，使圆弧 \(\overset{\frown}{AE}\) 和 \(\overset{\frown}{BF}\) 的圆心分别为 \(B\) 和 \(A\)。圆弧 \(\overset{\frown}{EF}\) 的圆心为 \(D\)。阴影“笑脸”区域 \(AEFBDA\) 的面积为

English（选项同上方）

Semicircle \(\overset{\frown}{AB}\) has center \(C\) and radius \(1.\) Point \(D\) is on \(\overset{\frown}{AB}\) and \(CD\perp AB.\) Extend \(\overline{BD}\) and \(\overline{AD}\) to \(E\) and \(F,\) respectively, so that circular arcs \(\overset{\frown}{AE}\) and \(\overset{\frown}{BF}\) have \(B\) and \(A\) as their respective centers. Circular arc \(\overset{\frown}{EF}\) has center \(D.\) The area of the shaded “smile,” \(AEFBDA,\) is

27单选

一个半径为 \(r\) 的圆有长为 \(10\) 的弦 \(\overline{AB}\) 和长为 \(7\) 的弦 \(\overline{CD}\)。分别沿经过 \(B\) 和 \(C\) 的方向延长 \(\overline{AB}\) 与 \(\overline{CD}\)，它们在圆外的点 \(P\) 相交。若 \(\angle APD=60^\circ\) 且 \(BP=8\)，则 \(r^2=\)

English（选项同上方）

A circle of radius \(r\) has chords \(\overline{AB}\) of length \(10\) and \(\overline{CD}\) of length \(7.\) When \(\overline{AB}\) and \(\overline{CD}\) are extended through \(B\) and \(C,\) respectively, they intersect at \(P,\) which is outside the circle. If \(\angle APD=60^\circ\) and \(BP=8,\) then \(r^2=\)

28单选

令 \(i=\sqrt{-1}\)。方程 \(z^2-z=5-5i\) 的两个根的实部之积为

English（选项同上方）

Let \(i=\sqrt{-1}.\) The product of the real parts of the roots of \(z^2-z=5-5i\) is

29单选

一枚“不均匀”硬币出现正面的概率为 \(\frac23\)。若将这枚硬币抛掷 \(50\) 次，正面出现总次数为偶数的概率是多少？

English（选项同上方）

An “unfair” coin has a \(\frac23\) probability of turning up heads. If this coin is tossed \(50\) times, what is the probability that the total number of heads is even?

30单选

设 \(ABCD\) 是一个等腰梯形，两底为 \(AB=92\) 和 \(CD=19\)。假设 \(AD=BC=x\)，且一个圆的圆心在 \(\overline{AB}\) 上，并与线段 \(\overline{AD}\) 和 \(\overline{BC}\) 都相切。若 \(m\) 是 \(x\) 的最小可能值，则 \(m^2=\)

English（选项同上方）

Let \(ABCD\) be an isosceles trapezoid with bases \(AB=92\) and \(CD=19.\) Suppose \(AD=BC=x\) and a circle with center on \(\overline{AB}\) is tangent to segments \(\overline{AD}\) and \(\overline{BC}.\) If \(m\) is the smallest possible value of \(x,\) then \(m^2=\)