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AMC12 2004 A

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AMC12 · 2004 (A)

Q1
Alicia earns 20 dollars per hour, of which $1.45\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?
Alicia 每小时赚 $20，其中 $1.45\%$ 被扣除用于支付地方税。Alicia 的工资每小时有多少美分用于支付地方税？
Q2
On the AMC 12, each correct answer is worth $6$ points, each incorrect answer is worth $0$ points, and each problem left unanswered is worth $2.5$ points. If Charlyn leaves $8$ of the $25$ problems unanswered, how many of the remaining problems must she answer correctly in order to score at least $100$?
在 AMC 12 中，每道正确答案得 $6$ 分，每道错误答案得 $0$ 分，每道未答题得 $2.5$ 分。如果 Charlyn 留下 $8$ 道（共 $25$ 道）题未答，她需要在剩余题目中答对多少道才能得分至少 $100$？
Q3
For how many ordered pairs of positive integers $(x,y)$ is $x + 2y = 100$?
有多少对正整数的有序对 $(x,y)$ 满足 $x + 2y = 100$？
Q4
Bertha has 6 daughters and no sons. Some of her daughters have 6 daughters, and the rest have none. Bertha has a total of 30 daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters?
Bertha 有 6 个女儿，没有儿子。她的一些女儿有 6 个女儿，其余的没有。Bertha 总共有 30 个女儿和孙女，没有曾孙女。Bertha 的女儿和孙女中有多少个没有女儿？
Q5
The graph of the line $y=mx+b$ is shown. Which of the following is true?
给出了直线 $y=mx+b$ 的图像。以下哪项是正确的？
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Q6
Let $U=2\cdot 2004^{2005}$, $V=2004^{2005}$, $W=2003\cdot 2004^{2004}$, $X=2\cdot 2004^{2004}$, $Y=2004^{2004}$ and $Z=2004^{2003}$. Which of the following is the largest?
设 $U=2\cdot 2004^{2005}$，$V=2004^{2005}$，$W=2003\cdot 2004^{2004}$，$X=2\cdot 2004^{2004}$，$Y=2004^{2004}$，$Z=2004^{2003}$。以下哪个最大？
Q7
A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile. The game ends when some player runs out of tokens. Players $A$, $B$, and $C$ start with $15$, $14$, and $13$ tokens, respectively. How many rounds will there be in the game?
一个游戏使用代币按照以下规则进行。每轮中，代币最多的玩家给每个其他玩家一个代币，并将一个代币放入弃牌堆。游戏在某个玩家代币用尽时结束。玩家 $A$、$B$ 和 $C$ 分别起始有 $15$、$14$ 和 $13$ 个代币。游戏有多少轮？
Q8
In the overlapping triangles $\triangle{ABC}$ and $\triangle{ABE}$ sharing common side $AB$, $\angle{EAB}$ and $\angle{ABC}$ are right angles, $AB=4$, $BC=6$, $AE=8$, and $\overline{AC}$ and $\overline{BE}$ intersect at $D$. What is the difference between the areas of $\triangle{ADE}$ and $\triangle{BDC}$?
在重叠的三角形 $\triangle{ABC}$ 和 $\triangle{ABE}$ 中，它们共有边 $AB$，$\angle{EAB}$ 和 $\angle{ABC}$ 是直角，$AB=4$，$BC=6$，$AE=8$，且 $\overline{AC}$ 与 $\overline{BE}$ 相交于 $D$。$\triangle{ADE}$ 与 $\triangle{BDC}$ 的面积之差是多少？
stem
Q9
A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars will increase sales. If the diameter of the jars is increased by $25\%$ without altering the volume, by what percent must the height be decreased?
一家公司用圆柱形罐子出售花生酱。市场研究表明，使用更宽的罐子会增加销量。如果罐子直径增加 $25\%$ 而体积不变，高度必须减少百分之多少？
Q10
The sum of $49$ consecutive integers is $7^5$. What is their median?
$49$ 个连续整数的和为 $7^5$。它们的中位数是多少？
Q11
The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is $20$ cents. If she had one more quarter, the average value would be $21$ cents. How many dimes does she have in her purse?
Paula 的钱包中所有便士、五分币、十分币和二十五分币的平均价值为 $20$ 美分。如果她再多一枚二十五分币，平均价值将变为 $21$ 美分。她钱包里有多少枚十分币？
Q12
Let $A = (0,9)$ and $B = (0,12)$. Points $A'$ and $B'$ are on the line $y = x$, and $\overline{AA'}$ and $\overline{BB'}$ intersect at $C = (2,8)$. What is the length of $\overline{A'B'}$?
设 $A = (0,9)$ 和 $B = (0,12)$。点 $A'$ 和 $B'$ 在直线 $y = x$ 上，且 $\overline{AA'}$ 和 $\overline{BB'}$ 相交于 $C = (2,8)$。$\overline{A'B'}$ 的长度是多少？
Q13
Let $S$ be the set of points $(a,b)$ in the coordinate plane, where each of $a$ and $b$ may be $- 1$, $0$, or $1$. How many distinct lines pass through at least two members of $S$?
设 $S$ 为坐标平面上的点集 $(a,b)$，其中 $a$ 和 $b$ 各可取 $- 1$、$0$ 或 $1$。通过 $S$ 中至少两点的不同直线共有多少条？
Q14
A sequence of three real numbers forms an arithmetic progression with a first term of $9$. If $2$ is added to the second term and $20$ is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term in the geometric progression?
三个实数构成一个等差数列，第一项为 $9$。如果把第二项加 $2$，把第三项加 $20$，则得到的三个数构成一个等比数列。求该等比数列第三项的最小可能值。
Q15
Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters?
Brenda 和 Sally 在一条环形跑道上朝相反方向跑步，从直径相对的两点出发。她们第一次相遇时，Brenda 已跑了 100 米。第二次相遇时，Sally 已从第一次相遇点继续跑了 150 米。两人速度恒定。求跑道的长度（米）。
Q16
The set of all real numbers $x$ for which \[\log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x})))\] is defined is $\{x\mid x > c\}$. What is the value of $c$?
所有使得 \[\log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x})))\] 有定义的实数 $x$ 的集合是 $\{x\mid x > c\}$。$c$ 的值是多少？
Q17
Let $f$ be a function with the following properties: (i) $f(1) = 1$, and (ii) $f(2n) = n \cdot f(n)$ for any positive integer $n$. What is the value of $f(2^{100})$?
设 $f$ 是一个具有以下性质的函数： (i) $f(1) = 1$，且 (ii) 对任意正整数 $n$，$f(2n) = n \cdot f(n)$。 $ f(2^{100})$ 的值是多少？
Q18
Square $ABCD$ has side length $2$. A semicircle with diameter $\overline{AB}$ is constructed inside the square, and the tangent to the semicircle from $C$ intersects side $\overline{AD}$ at $E$. What is the length of $\overline{CE}$?
正方形 $ABCD$ 的边长为 $2$。在正方形内部作以 $\overline{AB}$ 为直径的半圆，从 $C$ 点作该半圆的切线，与边 $\overline{AD}$ 交于 $E$。求 $\overline{CE}$ 的长度。
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Q19
Circles $A, B$ and $C$ are externally tangent to each other, and internally tangent to circle $D$. Circles $B$ and $C$ are congruent. Circle $A$ has radius $1$ and passes through the center of $D$. What is the radius of circle $B$?
圆 $A, B$ 和 $C$ 两两外切，并且都与圆 $D$ 内切。圆 $B$ 和 $C$ 全等。圆 $A$ 的半径为 $1$，且经过圆 $D$ 的圆心。圆 $B$ 的半径是多少？
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Q20
Select numbers $a$ and $b$ between $0$ and $1$ independently and at random, and let $c$ be their sum. Let $A, B$ and $C$ be the results when $a, b$ and $c$, respectively, are rounded to the nearest integer. What is the probability that $A + B = C$?
独立且随机地在 $0$ 与 $1$ 之间选取数 $a$ 和 $b$，令 $c$ 为它们的和。将 $a, b, c$ 分别四舍五入到最近的整数，得到结果 $A, B, C$。求 $A + B = C$ 的概率。
Q21
If $\sum_{n = 0}^{\infty}{\cos^{2n}}\theta = 5$, what is the value of $\cos{2\theta}$?
如果 $\sum_{n = 0}^{\infty}{\cos^{2n}}\theta = 5$，那么 $\cos{2\theta}$ 的值是多少？
Q22
Three mutually tangent spheres of radius $1$ rest on a horizontal plane. A sphere of radius $2$ rests on them. What is the distance from the plane to the top of the larger sphere?
三个半径为 $1$ 且两两相切的球放在一个水平平面上。一个半径为 $2$ 的球放在它们上面。从平面到较大球的顶部的距离是多少？
Q23
A polynomial \[P(x) = c_{2004}x^{2004} + c_{2003}x^{2003} + ... + c_1x + c_0\] has real coefficients with $c_{2004}\not = 0$ and $2004$ distinct complex zeroes $z_k = a_k + b_ki$, $1\leq k\leq 2004$ with $a_k$ and $b_k$ real, $a_1 = b_1 = 0$, and \[\sum_{k = 1}^{2004}{a_k} = \sum_{k = 1}^{2004}{b_k}.\] Which of the following quantities can be a non zero number?
多项式 \[P(x) = c_{2004}x^{2004} + c_{2003}x^{2003} + ... + c_1x + c_0\] 具有实系数，且 $c_{2004}\not = 0$，并且有 $2004$ 个互不相同的复零点 $z_k = a_k + b_ki$（$1\leq k\leq 2004$），其中 $a_k$ 和 $b_k$ 为实数，$a_1 = b_1 = 0$，并且 \[\sum_{k = 1}^{2004}{a_k} = \sum_{k = 1}^{2004}{b_k}.\] 下列哪个量可以是非零数？
Q24
A plane contains points $A$ and $B$ with $AB = 1$. Let $S$ be the union of all disks of radius $1$ in the plane that cover $\overline{AB}$. What is the area of $S$?
平面上有点 $A$ 和 $B$，且 $AB = 1$。设 $S$ 为平面中所有半径为 $1$ 且覆盖 $\overline{AB}$ 的圆盘的并集。求 $S$ 的面积。
Q25
For each integer $n\geq 4$, let $a_n$ denote the base-$n$ number $0.\overline{133}_n$. The product $a_4a_5\cdots a_{99}$ can be expressed as $\frac {m}{n!}$, where $m$ and $n$ are positive integers and $n$ is as small as possible. What is $m$?
对每个整数 $n\geq 4$，令 $a_n$ 表示 $n$ 进制数 $0.\overline{133}_n$。乘积 $a_4a_5\cdots a_{99}$ 可表示为 $\frac {m}{n!}$，其中 $m$ 和 $n$ 为正整数，且 $n$ 尽可能小。求 $m$。
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