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AMC10 2007 B

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AMC10 · 2007 (B)

Q1
Isabella's house has 3 bedrooms. Each bedroom is 12 feet long, 10 feet wide, and 8 feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy 60 square feet in each bedroom. How many square feet of walls must be painted?
Isabella 的房子有 3 间卧室。每间卧室长 12 英尺，宽 10 英尺，高 8 英尺。Isabella 必须粉刷所有卧室的墙壁。每间卧室中有门窗占用 60 平方英尺，这些地方不需要粉刷。需要粉刷多少平方英尺的墙壁？
Q2
Define the operation $\star$ by $a \star b = (a + b)b$. What is $(3 \star 5) - (5 \star 3)$?
定义运算 $\star$ 为 $a \star b = (a + b)b$。求 $(3 \star 5) - (5 \star 3)$ 的值？
Q3
A college student drove his compact car 120 miles home for the weekend and averaged 30 miles per gallon. On the return trip the student drove his parents' SUV and averaged only 20 miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?
一位大学生开车 120 英里回家过周末，平均油耗 30 英里/加仑。回程时他开父母的 SUV，平均油耗只有 20 英里/加仑。往返总油耗平均是多少英里/加仑？
Q4
The point $O$ is the center of the circle circumscribed about $\triangle ABC$, with $\angle BOC = 120^\circ$ and $\angle AOB = 140^\circ$, as shown. What is the degree measure of $\angle ABC$?
点 $O$ 是 $\triangle ABC$ 的外接圆圆心，已知 $\angle BOC = 120^\circ$，$\angle AOB = 140^\circ$，如图所示。求 $\angle ABC$ 的度数？
stem
Q5
In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these facts?
在某个国度，所有 Arogs 都是 Brafs，所有 Crups 都是 Brafs，所有 Dramps 都是 Arogs，所有 Crups 都是 Dramps。以下哪个陈述是由这些事实蕴含的？
Q6
The 2007 AMC 10 will be scored by awarding 6 points for each correct response, 0 points for each incorrect response, and 1.5 points for each problem left unanswered. After looking over the 25 problems, Sarah has decided to attempt the first 22 and leave only the last 3 unanswered. How many of the first 22 problems must she solve correctly in order to score at least 100 points?
2007年AMC 10的评分规则是：每题正确得6分，每题错误得0分，每题未答得1.5分。Sarah看过25道题后，决定尝试前22道，只留最后3道未答。为了得到至少100分，她必须在前22道题中正确解出多少道？
Q7
All sides of the convex pentagon ABCDE are of equal length, and $\angle A = \angle B = 90^\circ$. What is the degree measure of $\angle E$?
凸五边形ABCDE的所有边长相等，且$\angle A = \angle B = 90^\circ$。$\angle E$的度数是多少？
Q8
On the trip home from the meeting where this AMC10 was constructed, the Contest Chair noted that his airport parking receipt had digits of the form bbcac, where $0 \leq a < b < c \leq 9$, and $b$ was the average of $a$ and $c$. How many different five-digit numbers satisfy all these properties?
在构造本AMC10的会议回家的路上，Contest Chair注意到他的机场停车收据上的数字形式为bbcac，其中$0 \leq a < b < c \leq 9$，且$b$是$a$和$c$的平均数。满足所有这些性质的不同五位数有多少个？
Q9
A cryptographic code is designed as follows. The first time a letter appears in a given message it is replaced by the letter that is 1 place to its right in the alphabet (assuming that the letter A is one place to the right of the letter Z). The second time this same letter appears in the given message, it is replaced by the letter that is 1 + 2 places to the right, the third time it is replaced by the letter that is 1 + 2 + 3 places to the right, and so on. For example, with this code the word “banana” becomes “cboddg”. What letter will replace the last letter s in the message “Lee’s sis is a Mississippi miss, Chriss!”?
一种密码编码设计如下：给定消息中一个字母第一次出现时，用字母表中它右边1位的字母替换它（假设Z右边1位是A）。该字母第二次出现时，用右边1+2位替换，第三次用1+2+3位，以此类推。例如，用此编码，“banana”变成“cboddg”。消息“Lee’s sis is a Mississippi miss, Chriss!”中最后一个字母s将被什么字母替换？
Q10
Two points B and C are in a plane. Let S be the set of all points A in the plane for which $\triangle ABC$ has area 1. Which of the following describes S?
平面上有两点B和C。让S为平面中所有点A的集合，使得$\triangle ABC$的面积为1。以下哪项描述了S？
Q11
A circle passes through the three vertices of an isosceles triangle that has two sides of length 3 and a base of length 2. What is the area of this circle?
一个圆经过一个等腰三角形的三个顶点，该等腰三角形有两个边长为3，底边长为2。这个圆的面积是多少？
Q12
Tom’s age is T years, which is also the sum of the ages of his three children. His age N years ago was twice the sum of their ages then. What is T/N?
汤姆的年龄是T岁，这也是他三个孩子的年龄之和。T-N年前，他的年龄是当时他们三个孩子年龄之和的两倍。T/N是多少？
Q13
Two circles of radius 2 are centered at (2, 0) and at (0, 2). What is the area of the intersection of the interiors of the two circles?
两个半径为2的圆分别以(2, 0)和(0, 2)为圆心。两个圆内部交集的面积是多少？
Q14
Some boys and girls are having a car wash to raise money for a class trip to China. Initially 40% of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then 30% of the group are girls. How many girls were initially in the group?
一些男孩和女孩正在为班级去中国的旅行筹款洗车。最初小组的40%是女孩。不久之后，两个女孩离开，两个男孩到来，然后小组的30%是女孩。最初小组中有多少女孩？
Q15
The angles of quadrilateral ABCD satisfy $\angle A = 2\angle B = 3\angle C = 4\angle D$. What is the degree measure of $\angle A$, rounded to the nearest whole number?
四边形ABCD的内角满足$\angle A = 2\angle B = 3\angle C = 4\angle D$。$\angle A$的度数，四舍五入到最接近的整数是多少？
Q16
A teacher gave a test to a class in which 10% of the students are juniors and 90% are seniors. The average score on the test was 84. The juniors all received the same score, and the average score of the seniors was 83. What score did each of the juniors receive on the test?
一位老师给一个班级出了一次测试，班级中有10%的学生是低年级生，90%是高年级生。测试的平均分为84。所有低年级生的得分相同，高年级生的平均分为83。低年级生每人得了多少分？
Q17
Point P is inside equilateral $\triangle ABC$. Points Q, R, and S are the feet of the perpendiculars from P to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. Given that $PQ = 1$, $PR = 2$, and $PS = 3$, what is $AB$?
点P在等边$\triangle ABC$内部。点Q、R、S分别是P到$\overline{AB}$、$\overline{BC}$和$\overline{CA}$的垂足。已知$PQ = 1$，$PR = 2$，$PS = 3$，求$AB$？
Q18
A circle of radius 1 is surrounded by 4 circles of radius $r$ as shown. What is $r$?
如图，一个半径为1的圆被4个半径为$r$的圆包围。求$r$？
stem
Q19
The wheel shown is spun twice, and the randomly determined numbers opposite the pointer are recorded. The first number is divided by 4, and the second number is divided by 5. The first remainder designates a column, and the second remainder designates a row on the checkerboard shown. What is the probability that the pair of numbers designates a shaded square?
转动如图所示的轮盘两次，记录指针对面的随机数字。将第一个数字除以4，第二个数字除以5。第一余数指定一列，第二余数指定一行在如图棋盘上。指定着色方格的概率是多少？
stem
Q20
A set of 25 square blocks is arranged into a 5 × 5 square. How many different combinations of 3 blocks can be selected from that set so that no two are in the same row or column?
25个正方形方块排成5×5正方形。从中选3个方块的不同组合有多少种，使得没有两个在同一行或同一列？
Q21
Right $\triangle ABC$ has $AB = 3$, $BC = 4$, and $AC = 5$. Square $XYZW$ is inscribed in $\triangle ABC$ with $X$ and $Y$ on $AC$, $W$ on $AB$, and $Z$ on $BC$. What is the side length of the square?
直角 $\triangle ABC$ 有 $AB = 3$，$BC = 4$，$AC = 5$。正方形 $XYZW$ 铭刻在 $\triangle ABC$ 中，其中 $X$ 和 $Y$ 在 $AC$ 上，$W$ 在 $AB$ 上，$Z$ 在 $BC$ 上。正方形的边长是多少？
stem
Q22
A player chooses one of the numbers 1 through 4. After the choice has been made, two regular four-sided (tetrahedral) dice are rolled, with the sides of the dice numbered 1 through 4. If the number chosen appears on the bottom of exactly one die after it is rolled, then the player wins \$1. If the number chosen appears on the bottom of both of the dice, then the player wins \$2. If the number chosen does not appear on the bottom of either of the dice, the player loses \$1. What is the expected return to the player, in dollars, for one roll of the dice?
一名玩家选择 1 到 4 中的一个数字。选择后，掷两个正四面体骰子，骰子面编号 1 到 4。如果所选数字恰好出现在一个骰子的底部，玩家赢得 $1。如果出现在两个骰子的底部，赢得 $2。如果都不出现，输 $1$。掷一次骰子的玩家的期望收益是多少美元？
Q23
A pyramid with a square base is cut by a plane that is parallel to its base and is 2 units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. What is the altitude of the original pyramid?
一个底面为正方形的金字塔被一个与底面平行且距底面 2 单位的平面切割。从顶部切下的小金字塔的表面积是原金字塔表面积的一半。原金字塔的高度是多少？
Q24
Let $n$ denote the smallest positive integer that is divisible by both 4 and 9, and whose base-10 representation consists of only 4's and 9's, with at least one of each. What are the last four digits of $n$?
设 $n$ 表示最小的既能被 4 和 9 整除，其十进制表示只由 4 和 9 组成且至少各有一个的正整数。$n$ 的最后四位数字是什么？
Q25
How many pairs of positive integers $(a, b)$ are there such that $a$ and $b$ have no common factors greater than 1 and $\frac{a}{b} + \frac{14b}{9a}$ is an integer?
有多少对正整数对 $(a, b)$ 使得 $a$ 和 $b$ 没有大于 1 的公因数，且 $\frac{a}{b} + \frac{14b}{9a}$ 是整数？
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