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AMC10 2009 B

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AMC10 · 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上，买了一个50美分的松饼或一个75美分的百吉饼。她一周的总花费是一个整数组美元。她买了多少个百吉饼？
Q2
Which of the following is equal to （）
下列哪一项等于（）
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Q3
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?
画家Paula原本有刚好足够的油漆刷30间大小相同的房间。不幸的是，在上班路上，三个油漆罐从她的卡车上掉了下来，所以她只够刷25间房间。她刷这25间房间用了多少罐油漆？
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. 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全等、直角等腰三角形的花坛。院子剩余部分呈梯形。梯形的平行边长分别为15米和25米。花坛占院子的几分之几？
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Q5
Twenty percent less than 60 is one-third more than what number?
60的百分之二十少于某个数的三分之一多于多少？
Q6
Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages?
Kiana 有两个年龄相同的双胞胎哥哥。他们三人的年龄乘积是 128。他们三人的年龄和是多少？
Q7
By inserting parentheses, it is possible to give the expression $2 \times 3 + 4 \times 5$ several values. How many different values can be obtained?
通过插入括号，可以使表达式 $2 \times 3 + 4 \times 5$ 有几种不同的值？
Q8
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，汽油价格在一月份上涨 20%，二月份下跌 20%，三月份上涨 25%，四月份下跌 $x\%$ 。四月底的汽油价格与一月初相同。四月份下跌的百分比 $x$ 取整到最近整数是多少？
Q9
Segment $BD$ and $AE$ intersect at $C$, as shown, $AB = BC = CD = CE$, and $\angle A = \frac{3}{2}\angle B$. What is the degree measure of $\angle D$?
线段 $BD$ 和 $AE$ 相交于 $C$，如图所示，$AB = BC = CD = CE$，且 $\angle A = \frac{3}{2}\angle B$。$\angle D$ 的度数是多少？
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Q10
A flagpole is originally 5 meters tall. A hurricane snaps the flagpole at a point $x$ meters above the ground so that the upper part, still attached to the stump, touches the ground 1 meter away from the base. What is $x$?
一根旗杆原来高 5 米。飓风从离地面 $x$ 米处将其折断，上部仍附着在残桩上，触地时距基部 1 米远。$x$ 是多少？
Q11
How many 7 digit palindromes (numbers that read the same backward as forward) can be formed using the digits 2, 2, 3, 3, 5, 5, 5?
使用数字 2, 2, 3, 3, 5, 5, 5 可以形成多少个7位回文数（正读反读都相同的数字）？
Q12
Distinct points $A, B, C, D$ lie on a line, with $AB = BC = CD = 1$. Points $E$ and $F$ lie on a second line, parallel to the first, with $EF = 1$. A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of the triangle?
不同的点 $A, B, C, D$ 在一条直线上，且 $AB = BC = CD = 1$。点 $E$ 和 $F$ 在第二条与第一条平行的直线上，且 $EF = 1$。以六个点中的三个为顶点的三角形面积大于零。这样的三角形面积可能有几个不同的值？
Q13
As shown below, convex pentagon ABCDE has sides AB = 3, BC = 4, CD = 6, DE = 3, and EA = 7. The pentagon is originally positioned in the plane with vertex A at the origin and vertex B on the positive x-axis. The pentagon is then rolled clockwise to the right along the x-axis. Which side will touch the point x = 2009 on the x-axis?
如图所示，凸五边形ABCDE的边长AB = 3, BC = 4, CD = 6, DE = 3, EA = 7。五边形最初位于平面中，顶点A在原点，顶点B在正x轴上。然后五边形沿x轴向右顺时针滚动。哪条边会接触x轴上的点x = 2009？
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Q14
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，Millie向鸟食器中放入一夸脱种子，其中25%是小米。此后每天她再添加一夸脱相同混合的种子，而不移除剩余的种子。每天鸟儿只吃掉食器中25%的小米，但吃掉所有其他种子。在Millie放入种子后哪一天，鸟儿会发现食器中超过一半的种子是小米？
Q15
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?
当水桶装满水的三分之二时，水桶和水的总重量为a千克。当水桶装满水的一半时，总重量为b千克。用a和b表示水桶满水时的总重量（千克）？
Q16
Points A and C lie on a circle centered at O, each of \(\overline{BA}\) and \(\overline{BC}\) are tangent to the circle, and \(\triangle ABC\) is equilateral. The circle intersects BO at D. What is BD/BO?
点 A 和 C 位于以 O 为圆心的圆上，\(\overline{BA}\) 和 \(\overline{BC}\) 各切于该圆，且 \(\triangle ABC\) 是等边三角形。该圆与 BO 相交于 D。求 BD/BO？
Q17
Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $(a, 0)$ to $(3, 3)$, divides the entire region into two regions of equal area. What is $a$?
五个单位正方形如图在坐标平面中排列，左下角位于原点。从 $(a, 0)$ 到 $(3, 3)$ 的斜线将整个区域分为两个面积相等的区域。求 $a$？
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Q18
Rectangle $ABCD$ has $AB = 8$ and $BC = 6$. Point $M$ is the midpoint of diagonal $\overline{AC}$, and $E$ is on $\overline{AB}$ with $ME \perp \overline{AC}$. What is the area of $\triangle AME$?
矩形 $ABCD$ 有 $AB = 8$ 和 $BC = 6$。点 $M$ 是对角线 $\overline{AC}$ 的中点，$E$ 在 $\overline{AB}$ 上且 $ME \perp \overline{AC}$。求 \(\triangle AME\) 的面积？
Q19
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?
一个特殊的 12 小时数字时钟显示一天的小时和分钟。不幸的是，每当它应该显示 1 时，它错误地显示 9。例如，当是下午 1:16 时，时钟错误显示 9:96 PM。这一天中，时钟显示正确时间的几分之几？
Q20
Triangle $ABC$ has a right angle at $B$, $AB = 1$, and $BC = 2$. The bisector of $\angle BAC$ meets $\overline{BC}$ at $D$. What is $BD$?
\(\triangle ABC\) 在 B 处直角，$AB = 1$，$BC = 2$。\(\angle BAC\) 的平分线与 $\overline{BC}$ 相交于 D。求 $BD$？
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Q21
What is the remainder when $3^0 + 3^1 + 3^2 + \dots + 3^{2009}$ is divided by 8?
将 $3^0 + 3^1 + 3^2 + \dots + 3^{2009}$ 除以 8 所得的余数是多少？
Q22
A cubical cake with edge length 2 inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where M is the midpoint of a top edge. The piece whose top is triangle B contains c cubic inches of cake and s square inches of icing. What is $c + s$?
一个边长 2 英寸的立方体蛋糕在侧面和顶面涂了糖霜。如顶视图所示，它被垂直切成三块，其中 M 是顶边中点。三角形 B 的顶部那块蛋糕包含 $c$ 立方英寸的蛋糕和 $s$ 平方英寸的糖霜。求 $c + s$。
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Q23
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 start 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 秒完成一圈，Robert 顺时针跑，每 80 秒完成一圈。他们同时从起点开始。在他们开始跑后 10 分钟到 11 分钟之间的某个随机时刻，站在跑道内侧的摄影师拍了一张照片，照片显示以起点线为中心的一刻跑道。Rachel 和 Robert 同时出现在照片中的概率是多少？
Q24
The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with 9 trapezoids, let $x$ be the angle measure in degrees of the larger interior angle of the trapezoid. What is $x$?
拱顶是古老的建筑特征。它由沿非平行边拼合的全等等腰梯形组成，如图所示。两端梯形的底边是水平的。用 9 个梯形构成的拱顶中，设 $x$ 为梯形较大内角的度数。求 $x$。
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Q25
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of its 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?
立方体每个面上从一条边的中心画一条窄条纹到对边中心的中心。每面边对选择随机独立。存在一条环绕立方体的连续条纹的概率是多少？
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