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AMC10 2008 B

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AMC10 · 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。玩家的总得分可能有多少种不同的数值？
Q2
A $4 \times 4$ block of calendar dates is shown. 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?
展示了一个 $4 \times 4$ 的日历日期块。第二行的数字顺序将被反转。然后第四行的数字顺序将被反转。最后，将每个对角线的数字相加。两个对角线和的正差是多少？
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Q3
Assume that $x$ is a positive real number. Which is equivalent to $\sqrt[3]{x\sqrt{x}}$?
假设 $x$ 是一个正实数。以下哪个与 $\sqrt[3]{x\sqrt{x}}$ 等价？
Q4
A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least \$15,000, and that the total of all players' salaries for each team cannot exceed \$700,000. What is the maximum possible salary, in dollars, for a single player?
一个半职业棒球联盟的每个队有21名球员。联盟规则规定，每名球员的薪水至少为15000美元，每队的球员总薪水不得超过700000美元。一名球员的最大可能薪水是多少美元？
Q5
For real numbers $a$ and $b$, define $a \triangle b = (a - b)^2$. What is $(x \triangle y) \triangle (y \triangle x)$?
对于实数 $a$ 和 $b$，定义 $a \triangle b = (a - b)^2$。求 $(x \triangle y) \triangle (y \triangle x)$ 的值。
Q6
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，$\overline{AC}$ 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？
Q7
An equilateral triangle of side length 10 is completely filled in by non-overlapping equilateral triangles of side length 1. How many small triangles are required?
一个边长为 10 的等边三角形完全被不重叠的边长为 1 的等边小三角形填充。需要多少个小三角形？
Q8
A class collects \$50 to buy flowers for a classmate who is in the hospital. Roses cost \$3 each, and carnations cost \$2 each. No other flowers are to be used. How many different bouquets could be purchased for exactly $50$?
一个班级收集了 $50 来为住院的同学买花。玫瑰每朵 $3，康乃馨每朵 $2。不使用其他花。有多少种不同的花束可以用恰好 $50 购买？
Q9
A quadratic equation $ax^2 - 2ax + b = 0$ has two real solutions. What is the average of the solutions?
二次方程 $ax^2 - 2ax + b = 0$ 有两个实根。两个根的平均值是多少？
Q10
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 是较小弧 AB 的中点。线段 AC 的长度是多少？
Q11
Suppose that $(u_n)$ is a sequence of real numbers satisfying $u_{n+2} = 2u_{n+1} + u_n$, and that $u_3 = 9$ and $u_6 = 128$. What is $u_5$?
假设 $(u_n)$ 是一个实数序列，满足 $u_{n+2} = 2u_{n+1} + u_n$，且 $u_3 = 9$，$u_6 = 128$。求 $u_5$ 的值。
Q12
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?
邮递员皮特有一个计步器来记录他的步数。计步器最多记录到 99999 步，然后下一步就翻转到 00000。皮特计划计算一年的里程。1 月 1 日皮特将计步器设为 00000。在一年中，计步器从 99999 翻转到 00000 共 44 次。12 月 31 日计步器显示 50000。皮特每英里走 1800 步。以下哪项最接近皮特一年走的英里数？
Q13
For each positive integer $n$, the mean of the first $n$ terms of a sequence is $n$. What is the 2008th term of the sequence?
对于每个正整数 $n$，序列前 $n$ 项的平均值为 $n$。求该序列的第 2008 项。
Q14
Triangle OAB has O = (0, 0), B = (5, 0), and A in the first quadrant. In addition, $\angle ABO = 90^\circ$ and $\angle AOB = 30^\circ$. Suppose that OA is rotated $90^\circ$ counterclockwise about O. What are the coordinates of the image of A?
三角形 OAB 有 O = (0, 0)，B = (5, 0)，且 A 在第一象限。此外，$\angle ABO = 90^\circ$ 且 $\angle AOB = 30^\circ$。假设 OA 绕 O 逆时针旋转 $90^\circ$。A 的像的坐标是什么？
Q15
How many right triangles have integer leg lengths $a$ and $b$ and a hypotenuse of length $b + 1$, where $b < 100$?
有整数直角边长 $a$ 和 $b$，斜边长为 $b + 1$ 的直角三角形有多少个，其中 $b < 100$？
Q16
Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, the sum is 0.)
抛掷两枚公平硬币一次。每次出现正面，就抛掷一枚公平骰子。骰子点数之和为奇数的概率是多少？（注意：如果没有骰子被抛掷，和为 0。）
Q17
A poll shows that 70% of all voters approve of the mayor’s work. On three separate occasions a pollster selects a voter at random. What is the probability that on exactly one of these three occasions the voter approves of the mayor’s work?
一项民调显示，70%的选民赞成市长的表现。民调员在三个不同的场合随机选择一名选民。恰好在这三个场合中的一个场合该选民赞成市长的表现的概率是多少？
Q18
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?
砖瓦工 Brenda 独自砌烟囱需要 9 小时，Brandon 独自需要 10 小时。他们一起工作时聊天很多，每小时总产量减少 10 块砖。他们一起用 5 小时砌完了烟囱。烟囱中共有几块砖？
Q19
A cylindrical tank with radius 4 feet and height 9 feet is lying on its side. The tank is filled with water to a depth of 2 feet. What is the volume of the water, in cubic feet?
一个半径 4 英尺、高 9 英尺的圆柱形水箱侧卧着。水箱中水深 2 英尺。水的体积是多少立方英尺？
Q20
The faces of a cubical die are marked with the numbers 1, 2, 2, 3, 3, and 4. The faces of a second cubical die are marked with the numbers 1, 3, 4, 5, 6, and 8. Both dice are thrown. What is the probability that the sum of the two top numbers will be 5, 7, or 9?
第一枚立方体骰子的面标有数字 1、2、2、3、3 和 4。第二枚立方体骰子的面标有数字 1、3、4、5、6 和 8。抛掷这两枚骰子。两个上面数字之和为 5、7 或 9 的概率是多少？
Q21
Ten chairs are evenly spaced around a round table and numbered clockwise from 1 through 10. Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or directly across from his or her spouse. How many seating arrangements are possible?
十把椅子均匀地围成一圈摆放，沿顺时针方向编号为1到10。五对夫妻要坐在椅子上，要求男女交替坐，并且没有人坐在配偶的旁边或正对面。有多少种可能的座位安排？
Q22
Three red beads, two white beads, and one blue bead are placed in a line in random order. What is the probability that no two neighboring beads are the same color?
三个红珠子、两个白珠子和一个蓝珠子随机排成一行。邻近的珠子没有相同颜色的概率是多少？
Q23
A rectangular floor measures $a$ feet 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 the area of the entire floor. How many possibilities are there for the ordered pair $(a, b)$?
一个矩形地板尺寸为$a$英尺乘$b$英尺，其中$a$和$b$是正整数且$b>a$。艺术家在地板上画一个矩形，其边与地板边平行。未涂部分形成宽度1英尺的边框，占据整个地板面积的一半。有多少种可能的有序对$(a,b)$？
Q24
Quadrilateral ABCD has AB = BC = CD, $\angle ABC = 70^\circ$, and $\angle BCD = 170^\circ$. What is the degree measure of $\angle BAD$?
四边形ABCD有AB = BC = CD，$\angle ABC = 70^\circ$，$\angle BCD = 170^\circ$。$\angle BAD$的度量是多少度？
Q25
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 travels at 10 feet per second in the same direction as Michael and 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英尺的速度在一长直路径上行走。垃圾桶沿路径每200英尺放置一个。垃圾车以每秒10英尺的速度朝同一方向行驶，并在每个垃圾桶停30秒。当迈克尔经过一个垃圾桶时，他注意到前方的卡车刚离开下一个垃圾桶。迈克尔和卡车会相遇多少次？
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