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AMC10 2002 A

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AMC10 · 2002 (A)

Q1
The ratio $\frac{10^{2000} + 10^{2002}}{10^{2001} + 10^{2001}}$ is closest to which of the following numbers?
比例 $\frac{10^{2000} + 10^{2002}}{10^{2001} + 10^{2001}}$ 最接近于下列哪个数？
Q2
For the nonzero numbers $a$, $b$, and $c$, define $(a, b, c) = \frac{a}{b} + \frac{b}{c} + \frac{c}{a}$. Find $(2, 12, 9)$.
对于非零数 $a$、$b$ 和 $c$，定义 $(a, b, c) = \frac{a}{b} + \frac{b}{c} + \frac{c}{a}$。求 $(2, 12, 9)$ 的值。
Q3
According to the standard convention for exponentiation, $2^{2^{2^2}} = 2^{(2^{(2^2)})} = 2^{16} = 65,536$. If the order in which the exponentiations are performed is changed, how many other values are possible?
根据指数运算的标准约定，$2^{2^{2^2}} = 2^{(2^{(2^2)})} = 2^{16} = 65,536$。如果改变指数运算的顺序，可能得到多少其他值？
Q4
For how many positive integers $m$ does there exist at least one positive integer $n$ such that $m \cdot n \leq m + n$?
有且仅有有多少个正整数 $m$，使得存在至少一个正整数 $n$ 满足 $m \cdot n \leq m + n$？
Q5
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.
图中小圆的半径均为 1。最内侧的圆与围绕它的六个圆相切，每个那些圆与大圆及其小圆邻居相切。求阴影区域的面积。
stem
Q6
Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly?
老师要求Cindy从某个数中减去3，然后将结果除以9。但她却先减去了9，然后将结果除以3，得到答案43。如果她正确计算，这个答案应该是多少？
Q7
If an arc of 45° on circle A has the same length as an arc of 30° on circle B, then the ratio of the area of circle A to the area of circle B is
如果圆A上45°的弧长与圆B上30°的弧长相等，则圆A的面积与圆B的面积之比是
Q8
Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let $B$ be the total area of the blue triangles, $W$ the total area of the white squares, and $R$ the area of the red square. Which of the following is correct?
Betsy设计了一面旗帜，使用蓝色三角形、小白色方块和一个红色中心方块，如图所示。设 $B$ 为蓝色三角形的总面积，$W$ 为白色方块的总面积，$R$ 为红色方块的面积。以下哪个正确？
stem
Q9
Suppose $A$, $B$, and $C$ are three numbers for which $1001C - 2002A = 4004$ and $1001B + 3003A = 5005$. The average of the three numbers $A$, $B$, and $C$ is
假设 $A$、$B$ 和 $C$ 是三个数，使得 $1001C - 2002A = 4004$ 和 $1001B + 3003A = 5005$。这三个数 $A$、$B$ 和 $C$ 的平均值是
Q10
Compute the sum of all the roots of $(2x + 3)(x - 4) + (2x + 3)(x - 6) = 0$.
计算 $(2x + 3)(x - 4) + (2x + 3)(x - 6) = 0$ 所有根之和。
Q11
Jamal wants to store 30 computer files on floppy disks, each of which has a capacity of 1.44 megabytes (mb). Three of his files require 0.8 mb of memory each, 12 more require 0.7 mb each, and the remaining 15 require 0.4 mb each. No file can be split between floppy disks. What is the minimal number of floppy disks that will hold all the files?
Jamal 想要将 30 个计算机文件存储在软盘上，每个软盘容量为 1.44 兆字节 (mb)。其中 3 个文件每个需要 0.8 mb 内存，另外 12 个每个需要 0.7 mb，剩余 15 个每个需要 0.4 mb。文件不能分割存储在软盘之间。需要的最少软盘数量是多少？
Q12
Mr. Earl E. Bird leaves his house for work at exactly 8:00 A.M. every morning. When he averages 40 miles per hour, he arrives at his workplace three minutes late. When he averages 60 miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace precisely on time?
Earl E. Bird 先生每天早上正好 8:00 离开家去上班。当他平均速度为 40 英里/小时时，到达工作地点晚了 3 分钟。当平均速度为 60 英里/小时时，早到了 3 分钟。Bird 先生应该以多少平均速度（英里/小时）开车才能准时到达工作地点？
Q13
The sides of a triangle have lengths of 15, 20, and 25. Find the length of the shortest altitude.
一个三角形的边长为 15、20 和 25。求最短的高的长度。
Q14
Both roots of the quadratic equation $x^2 - 63x + k = 0$ are prime numbers. The number of possible values of $k$ is
二次方程 $x^2 - 63x + k = 0$ 的两个根均为素数。$k$ 的可能值的个数是
Q15
The digits 1, 2, 3, 4, 5, 6, 7, and 9 are used to form four two-digit prime numbers, with each digit used exactly once. What is the sum of these four primes?
用数字 1、2、3、4、5、6、7 和 9 构成四个两位素数，每个数字恰好使用一次。这四个素数的和是多少？
Q16
If $a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5$, then $a + b + c + d$ is
如果 $a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5$，则 $a + b + c + d$ 是
Q17
Sarah pours four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then transfers half the coffee from the first cup to the second and, after stirring thoroughly, transfers half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream?
Sarah 将四盎司咖啡倒入一个八盎司的杯子中，并将四盎司奶油倒入另一个同等大小的杯子中。然后她将第一杯中一半的咖啡转移到第二杯中，彻底搅拌后，将第二杯中一半的液体转移回第一杯。第一杯中的液体现在有多少是奶油？
Q18
A $3 \times 3 \times 3$ cube is formed by gluing together 27 standard cubical dice. (On a standard die, the sum of the numbers on any pair of opposite faces is 7.) The smallest possible sum of all the numbers showing on the surface of the $3 \times 3 \times 3$ cube is
一个 $3 \times 3 \times 3$ 立方体由粘合在一起的 27 个标准骰子构成。（标准骰子上，任意一对相对面的数字和为 7。）$3 \times 3 \times 3$ 立方体表面上所有数字的最小可能总和是
Q19
Spot’s doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside the doghouse that Spot can reach?
Spot 的狗屋有一个边长一码的正六边形底座。他被一根两码长的绳子拴在一个顶点上。Spot 能到达的狗屋外部区域面积有多少平方码？
Q20
Points A, B, C, D, E, and F lie, in that order, on AF, dividing it into five segments, each of length 1. Point G is not on line AF. Point H lies on GD, and point J lies on GF. The line segments HC, JE, and AG are parallel. Find HC/JE.
点 A、B、C、D、E 和 F 按顺序位于 AF 上，将其分成五个长度均为 1 的线段。点 G 不在 AF 线上。点 H 在 GD 上，点 J 在 GF 上。线段 HC、JE 和 AG 平行。求 HC/JE。
stem
Q21
The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is
一组八个整数的平均数、中位数、唯一众数和范围均为8。该集合中可能的最大整数是
Q22
A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?
一组编号为1至100的瓦片通过以下操作反复修改：移除所有完美平方编号的瓦片，并将剩余瓦片从1开始连续重新编号。要将瓦片数量减少到1次，需要执行该操作多少次？
Q23
Points A, B, C, and D lie on a line, in that order, with AB = CD and BC = 12. Point E is not on the line, and BE = CE = 10. The perimeter of $\triangle AED$ is twice the perimeter of $\triangle BEC$. Find AB.
点 A、B、C、D 在一条直线上，按此顺序，AB = CD，BC = 12。点 E 不在直线上，且 BE = CE = 10。$ riangle AED$ 的周长是 $\triangle BEC$ 周长的两倍。求 AB。
stem
Q24
Tina randomly selects two distinct numbers from the set {1, 2, 3, 4, 5}, and Sergio randomly selects a number from the set {1, 2, ..., 10}. The probability that Sergio’s number is larger than the sum of the two numbers chosen by Tina is
Tina 从集合 {1, 2, 3, 4, 5} 中随机选取两个不同的数，Sergio 从 {1, 2, ..., 10} 中随机选取一个数。Sergio 的数大于 Tina 选取两个数之和的概率是
Q25
In trapezoid ABCD with bases AB and CD, we have AB = 52, BC = 12, CD = 39, and DA = 5. The area of ABCD is
梯形 ABCD 底边 AB 和 CD，有 AB = 52，BC = 12，CD = 39，DA = 5。ABCD 的面积是
stem
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