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AMC12 2009 B

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AMC12 · 2009 (B)

Q1
Each morning of her five-day workweek, Jane bought either a $50$-cent muffin or a $75$-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?
Jane 在她五天工作周的每个早晨，都会买一个 $50$ 美分的松饼或一个 $75$ 美分的百吉饼。她这一周的总花费是整数美元。她买了多少个百吉饼？
Q2
Paula the painter had just enough paint for 30 identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for 25 rooms. How many cans of paint did she use for the 25 rooms?
油漆工 Paula 原本有刚好够刷 $30$ 间大小相同房间的油漆。不幸的是，在去上班的路上，有三罐油漆从她的卡车上掉了下来，所以她现在只够刷 $25$ 间房。她刷这 $25$ 间房用了多少罐油漆？
Q3
Twenty percent less than 60 is one-third more than what number?
比 $60$ 少 $20\%$ 的数，比哪个数多三分之一？
Q4
A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the flower beds?
一个矩形院子里有两个形状为全等的等腰直角三角形的花坛。院子剩余部分呈梯形，如图所示。梯形的两条平行边长分别为 $15$ 米和 $25$ 米。花坛占院子面积的几分之几？
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Q5
Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages?
Kiana 有两个年长的双胞胎哥哥。他们三个人年龄的乘积是 $128$。他们三个人年龄的和是多少？
Q6
By inserting parentheses, it is possible to give the expression \[2\times3 + 4\times5\] several values. How many different values can be obtained?
通过插入括号，可以使表达式 \[2\times3 + 4\times5\] 具有多个值。可以得到多少个不同的值？
Q7
In a certain year the price of gasoline rose by $20\%$ during January, fell by $20\%$ during February, rose by $25\%$ during March, and fell by $x\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is $x$
在某一年，汽油价格在一月上涨 $20\%$，二月下跌 $20\%$，三月上涨 $25\%$，四月下跌 $x\%$。四月底的汽油价格与一月初相同。$x$ 的最近整数是多少？
Q8
When a bucket is two-thirds full of water, the bucket and water weigh $a$ kilograms. When the bucket is one-half full of water the total weight is $b$ kilograms. In terms of $a$ and $b$, what is the total weight in kilograms when the bucket is full of water?
当一个水桶装了三分之二桶水时，水桶和水的总重量为 $a$ 千克。当水桶装了二分之一桶水时，总重量为 $b$ 千克。用 $a$ 和 $b$ 表示，当水桶装满水时的总重量（千克）是多少？
Q9
Triangle $ABC$ has vertices $A = (3,0)$, $B = (0,3)$, and $C$, where $C$ is on the line $x + y = 7$. What is the area of $\triangle ABC$?
三角形 $ABC$ 的顶点为 $A = (3,0)$、$B = (0,3)$，以及 $C$，其中 $C$ 在直线 $x + y = 7$ 上。$\triangle ABC$ 的面积是多少？
Q10
A particular $12$-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a $1$, it mistakenly displays a $9$. For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time?
某种 $12$ 小时制的数字时钟显示一天中的小时和分钟。不幸的是，每当它应该显示一个 $1$ 时，它会错误地显示成一个 $9$。例如，当时间是下午 1:16 时，时钟会错误显示为下午 9:96。一天中有多少比例的时间该时钟会显示正确时间？
Q11
On Monday, Millie puts a quart of seeds, $25\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $25\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?
周一，米莉向一个鸟食器中放入一夸脱种子，其中 $25\%$ 是小米。在接下来的每一天，她都会再加入一夸脱相同配比的种子，并且不取出任何剩下的种子。每天鸟只吃掉食器中小米的 $25\%$，但会吃掉所有其他种子。问在哪一天，米莉刚放入种子后，鸟会发现食器中超过一半的种子是小米？
Q12
The fifth and eighth terms of a geometric sequence of real numbers are $7!$ and $8!$ respectively. What is the first term?
一个实数等比数列的第 5 项和第 8 项分别是 $7!$ 和 $8!$。首项是多少？
Q13
Triangle $ABC$ has $AB = 13$ and $AC = 15$, and the altitude to $\overline{BC}$ has length $12$. What is the sum of the two possible values of $BC$?
三角形 $ABC$ 中 $AB = 13$ 且 $AC = 15$，到 $\overline{BC}$ 的高长为 $12$。$BC$ 的两个可能取值之和是多少？
Q14
Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $(c,0)$ to $(3,3)$, divides the entire region into two regions of equal area. What is $c$?
如图，坐标平面上摆放了五个单位正方形，左下角在原点。连接 $(c,0)$ 与 $(3,3)$ 的斜线将整个区域分成面积相等的两部分。求 $c$。
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Q15
Assume $0 < r < 3$. Below are five equations for $x$. Which equation has the largest solution $x$?
设 $0 < r < 3$。下面给出五个关于 $x$ 的方程。哪个方程的解 $x$ 最大？
Q16
Trapezoid $ABCD$ has $AD||BC$, $BD = 1$, $\angle DBA = 23^{\circ}$, and $\angle BDC = 46^{\circ}$. The ratio $BC: AD$ is $9: 5$. What is $CD$?
梯形 $ABCD$ 满足 $AD||BC$，$BD = 1$，$\angle DBA = 23^{\circ}$，且 $\angle BDC = 46^{\circ}$。比值 $BC: AD$ 为 $9: 5$。求 $CD$。
Q17
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube?
一个立方体的每个面上都画有一条从一条边的中心到其对边中心的窄条纹。每个面的边配对选择是随机且相互独立的。求存在一条连续条纹环绕立方体的概率。
Q18
Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture?
Rachel 和 Robert 在一个圆形跑道上跑步。Rachel 逆时针跑，每 90 秒跑完一圈；Robert 顺时针跑，每 80 秒跑完一圈。两人同时从同一起跑线出发。在他们开始跑后的 10 分钟到 11 分钟之间的某个随机时刻，站在跑道内的摄影师拍了一张照片，照片显示跑道的四分之一，且以起跑线为中心。求 Rachel 和 Robert 都在照片中的概率。
Q19
For each positive integer $n$, let $f(n) = n^4 - 360n^2 + 400$. What is the sum of all values of $f(n)$ that are prime numbers?
对每个正整数 $n$，令 $f(n) = n^4 - 360n^2 + 400$。所有使得 $f(n)$ 为素数的取值中，$f(n)$ 的总和是多少？
Q20
A convex polyhedron $Q$ has vertices $V_1,V_2,\ldots,V_n$, and $100$ edges. The polyhedron is cut by planes $P_1,P_2,\ldots,P_n$ in such a way that plane $P_k$ cuts only those edges that meet at vertex $V_k$. In addition, no two planes intersect inside or on $Q$. The cuts produce $n$ pyramids and a new polyhedron $R$. How many edges does $R$ have?
一个凸多面体 $Q$ 有顶点 $V_1,V_2,\ldots,V_n$，并且有 $100$ 条棱。用平面 $P_1,P_2,\ldots,P_n$ 切割该多面体，使得平面 $P_k$ 只切割那些在顶点 $V_k$ 处相交的棱。此外，任意两平面都不在 $Q$ 的内部或表面相交。切割产生 $n$ 个棱锥和一个新的多面体 $R$。求 $R$ 有多少条棱。
Q21
Ten women sit in $10$ seats in a line. All of the $10$ get up and then reseat themselves using all $10$ seats, each sitting in the seat she was in before or a seat next to the one she occupied before. In how many ways can the women be reseated?
十位女士坐在一条直线上的 $10$ 个座位上。她们全部站起来，然后重新坐下，使用所有 $10$ 个座位，并且每位女士坐在她之前坐的座位或与其相邻的座位上。她们重新坐下的方式有多少种？
Q22
Parallelogram $ABCD$ has area $1,\!000,\!000$. Vertex $A$ is at $(0,0)$ and all other vertices are in the first quadrant. Vertices $B$ and $D$ are lattice points on the lines $y = x$ and $y = kx$ for some integer $k > 1$, respectively. How many such parallelograms are there? (A lattice point is any point whose coordinates are both integers.)
平行四边形 $ABCD$ 的面积为 $1,\!000,\!000$。顶点 $A$ 在 $(0,0)$，其余顶点都在第一象限。顶点 $B$ 和 $D$ 分别是直线 $y=x$ 与 $y=kx$（其中 $k>1$ 为整数）上的格点。这样的平行四边形有多少个？（格点指坐标均为整数的点。）
Q23
A region $S$ in the complex plane is defined by \[S = \{x + iy: - 1\le x\le1, - 1\le y\le1\}.\] A complex number $z = x + iy$ is chosen uniformly at random from $S$. What is the probability that $\left(\frac34 + \frac34i\right)z$ is also in $S$?
复平面中的区域 $S$ 定义为 \[S = \{x + iy: - 1\le x\le1, - 1\le y\le1\}.\] 从 $S$ 中均匀随机选取一个复数 $z = x + iy$。求 $\left(\frac34 + \frac34i\right)z$ 也在 $S$ 中的概率。
Q24
For how many values of $x$ in $[0,\pi]$ is $\sin^{ - 1}(\sin 6x) = \cos^{ - 1}(\cos x)$? Note: The functions $\sin^{ - 1} = \arcsin$ and $\cos^{ - 1} = \arccos$ denote inverse trigonometric functions.
在 $[0,\pi]$ 内，有多少个 $x$ 满足 $\sin^{ - 1}(\sin 6x) = \cos^{ - 1}(\cos x)$？ 注意：函数 $\sin^{ - 1} = \arcsin$ 与 $\cos^{ - 1} = \arccos$ 表示反三角函数。
Q25
The set $G$ is defined by the points $(x,y)$ with integer coordinates, $3\le|x|\le7$, $3\le|y|\le7$. How many squares of side at least $6$ have their four vertices in $G$?
集合 $G$ 由整数坐标点 $(x,y)$ 构成，满足 $3\le|x|\le7$，$3\le|y|\le7$。有多少个边长至少为 $6$ 的正方形，其四个顶点都在 $G$ 中？
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