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AMC12 2005 B

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AMC12 · 2005 (B)

Q1
A scout troop buys $1000$ candy bars at a price of five for $2$ dollars. They sell all the candy bars at the price of two for $1$ dollar. What was their profit, in dollars?
一个童子军小队以每5根2美元的价格购买了$1000$根糖果棒。他们以每2根1美元的价格卖出了所有糖果棒。他们的利润是多少美元？
Q2
A positive number $x$ has the property that $x\%$ of $x$ is $4$. What is $x$?
一个正数 $x$ 具有这样的性质：$x$ 的 $x\%$ 是 $4$。$x$ 是多少？
Q3
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?
布里安娜用她周末工作赚的一部分钱购买了若干张价格相同的CD。她用她钱的五分之一买了CD的三分之一。她买完所有CD后，还剩下她钱的几分之几？
Q4
At the beginning of the school year, Lisa's goal was to earn an $A$ on at least $80\%$ of her $50$ quizzes for the year. She earned an $A$ on $22$ of the first $30$ quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an $A$?
在学年开始时，丽莎的目标是在全年$50$次小测验中至少有$80\%$获得$A$。她在前$30$次小测验中有$22$次获得$A$。如果她要实现目标，那么在剩下的小测验中，最多有多少次她可以得到低于$A$的成绩？
Q5
An $8$-foot by $10$-foot bathroom floor is tiled with square tiles of size $1$ foot by $1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $1/2$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?
一个$8$英尺乘$10$英尺的浴室地面用边长为$1$英尺的正方形瓷砖铺设。每块瓷砖上有一个图案：在瓷砖的每个角以该角为圆心、半径为$1/2$英尺画一个白色的四分之一圆，共四个。瓷砖剩余部分为阴影。地面上共有多少平方英尺的面积是阴影部分？
stem
Q6
In $\triangle ABC$, we have $AC=BC=7$ and $AB=2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD=8$. What is $BD$?
在 $\triangle ABC$ 中，$AC=BC=7$ 且 $AB=2$。设 $D$ 是直线 $AB$ 上的一点，使得 $B$ 在 $A$ 与 $D$ 之间，并且 $CD=8$。求 $BD$。
Q7
What is the area enclosed by the graph of $|3x|+|4y|=12$?
由方程 $|3x|+|4y|=12$ 的图象所围成的面积是多少？
Q8
For how many values of $a$ is it true that the line $y = x + a$ passes through the vertex of the parabola $y = x^2 + a^2$ ?
有多少个 $a$ 的取值使得直线 $y = x + a$ 经过抛物线 $y = x^2 + a^2$ 的顶点？
Q9
On a certain math exam, $10\%$ of the students got $70$ points, $25\%$ got $80$ points, $20\%$ got $85$ points, $15\%$ got $90$ points, and the rest got $95$ points. What is the difference between the mean and the median score on this exam?
在某次数学考试中，$10\%$ 的学生得 $70$ 分，$25\%$ 得 $80$ 分，$20\%$ 得 $85$ 分，$15\%$ 得 $90$ 分，其余得 $95$ 分。这次考试的平均分与中位数之差是多少？
Q10
The first term of a sequence is $2005$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the ${2005}^{\text{th}}$ term of the sequence?
一个数列的第一项是 $2005$。其后每一项等于前一项各位数字的立方和。求该数列的第 ${2005}^{\text{th}}$ 项。
Q11
An envelope contains eight bills: $2$ ones, $2$ fives, $2$ tens, and $2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $\$20$ or more?
一个信封里装有八张钞票：$2$ 张 1 美元、$2$ 张 5 美元、$2$ 张 10 美元和 $2$ 张 20 美元。随机不放回地抽取两张钞票。它们的总和为 $\$20$ 或更多的概率是多少？
Q12
The quadratic equation $x^2+mx+n$ has roots twice those of $x^2+px+m$, and none of $m,n,$ and $p$ is zero. What is the value of $n/p$?
二次方程 $x^2+mx+n$ 的根是 $x^2+px+m$ 的根的两倍，且 $m,n,$ 和 $p$ 都不为零。$n/p$ 的值是多少？
Q13
Suppose that $4^{x_1}=5$, $5^{x_2}=6$, $6^{x_3}=7$, ... , $127^{x_{124}}=128$. What is $x_1x_2...x_{124}$?
假设 $4^{x_1}=5$, $5^{x_2}=6$, $6^{x_3}=7$, ... , $127^{x_{124}}=128$。$x_1x_2...x_{124}$ 等于多少？
Q14
A circle having center $(0,k)$, with $k>6$, is tangent to the lines $y=x$, $y=-x$ and $y=6$. What is the radius of this circle?
一个圆心为 $(0,k)$ 的圆，其中 $k>6$，与直线 $y=x$、$y=-x$ 和 $y=6$ 相切。这个圆的半径是多少？
Q15
The sum of four two-digit numbers is $221$. None of the eight digits is $0$ and no two of them are the same. Which of the following is not included among the eight digits?
四个两位数的和为 $221$。这八个数字中没有 $0$，且没有两个相同。以下哪个数字不在这八个数字之中？
Q16
Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres? $\mathrm {
有8个半径为$1$的球，每个球位于一个八分体中，且每个球都与坐标平面相切。求以原点为中心的最小球的半径，该球包含这8个球。 $\mathrm {
Q17
How many distinct four-tuples $(a,b,c,d)$ of rational numbers are there with \[a\cdot\log_{10}2+b\cdot\log_{10}3+c\cdot\log_{10}5+d\cdot\log_{10}7=2005?\]
有多少个不同的有理数四元组$(a,b,c,d)$满足 \[a\cdot\log_{10}2+b\cdot\log_{10}3+c\cdot\log_{10}5+d\cdot\log_{10}7=2005?\]
Q18
Let $A(2,2)$ and $B(7,7)$ be points in the plane. Define $R$ as the region in the first quadrant consisting of those points $C$ such that $\triangle ABC$ is an acute triangle. What is the closest integer to the area of the region $R$?
设平面上的点$A(2,2)$和$B(7,7)$。定义$R$为第一象限中那些点$C$组成的区域，使得$\triangle ABC$为锐角三角形。求区域$R$面积的最接近整数。
Q19
Let $x$ and $y$ be two-digit integers such that $y$ is obtained by reversing the digits of $x$. The integers $x$ and $y$ satisfy $x^{2}-y^{2}=m^{2}$ for some positive integer $m$. What is $x+y+m$?
设$x$和$y$为两位整数，$y$由$x$的各位数字反转得到。整数$x$和$y$满足$x^{2}-y^{2}=m^{2}$，其中$m$为某个正整数。求$x+y+m$？
Q20
Let $a,b,c,d,e,f,g$ and $h$ be distinct elements in the set $\{-7,-5,-3,-2,2,4,6,13\}.$ What is the minimum possible value of $(a+b+c+d)^{2}+(e+f+g+h)^{2}?$
设$a,b,c,d,e,f,g$和$h$是集合$\{-7,-5,-3,-2,2,4,6,13\}.$中不同的元素。 $(a+b+c+d)^{2}+(e+f+g+h)^{2}$的最小可能值是多少？
Q21
A positive integer $n$ has $60$ divisors and $7n$ has $80$ divisors. What is the greatest integer $k$ such that $7^k$ divides $n$?
一个正整数 $n$ 有 $60$ 个因数，且 $7n$ 有 $80$ 个因数。求最大的整数 $k$ 使得 $7^k$ 整除 $n$？
Q22
A sequence of complex numbers $z_{0}, z_{1}, z_{2}, ...$ is defined by the rule \[z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},\] where $\overline {z_{n}}$ is the complex conjugate of $z_{n}$ and $i^{2}=-1$. Suppose that $|z_{0}|=1$ and $z_{2005}=1$. How many possible values are there for $z_{0}$?
复数序列 $z_{0}, z_{1}, z_{2}, ...$ 由规则定义 \[z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},\] 其中 $\overline {z_{n}}$ 是 $z_{n}$ 的共轭复数，且 $i^{2}=-1$。假设 $|z_{0}|=1$ 且 $z_{2005}=1$。$z_{0}$ 有多少可能值？
Q23
Let $S$ be the set of ordered triples $(x,y,z)$ of real numbers for which \[\log_{10}(x+y) = z \text{ and } \log_{10}(x^{2}+y^{2}) = z+1.\] There are real numbers $a$ and $b$ such that for all ordered triples $(x,y,z)$ in $S$ we have $x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.$ What is the value of $a+b?$
令 $S$ 为满足以下条件的实数有序三元组 $(x,y,z)$ 的集合： \[\log_{10}(x+y) = z \text{ and } \log_{10}(x^{2}+y^{2}) = z+1.\] 存在实数 $a$ 和 $b$，使得对于 $S$ 中所有有序三元组 $(x,y,z)$，有 $x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.$ 求 $a+b$ 的值。
Q24
All three vertices of an equilateral triangle are on the parabola $y = x^2$, and one of its sides has a slope of $2$. The $x$-coordinates of the three vertices have a sum of $m/n$, where $m$ and $n$ are relatively prime positive integers. What is the value of $m + n$?
一个正三角形的三个顶点都在抛物线 $y = x^2$ 上，其中一边斜率为 $2$。三顶点的 $x$ 坐标之和为 $m/n$，其中 $m$ 和 $n$ 互质正整数。求 $m + n$ 的值。
Q25
Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. What is the probability that no two ants arrive at the same vertex?
六只蚂蚁同时站在正八面体的六个顶点上，每只蚂蚁在不同的顶点。同时且独立地，每只蚂蚁从其顶点移动到四个相邻顶点之一，每种选择等概率。求没有两只蚂蚁到达同一顶点的概率。
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