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AMC12 2005 A

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AMC12 · 2005 (A)

Q1
Two is $10 \%$ of $x$ and $20 \%$ of $y$. What is $x - y$?
2 是 $x$ 的 $10 \%$，也是 $y$ 的 $20 \%$。$x - y$ 是多少？
Q2
The equations $2x + 7 = 3$ and $bx - 10 = -2$ have the same solution $x$. What is the value of $b$?
方程 $2x + 7 = 3$ 和 $bx - 10 = -2$ 有相同的解 $x$。$b$ 的值是多少？
Q3
A rectangle with a diagonal of length $x$ is twice as long as it is wide. What is the area of the rectangle?
一个对角线长度为 $x$ 的矩形，其长是宽的两倍。该矩形的面积是多少？
Q4
A store normally sells windows at $\$100$ each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?
一家商店通常以 $\$100$ 的价格出售每个窗户。本周商店提供每购买四个窗户赠送一个免费窗户。Dave 需要七个窗户，Doug 需要八个窗户。如果他们一起购买而不是分开购买，能节省多少美元？
Q5
The average (mean) of $20$ numbers is $30$, and the average of $30$ other numbers is $20$. What is the average of all $50$ numbers?
$20$ 个数的平均数（均值）是 $30$，另外 $30$ 个数的平均数是 $20$。这 $50$ 个数的平均数是多少？
Q6
Josh and Mike live $13$ miles apart. Yesterday Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met?
Josh 和 Mike 相距 $13$ 英里。昨天 Josh 开始骑自行车朝 Mike 家骑去。过了一会儿 Mike 开始骑自行车朝 Josh 家骑去。当他们相遇时，Josh 骑行的时间是 Mike 的两倍，并且速度是 Mike 的五分之四。他们相遇时 Mike 骑了多少英里？
Q7
Square $EFGH$ is inside square $ABCD$ so that each side of $EFGH$ can be extended to pass through a vertex of $ABCD$. Square $ABCD$ has side length $\sqrt{50}$, $E$ is between $B$ and $H$, and $BE = 1$. What is the area of the inner square $EFGH$?
正方形 $EFGH$ 在正方形 $ABCD$ 内部，使得 $EFGH$ 的每条边延长后都能通过 $ABCD$ 的一个顶点。正方形 $ABCD$ 的边长为 $\sqrt{50}$，$E$ 在 $B$ 和 $H$ 之间，且 $BE = 1$。内正方形 $EFGH$ 的面积是多少？
stem
Q8
Let $A,M$, and $C$ be digits with \[(100A+10M+C)(A+M+C) = 2005.\] What is $A$?
设 $A,M$, 和 $C$ 是数字，使得 \[(100A+10M+C)(A+M+C) = 2005.\] $A$ 是多少？
Q9
There are two values of $a$ for which the equation $4x^2 + ax + 8x + 9 = 0$ has only one solution for $x$. What is the sum of those values of $a$?
方程 $4x^2 + ax + 8x + 9 = 0$ 只有一个 $x$ 解时，$a$ 有两个取值。这两个 $a$ 值的和是多少？
Q10
A wooden cube $n$ units on a side is painted red on all six faces and then cut into $n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $n$?
一个边长为 $n$ 单位的木立方体，六个面都涂成红色，然后切成 $n^3$ 个单位立方体。单位立方体所有面的总数中，恰好有四分之一是红色的。$n$ 是多少？
Q11
How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?
有多少个三位数满足中间的数位是首位和末位数位的平均数？
Q12
A line passes through $A(1,1)$ and $B(100,1000)$. How many other points with integer coordinates are on the line and strictly between $A$ and $B$?
一条直线经过点 $A(1,1)$ 和 $B(100,1000)$。在这条直线上，严格位于 $A$ 和 $B$ 之间且具有整数坐标的其它点有多少个？
Q13
In the five-sided star shown, the letters $A$, $B$, $C$, $D$ and $E$ are replaced by the numbers $3$, $5$, $6$, $7$ and $9$, although not necessarily in that order. The sums of the numbers at the ends of the line segments $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, $\overline{DE}$ and $\overline{EA}$ form an arithmetic sequence, although not necessarily in that order. What is the middle term of the arithmetic sequence?
在所示的五角星中，字母 $A$, $B$, $C$, $D$ 和 $E$ 被数字 $3$, $5$, $6$, $7$ 和 $9$ 替换（不一定按此顺序）。线段 $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, $\overline{DE}$ 和 $\overline{EA}$ 两端数字之和形成一个等差数列（不一定按此顺序）。这个等差数列的中间项是多少？
stem
Q14
On a standard die one of the dots is removed at random with each dot equally likely to be chosen. The die is then rolled. What is the probability that the top face has an odd number of dots?
在一个标准骰子上，随机移除一个点，每个点被选中的概率相等。然后掷骰子。顶面点数为奇数的概率是多少？
Q15
Let $\overline{AB}$ be a diameter of a circle and $C$ be a point on $\overline{AB}$ with $2 \cdot AC = BC$. Let $D$ and $E$ be points on the circle such that $\overline{DC} \perp \overline{AB}$ and $\overline{DE}$ is a second diameter. What is the ratio of the area of $\triangle DCE$ to the area of $\triangle ABD$?
设 $\overline{AB}$ 为圆的直径，$C$ 为 $\overline{AB}$ 上的点且 $2 \cdot AC = BC$。$D$ 和 $E$ 为圆上的点，使得 $\overline{DC} \perp \overline{AB}$ 且 $\overline{DE}$ 为另一条直径。$\triangle DCE$ 的面积与 $\triangle ABD$ 的面积之比是多少？
stem
Q16
Three circles of radius $s$ are drawn in the first quadrant of the $xy$-plane. The first circle is tangent to both axes, the second is tangent to the first circle and the $x$-axis, and the third is tangent to the first circle and the $y$-axis. A circle of radius $r > s$ is tangent to both axes and to the second and third circles. What is $r/s$?
在 $xy$ 平面的第一象限中画三个半径为 $s$ 的圆。第一个圆与两条坐标轴都相切，第二个圆与第一个圆和 $x$ 轴相切，第三个圆与第一个圆和 $y$ 轴相切。一个半径为 $r > s$ 的圆与两条坐标轴以及第二、第三个圆都相切。求 $r/s$。
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Q17
A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex $W$?
一个单位立方体被切割两次，形成三个三角柱体，其中两个全等，如图 1 所示。然后按照图 2 中虚线所示的方式，用同样的方法再次切割该立方体。这将产生九个部分。包含顶点 $W$ 的那一块的体积是多少？
stem
Q18
Call a number "prime-looking" if it is composite but not divisible by $2$, $3$, or $5$. The three smallest prime-looking numbers are $49$, $77$, and $91$. There are $168$ prime numbers less than $1000$. How many prime-looking numbers are there less than $1000$?
称一个数为“质数样”数，如果它是合数但不被 $2$、$3$ 或 $5$ 整除。最小的三个质数样数是 $49$、$77$ 和 $91$。小于 $1000$ 的质数有 $168$ 个。小于 $1000$ 的质数样数有多少个？
Q19
A faulty car odometer proceeds from digit $3$ to digit $5$, always skipping the digit $4$, regardless of position. For example, after traveling one mile the odometer changed from $000039$ to $000050$. If the odometer now reads $002005$, how many miles has the car actually traveled?
一个有故障的汽车里程表从数字 $3$ 直接跳到数字 $5$，总是跳过数字 $4$，无论在何位置。例如，行驶一英里后，里程表从 $000039$ 变为 $000050$。如果里程表现在显示 $002005$，汽车实际行驶了多少英里？
Q20
For each $x$ in $[0,1]$, define \[f(x) = \begin{cases} 2x, \qquad\qquad \mathrm{if} \quad 0 \leq x \leq \frac{1}{2}\\ 2-2x, \qquad \mathrm{if} \quad \frac{1}{2} < x \leq 1. \end{cases}\] Let $f^{[2]}(x) = f(f(x))$, and $f^{[n + 1]}(x) = f^{[n]}(f(x))$ for each integer $n \geq 2$. For how many values of $x$ in $[0,1]$ is $f^{[2005]}(x) = 1/2$?
对每个 $x \in [0,1]$，定义 \[f(x) = \begin{cases} 2x, \qquad\qquad \mathrm{if} \quad 0 \leq x \leq \frac{1}{2}\\ 2-2x, \qquad \mathrm{if} \quad \frac{1}{2} < x \leq 1. \end{cases}\] 令 $f^{[2]}(x) = f(f(x))$，并且对每个整数 $n \geq 2$，令 $f^{[n + 1]}(x) = f^{[n]}(f(x))$。在 $[0,1]$ 中有多少个 $x$ 满足 $f^{[2005]}(x) = 1/2$？
Q21
How many ordered triples of integers $(a,b,c)$, with $a \geq 2$, $b \geq 1$, and $c \geq 0$, satisfy both $\log_{a}b = c^{2005}$ and $a + b + c = 2005$?
有整数的有序三元组 $(a,b,c)$ 多少个，其中 $a \geq 2$，$b \geq 1$，$c \geq 0$，满足 $\log_{a}b = c^{2005}$ 和 $a + b + c = 2005$？
Q22
A rectangular box $P$ is inscribed in a sphere of radius $r$. The surface area of $P$ is 384, and the sum of the lengths of its $12$ edges is $112$. What is $r$?
一个长方体盒子 $P$ 内接于半径为 $r$ 的球中。$P$ 的表面积是 384，其 12 条棱的长度和是 112。$r$ 是多少？
Q23
Two distinct numbers $a$ and $b$ are chosen randomly from the set $\{2, 2^2, 2^3, \ldots, 2^{25}\}$. What is the probability that $\log_{a}b$ is an integer?
从集合 $\{2, 2^2, 2^3, \dots , 2^{25}\}$ 中随机选择两个不同的数 $a$ 和 $b$。$\log_a b$ 是整数的概率是多少？
Q24
Let $P(x)=(x-1)(x-2)(x-3)$. For how many polynomials $Q(x)$ does there exist a polynomial $R(x)$ of degree $3$ such that $P(Q(x)) = P(x) \cdot R(x)$?
设 $P(x)=(x-1)(x-2)(x-3)$。有多少个多项式 $Q(x)$ 存在，使得存在次数为 $3$ 的多项式 $R(x)$ 满足 $P(Q(x)) = P(x) \cdot R(x)$？
Q25
Let $S$ be the set of all points with coordinates $(x,y,z)$, where $x$, $y$, and $z$ are each chosen from the set $\{0,1,2\}$. How many equilateral triangles all have their vertices in $S$?
设 $S$ 是所有点 $(x,y,z)$ 的集合，其中 $x$, $y$, $z$ 都从集合 $\{0,1,2\}$ 中选取。有多少个等边三角形的三个顶点都在 $S$ 中？
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