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AMC12 2007 B

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AMC12 · 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 $3$ 个卧室。每个卧室长 $12$ 英尺，宽 $10$ 英尺，高 $8$ 英尺。Isabella 必须粉刷所有卧室的墙壁。每个卧室的门窗（不需粉刷）占用 $60$ 平方英尺。必须粉刷多少平方英尺的墙壁？
Q2
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?
一名大学生开他的紧凑型汽车行驶 $120$ 英里回家过周末，平均油耗为每加仑 $30$ 英里。返程时该学生开父母的 SUV，平均油耗只有每加仑 $20$ 英里。往返行程的平均油耗（英里/加仑）是多少？
Q3
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 $O$ 是三角形 $ABC$ 的外接圆圆心，且 $\angle BOC = 120^{\circ}$、$\angle AOB = 140^{\circ}$，如图所示。$\angle ABC$ 的度数是多少？
stem
Q4
At Frank's Fruit Market, $3$ bananas cost as much as $2$ apples, and $6$ apples cost as much as $4$ oranges. How many oranges cost as much as $18$ bananas?
在 Frank 的水果市场，$3$ 根香蕉的价格与 $2$ 个苹果相同，且 $6$ 个苹果的价格与 $4$ 个橙子相同。多少个橙子的价格与 $18$ 根香蕉相同？
Q5
The 2007 AMC 12 contests 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 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 12 比赛的计分方式为：每题答对得 6 分，答错得 0 分，不作答得 1.5 分。看完 25 道题后，Sarah 决定尝试前 22 题，并将最后 3 题留空不答。她在前 22 题中至少需要答对多少题才能得到至少 100 分？
Q6
Triangle $ABC$ has side lengths $AB = 5$, $BC = 6$, and $AC = 7$. Two bugs start simultaneously from $A$ and crawl along the sides of the triangle in opposite directions at the same speed. They meet at point $D$. What is $BD$?
三角形 $ABC$ 的边长 $AB = 5$，$BC = 6$，$AC = 7$。两只虫子同时从 $A$ 点出发，沿着三角形的边以相同的速度向相反方向爬行。它们在点 $D$ 相遇。$BD$ 等于多少？
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
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 $T/N$?
Tom 的年龄是 $T$ 岁，这也是他三个孩子的年龄之和。$N$ 年前，他的年龄是当时他们年龄之和的两倍。$T/N$ 等于多少？
Q9
A function $f$ has the property that $f(3x-1)=x^2+x+1$ for all real numbers $x$. What is $f(5)$?
函数 $f$ 满足对于所有实数 $x$，$f(3x-1)=x^2+x+1$。$f(5)$ 等于多少？
Q10
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?
一些男孩和女孩在为班级中国之旅筹款而洗车。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？
Q11
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 $\angle A=2 \angle B=3 \angle C=4 \angle D.$ 求 $\angle A$ 的度数，并将结果四舍五入到最接近的整数。
Q12
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进行了一次测试，其中 $10\%$ 的学生是低年级生，$90\%$ 是高年级生。测试的平均分是 $84.$ 所有低年级生的得分都相同，而高年级生的平均分是 $83.$ 问每个低年级生在测试中得了多少分？
Q13
A traffic light runs repeatedly through the following cycle: green for $30$ seconds, then yellow for $3$ seconds, and then red for $30$ seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching?
一个交通信号灯反复循环以下周期：绿灯亮 $30$ 秒，然后黄灯亮 $3$ 秒，然后红灯亮 $30$ 秒。Leah 随机选择一个 $3$ 秒的时间区间来观察信号灯。她观察期间信号灯发生变色的概率是多少？
Q14
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。从 $P$ 向 $\overline{AB}$、$\overline{BC}$ 和 $\overline{CA}$ 作垂线，垂足分别为 $Q$、$R$ 和 $S$。已知 $PQ=1$、$PR=2$、$PS=3$，求 $AB$。
Q15
The geometric series $a+ar+ar^2\ldots$ has a sum of $7$, and the terms involving odd powers of $r$ have a sum of $3$. What is $a+r$?
几何级数 $a+ar+ar^2\ldots$ 的和为 $7$，且含有 $r$ 的奇数次幂的项的和为 $3$。求 $a+r$。
Q16
Each face of a regular tetrahedron is painted either red, white, or blue. Two colorings are considered indistinguishable if two congruent tetrahedra with those colorings can be rotated so that their appearances are identical. How many distinguishable colorings are possible?
一个正四面体的每个面都被涂成红色、白色或蓝色。如果两个着色被认为不可区分，当且仅当两个带这些着色的全等四面体可以通过旋转使它们的外观完全相同。有多少种可区分的着色方式？
Q17
If $a$ is a nonzero integer and $b$ is a positive number such that $ab^2=\log_{10}b$, what is the median of the set $\{0,1,a,b,1/b\}$?
若$a$为非零整数，$b$为正数，且满足 $ab^2=\log_{10}b$，则集合 $\{0,1,a,b,1/b\}$ 的中位数是多少？
Q18
Let $a$, $b$, and $c$ be digits with $a\ne 0$. The three-digit integer $abc$ lies one third of the way from the square of a positive integer to the square of the next larger integer. The integer $acb$ lies two thirds of the way between the same two squares. What is $a+b+c$?
设$a$、$b$、$c$为数字且$a\ne 0$。三位整数$abc$位于某个正整数的平方与下一个更大整数的平方之间的三分之一处。整数$acb$位于同一对平方之间的三分之二处。求$a+b+c$。
Q19
Rhombus $ABCD$, with side length $6$, is rolled to form a cylinder of volume $6$ by taping $\overline{AB}$ to $\overline{DC}$. What is $\sin(\angle ABC)$?
边长为$6$的菱形$ABCD$通过将$\overline{AB}$与$\overline{DC}$粘贴卷成一个体积为$6$的圆柱。求$\sin(\angle ABC)$。
Q20
The parallelogram bounded by the lines $y=ax+c$, $y=ax+d$, $y=bx+c$, and $y=bx+d$ has area $18$. The parallelogram bounded by the lines $y=ax+c$, $y=ax-d$, $y=bx+c$, and $y=bx-d$ has area $72$. Given that $a$, $b$, $c$, and $d$ are positive integers, what is the smallest possible value of $a+b+c+d$?
由直线 $y=ax+c$, $y=ax+d$, $y=bx+c$, $y=bx+d$ 围成的平行四边形面积为$18$。由直线 $y=ax+c$, $y=ax-d$, $y=bx+c$, $y=bx-d$ 围成的平行四边形面积为$72$。已知$a$、$b$、$c$、$d$为正整数，求$a+b+c+d$的最小可能值。
Q21
The first $2007$ positive integers are each written in base $3$. How many of these base-$3$ representations are palindromes? (A palindrome is a number that reads the same forward and backward.)
前 $2007$ 个正整数都用 $3$ 进制表示。其中有多少个 $3$ 进制表示是回文数？（回文数是指从左到右与从右到左读起来相同的数。）
Q22
Two particles move along the edges of equilateral $\triangle ABC$ in the direction \[A\Rightarrow B\Rightarrow C\Rightarrow A,\] starting simultaneously and moving at the same speed. One starts at $A$, and the other starts at the midpoint of $\overline{BC}$. The midpoint of the line segment joining the two particles traces out a path that encloses a region $R$. What is the ratio of the area of $R$ to the area of $\triangle ABC$?
两个粒子沿正三角形 $\triangle ABC$ 的边按方向 \[A\Rightarrow B\Rightarrow C\Rightarrow A,\] 同时出发并以相同速度运动。一个从 $A$ 出发，另一个从 $\overline{BC}$ 的中点出发。连接这两个粒子的线段的中点所描出的轨迹围成一个区域 $R$。求 $R$ 的面积与 $\triangle ABC$ 的面积之比。
Q23
How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $3$ times their perimeters?
有多少个两条直角边长为正整数且互不全等的直角三角形，使得它们的面积在数值上等于其周长的 $3$ 倍？
Q24
How many pairs of positive integers $(a,b)$ are there such that $\text{gcd}(a,b)=1$ and $\frac{a}{b} + \frac{14b}{9a}$ is an integer?
有多少对正整数 $(a,b)$ 满足 $\text{gcd}(a,b)=1$ 且 $\frac{a}{b} + \frac{14b}{9a}$ 是整数？
Q25
Points $A,B,C,D$ and $E$ are located in 3-dimensional space with $AB=BC=CD=DE=EA=2$ and $\angle ABC=\angle CDE=\angle DEA=90^o$. The plane of $\triangle ABC$ is parallel to $\overline{DE}$. What is the area of $\triangle BDE$?
点 $A,B,C,D$ 和 $E$ 位于三维空间中，满足 $AB=BC=CD=DE=EA=2$ 且 $\angle ABC=\angle CDE=\angle DEA=90^o$。$\triangle ABC$ 所在平面与 $\overline{DE}$ 平行。求 $\triangle BDE$ 的面积。
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