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AMC12 2008 B

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AMC12 · 2008 (B)

Q1
A basketball player made $5$ baskets during a game. Each basket was worth either $2$ or $3$ points. How many different numbers could represent the total points scored by the player?
一名篮球运动员在一场比赛中投中了 $5$ 个篮。每个篮得分要么是 $2$ 分，要么是 $3$ 分。该运动员的总得分可能有多少种不同的数值？
Q2
A $4\times4$ block of calendar dates is shown. \[ \begin{array}{|c|c|c|c|} \hline 1 & 2 & 3 & 4\\ \hline 8 & 9 & 10 & 11\\ \hline 15 & 16 & 17 & 18\\ \hline 22 & 23 & 24 & 25\\ \hline \end{array} \] The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums?
给出一个日历日期组成的 $4\times4$ 方块： \[ \begin{array}{|c|c|c|c|} \hline 1 & 2 & 3 & 4\\ \hline 8 & 9 & 10 & 11\\ \hline 15 & 16 & 17 & 18\\ \hline 22 & 23 & 24 & 25\\ \hline \end{array} \] 将第二行的数字顺序反转；然后将第四行的数字顺序反转。最后，把两条对角线上的数字分别相加。两条对角线和的正差是多少？
Q3
A semipro baseball league has teams with $21$ players each. League rules state that a player must be paid at least $15,000$ dollars, and that the total of all players' salaries for each team cannot exceed $700,000$ dollars. What is the maximum possiblle salary, in dollars, for a single player?
一个半职业棒球联盟的每支球队有 $21$ 名球员。联盟规则规定，每名球员的薪水至少为 $15,000$ 美元，并且每支球队所有球员薪水总和不得超过 $700,000$ 美元。单个球员的最大可能薪水（美元）是多少？
Q4
On circle $O$, points $C$ and $D$ are on the same side of diameter $\overline{AB}$, $\angle AOC = 30^\circ$, and $\angle DOB = 45^\circ$. What is the ratio of the area of the smaller sector $COD$ to the area of the circle?
在圆 $O$ 上，点 $C$ 和 $D$ 位于直径 $\overline{AB}$ 的同一侧，$\angle AOC = 30^\circ$，且 $\angle DOB = 45^\circ$。较小的扇形 $COD$ 的面积与圆面积之比是多少？
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Q5
A class collects $50$ dollars to buy flowers for a classmate who is in the hospital. Roses cost $3$ dollars each, and carnations cost $2$ dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly $50$ dollars?
一个班级筹集了 $50$ 美元，为住院的同学买花。玫瑰每朵 $3$ 美元，康乃馨每朵 $2$ 美元。不使用其他花。恰好花 $50$ 美元可以购买多少种不同的花束？
Q6
Postman Pete has a pedometer to count his steps. The pedometer records up to $99999$ steps, then flips over to $00000$ on the next step. Pete plans to determine his mileage for a year. On January $1$ Pete sets the pedometer to $00000$. During the year, the pedometer flips from $99999$ to $00000$ forty-four times. On December $31$ the pedometer reads $50000$. Pete takes $1800$ steps per mile. Which of the following is closest to the number of miles Pete walked during the year?
邮递员 Pete 有一个计步器来记录他的步数。计步器最多记录到 $99999$ 步，然后在下一步翻转到 $00000$。Pete 计划计算他一年的里程数。1 月 $1$ 日 Pete 将计步器设为 $00000$。在这一年中，计步器从 $99999$ 翻转到 $00000$ 共四十四次。12 月 $31$ 日计步器显示 $50000$。Pete 每英里走 $1800$ 步。以下哪个最接近 Pete 这一年走的英里数？
Q7
For real numbers $a$ and $b$, define $a \$ b = (a - b)^2$. What is $(x - y)^2\$(y - x)^2$?
对实数 $a$ 和 $b$，定义 $a\$ b = (a - b)^2$。求 $(x - y)^2\$(y - x)^2$ 的值。
Q8
Points $B$ and $C$ lie on $\overline{AD}$. The length of $\overline{AB}$ is $4$ times the length of $\overline{BD}$, and the length of $\overline{AC}$ is $9$ times the length of $\overline{CD}$. The length of $\overline{BC}$ is what fraction of the length of $\overline{AD}$?
点 $B$ 和 $C$ 在 $\overline{AD}$ 上。$\overline{AB}$ 的长度是 $\overline{BD}$ 长度的 $4$ 倍，且 $\overline{AC}$ 的长度是 $\overline{CD}$ 长度的 $9$ 倍。$\overline{BC}$ 的长度是 $\overline{AD}$ 长度的几分之几？
Q9
Points $A$ and $B$ are on a circle of radius $5$ and $AB = 6$. Point $C$ is the midpoint of the minor arc $AB$. What is the length of the line segment $AC$?
点 $A$ 和 $B$ 在半径为 $5$ 的圆上，且 $AB = 6$。点 $C$ 是小弧 $AB$ 的中点。线段 $AC$ 的长度是多少？
Q10
Bricklayer Brenda would take $9$ hours to build a chimney alone, and bricklayer Brandon would take $10$ hours to build it alone. When they work together they talk a lot, and their combined output is decreased by $10$ bricks per hour. Working together, they build the chimney in $5$ hours. How many bricks are in the chimney?
砌砖工 Brenda 独自砌一个烟囱需要 $9$ 小时，砌砖工 Brandon 独自砌需要 $10$ 小时。他们一起工作时话很多，导致他们的合计产量每小时减少 $10$ 块砖。他们一起用 $5$ 小时砌完了烟囱。烟囱共有多少块砖？
Q11
A cone-shaped mountain has its base on the ocean floor and has a height of 8000 feet. The top $\frac{1}{8}$ of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain in feet?
一座锥形山以其底面位于海底，高度为8000英尺。山体体积的顶端 $\frac{1}{8}$ 在水上。该山的底面处海洋深度是多少英尺？
Q12
For each positive integer $n$, the mean of the first $n$ terms of a sequence is $n$. What is the $2008$th term of the sequence?
对于每个正整数 $n$，该数列前 $n$ 项的平均数为 $n$。该数列的第 $2008$ 项是多少？
Q13
Vertex $E$ of equilateral $\triangle{ABE}$ is in the interior of unit square $ABCD$. Let $R$ be the region consisting of all points inside $ABCD$ and outside $\triangle{ABE}$ whose distance from $AD$ is between $\frac{1}{3}$ and $\frac{2}{3}$. What is the area of $R$?
正方形 $ABCD$（边长为1）的内部有正三角形 $\triangle{ABE}$ 的顶点 $E$。令 $R$ 为方形 $ABCD$ 内且 $\triangle{ABE}$ 外、距边 $AD$ 的距离在 $\frac{1}{3}$ 与 $\frac{2}{3}$ 之间的所有点的区域。$R$ 的面积是多少？
Q14
A circle has a radius of $\log_{10}{(a^2)}$ and a circumference of $\log_{10}{(b^4)}$. What is $\log_{a}{b}$?
一个圆的半径为 $\log_{10}{(a^2)}$，周长为 $\log_{10}{(b^4)}$。求 $\log_{a}{b}$。
Q15
On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let $R$ be the region formed by the union of the square and all the triangles, and $S$ be the smallest convex polygon that contains $R$. What is the area of the region that is inside $S$ but outside $R$?
在单位正方形的每条边上构造一个边长为1的正三角形。在每个正三角形的新边上再构造另一个边长为1的正三角形。正方形和12个三角形的内部没有公共点。令 $R$ 为正方形与所有三角形的并集所形成的区域，$S$ 为包含 $R$ 的最小凸多边形。$S$ 内而 $R$ 外的区域面积是多少？
Q16
A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $1$ foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair $(a,b)$?
一个矩形地板的尺寸为 $a$ 英尺乘 $b$ 英尺，其中 $a$ 和 $b$ 为正整数且 $b > a$。一位艺术家在地板上涂画了一个矩形，且该矩形的边与地板的边平行。未涂画的部分在涂画矩形周围形成宽度为 $1$ 英尺的边框，并且其面积占整个地板面积的一半。有多少种有序对 $(a,b)$ 的可能？
Q17
Let $A$, $B$ and $C$ be three distinct points on the graph of $y=x^2$ such that line $AB$ is parallel to the $x$-axis and $\triangle ABC$ is a right triangle with area $2008$. What is the sum of the digits of the $y$-coordinate of $C$?
设 $A$、$B$、$C$ 是抛物线 $y=x^2$ 上三个不同的点，使得直线 $AB$ 平行于 $x$ 轴，且 $\triangle ABC$ 是面积为 $2008$ 的直角三角形。点 $C$ 的 $y$ 坐标的各位数字之和是多少？
Q18
A pyramid has a square base $ABCD$ and vertex $E$. The area of square $ABCD$ is $196$, and the areas of $\triangle ABE$ and $\triangle CDE$ are $105$ and $91$, respectively. What is the volume of the pyramid?
一个棱锥的底面是正方形 $ABCD$，顶点为 $E$。正方形 $ABCD$ 的面积为 $196$，并且 $\triangle ABE$ 与 $\triangle CDE$ 的面积分别为 $105$ 和 $91$。该棱锥的体积是多少？
Q19
A function $f$ is defined by $f(z) = (4 + i) z^2 + \alpha z + \gamma$ for all complex numbers $z$, where $\alpha$ and $\gamma$ are complex numbers and $i^2 = - 1$. Suppose that $f(1)$ and $f(i)$ are both real. What is the smallest possible value of $| \alpha | + |\gamma |$ ?
函数 $f$ 定义为对所有复数 $z$，$f(z) = (4 + i) z^2 + \alpha z + \gamma$，其中 $\alpha$ 和 $\gamma$ 为复数，且 $i^2 = - 1$。已知 $f(1)$ 和 $f(i)$ 都是实数。$| \alpha | + |\gamma |$ 的最小可能值是多少？
Q20
Michael walks at the rate of $5$ feet per second on a long straight path. Trash pails are located every $200$ feet along the path. A garbage truck traveling at $10$ feet per second in the same direction as Michael stops for $30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet?
迈克尔以每秒 $5$ 英尺的速度沿一条很长的直线路径行走。路径上每隔 $200$ 英尺放置一个垃圾桶。一辆垃圾车以每秒 $10$ 英尺的速度与迈克尔同向行驶，并在每个垃圾桶处停留 $30$ 秒。当迈克尔经过一个垃圾桶时，他注意到垃圾车在他前方刚刚离开下一个垃圾桶。迈克尔与垃圾车将相遇多少次？
Q21
Two circles of radius 1 are to be constructed as follows. The center of circle $A$ is chosen uniformly and at random from the line segment joining $(0,0)$ and $(2,0)$. The center of circle $B$ is chosen uniformly and at random, and independently of the first choice, from the line segment joining $(0,1)$ to $(2,1)$. What is the probability that circles $A$ and $B$ intersect?
按如下方式构造两个半径为 1 的圆。圆 $A$ 的圆心从连接 $(0,0)$ 与 $(2,0)$ 的线段上均匀随机选取。圆 $B$ 的圆心从连接 $(0,1)$ 与 $(2,1)$ 的线段上均匀随机选取，并且与第一次选取相互独立。求圆 $A$ 与圆 $B$ 相交的概率。
Q22
A parking lot has 16 spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers chose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park?
一个停车场有一排 16 个车位。12 辆车到达，每辆车需要 1 个车位，司机从仍然空着的车位中随机选择。随后 Em 阿姨开着她的 SUV 到达，这辆车需要 2 个相邻车位。她能够停车的概率是多少？
Q23
The sum of the base-$10$ logarithms of the divisors of $10^n$ is $792$. What is $n$?
$10^n$ 的所有因子的以 $10$ 为底的对数之和为 $792$。求 $n$。
Q24
Let $A_0=(0,0)$. Distinct points $A_1,A_2,\dots$ lie on the $x$-axis, and distinct points $B_1,B_2,\dots$ lie on the graph of $y=\sqrt{x}$. For every positive integer $n,\ A_{n-1}B_nA_n$ is an equilateral triangle. What is the least $n$ for which the length $A_0A_n\geq100$?
设 $A_0=(0,0)$。互不相同的点 $A_1,A_2,\dots$ 在 $x$ 轴上，互不相同的点 $B_1,B_2,\dots$ 在曲线 $y=\sqrt{x}$ 上。对每个正整数 $n$，$A_{n-1}B_nA_n$ 为等边三角形。求满足 $A_0A_n\geq100$ 的最小 $n$。
Q25
Let $ABCD$ be a trapezoid with $AB||CD, AB=11, BC=5, CD=19,$ and $DA=7$. Bisectors of $\angle A$ and $\angle D$ meet at $P$, and bisectors of $\angle B$ and $\angle C$ meet at $Q$. What is the area of hexagon $ABQCDP$?
设 $ABCD$ 为梯形，满足 $AB||CD, AB=11, BC=5, CD=19,$ 且 $DA=7$。$\angle A$ 与 $\angle D$ 的角平分线交于 $P$，$\angle B$ 与 $\angle C$ 的角平分线交于 $Q$。求六边形 $ABQCDP$ 的面积。
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