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AMC12 2002 A

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AMC12 · 2002 (A)

Q1
Compute the sum of all the roots of $(2x+3)(x-4)+(2x+3)(x-6)=0$
计算方程 $(2x+3)(x-4)+(2x+3)(x-6)=0$ 的所有根之和。
Q2
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。但她先减去了 9，然后将结果除以 3，得到答案 43。如果她正确完成问题，答案应该是多少？
Q3
According to the standard convention for exponentiation, \[2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.\] 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， \[2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.\] 如果改变进行指数运算的顺序，还可能得到多少个其他的值？
Q4
Find the degree measure of an angle whose complement is 25% of its supplement.
求一个角的度数，使得它的余角是它的补角的 25%。
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.
图中每个小圆的半径均为 1。最内侧的圆与围绕它的六个圆相切，而每个外侧小圆都与大圆及其相邻的小圆相切。求阴影区域的面积。
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Q6
For how many positive integers $m$ does there exist at least one positive integer n such that $m \cdot n \le m + n$?
有多少个正整数 $m$，使得存在至少一个正整数 $n$ 满足 $m \cdot n \le m + n$？
Q7
A $45^\circ$ arc of circle A is equal in length to a $30^\circ$ arc of circle B. What is the ratio of circle A's area and circle B's area?
圆 A 上的一个 $45^\circ$ 弧长与圆 B 上的一个 $30^\circ$ 弧长相等。圆 A 的面积与圆 B 的面积之比是多少？
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 $P$ the area of the red square. Which of the following is correct?
Betsy 用蓝色三角形、小白色正方形和一个红色中心正方形设计了一面旗帜，如图所示。设 $B$ 为蓝色三角形的总面积，$W$ 为白色正方形的总面积，$P$ 为红色正方形的面积。以下哪项正确？
stem
Q9
Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?
Jamal 想把 30 个文件存到软盘中，每张软盘容量为 1.44 MB。3 个文件各占 0.8 MB，12 个文件各占 0.7 MB，其余文件各占 0.4 MB。文件不能拆分存到两张不同的软盘上。存下全部 30 个文件所需的最少软盘数是多少？
Q10
Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then pours half the coffee from the first cup to the second and, after stirring thoroughly, pours 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样大小的杯子中。然后她把第一个杯子中一半的咖啡倒入第二个杯子，充分搅拌后，再把第二个杯子中一半的液体倒回第一个杯子。现在第一个杯子中的液体有多少分数是奶油？
Q11
Mr. Earl E. Bird gets up every day at 8:00 AM to go to work. If he drives at an average speed of 40 miles per hour, he will be late by 3 minutes. If he drives at an average speed of 60 miles per hour, he will be early by 3 minutes. How many miles per hour does Mr. Bird need to drive to get to work exactly on time?
Earl E. Bird 先生每天早上 8:00 起床去上班。如果他以平均速度 40 英里/小时开车，他会迟到 3 分钟。如果他以平均速度 60 英里/小时开车，他会早到 3 分钟。Bird 先生需要以多少英里/小时的速度开车才能恰好准时到达？
Q12
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 $x^2 - 63x + k = 0$ 的两个根都是质数。$k$ 的可能取值个数是
Q13
Two different positive numbers $a$ and $b$ each differ from their reciprocals by $1$. What is $a+b$?
两个不同的正数 $a$ 和 $b$ 各自与其倒数之差均为 $1$。求 $a+b$。
Q14
For all positive integers $n$, let $f(n)=\log_{2002} n^2$. Let $N=f(11)+f(13)+f(14)$. Which of the following relations is true?
对所有正整数 $n$，定义 $f(n)=\log_{2002} n^2$。令 $N=f(11)+f(13)+f(14)$。下列哪个关系成立？
Q15
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
由八个整数构成的一组数，其平均数、中位数、唯一众数和极差都等于 8。该组数中可能出现的最大整数是
Q16
Tina randomly selects two distinct numbers from the set $\{1,2,3,4,5\}$, and Sergio randomly selects a number from the set $\{1,2,\ldots,10\}$. What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?
蒂娜从集合 $\{1,2,3,4,5\}$ 中随机选取两个不同的数，塞尔吉奥从集合 $\{1,2,\ldots,10\}$ 中随机选取一个数。求塞尔吉奥选到的数大于蒂娜所选两个数之和的概率。
Q17
Several sets of prime numbers, such as $\{7,83,421,659\}$ use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?
一些质数集合（例如 $\{7,83,421,659\}$）恰好把九个非零数字各用一次。这样的质数集合可能的最小和是多少？
Q18
Let $C_1$ and $C_2$ be circles defined by $(x-10)^2 + y^2 = 36$ and $(x+15)^2 + y^2 = 81$ respectively. What is the length of the shortest line segment $PQ$ that is tangent to $C_1$ at $P$ and to $C_2$ at $Q$?
设圆 $C_1$ 和 $C_2$ 分别由 $(x-10)^2 + y^2 = 36$ 和 $(x+15)^2 + y^2 = 81$ 定义。与 $C_1$ 在点 $P$ 相切且与 $C_2$ 在点 $Q$ 相切的最短线段 $PQ$ 的长度是多少？
Q19
The graph of the function $f$ is shown below. How many solutions does the equation $f(f(x))=6$ have?
函数 $f$ 的图像如下所示。方程 $f(f(x))=6$ 有多少个解？
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Q20
Suppose that $a$ and $b$ are digits, not both nine and not both zero, and the repeating decimal $0.\overline{ab}$ is expressed as a fraction in lowest terms. How many different denominators are possible?
设 $a$ 和 $b$ 是数字，不同时都为 9 且不同时都为 0，将循环小数 $0.\overline{ab}$ 表示为最简分数。可能出现多少种不同的分母？
Q21
Consider the sequence of numbers: $4,7,1,8,9,7,6,\dots$ For $n>2$, the $n$-th term of the sequence is the units digit of the sum of the two previous terms. Let $S_n$ denote the sum of the first $n$ terms of this sequence. The smallest value of $n$ for which $S_n>10,000$ is:
考虑数列：$4,7,1,8,9,7,6,\dots$。对于 $n>2$，该数列的第 $n$ 项是前两项之和的个位数。设 $S_n$ 表示该数列前 $n$ 项的和。使得 $S_n>10,000$ 的最小 $n$ 是：
Q22
Triangle $ABC$ is a right triangle with $\angle ACB$ as its right angle, $m\angle ABC = 60^\circ$ , and $AB = 10$. Let $P$ be randomly chosen inside $ABC$ , and extend $\overline{BP}$ to meet $\overline{AC}$ at $D$. What is the probability that $BD > 5\sqrt2$?
三角形 $ABC$ 是直角三角形，且 $\angle ACB$ 为直角，$m\angle ABC = 60^\circ$，并且 $AB = 10$。在 $ABC$ 内随机选取点 $P$，将 $\overline{BP}$ 延长与 $\overline{AC}$ 相交于 $D$。$BD > 5\sqrt2$ 的概率是多少？
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Q23
In triangle $ABC$, side $AC$ and the perpendicular bisector of $BC$ meet in point $D$, and $BD$ bisects $\angle ABC$. If $AD=9$ and $DC=7$, what is the area of triangle $ABD$?
在三角形 $ABC$ 中，边 $AC$ 与 $BC$ 的垂直平分线相交于点 $D$，且 $BD$ 平分 $\angle ABC$。若 $AD=9$ 且 $DC=7$，求三角形 $ABD$ 的面积。
Q24
Find the number of ordered pairs of real numbers $(a,b)$ such that $(a+bi)^{2002} = a-bi$.
求满足 $(a+bi)^{2002} = a-bi$ 的实数有序对 $(a,b)$ 的个数。
Q25
The nonzero coefficients of a polynomial $P$ with real coefficients are all replaced by their mean to form a polynomial $Q$. Which of the following could be a graph of $y=P(x)$ and $y=Q(x)$ over the interval $-4\le x\le 4$?
将一个实系数多项式 $P$ 的所有非零系数都替换为它们的平均值，从而得到多项式 $Q$。下列哪一个可能是区间 $-4\le x\le 4$ 上 $y=P(x)$ 与 $y=Q(x)$ 的图像？
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