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AMC10 2009 A

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AMC10 · 2009 (A)

Q1
One can holds 12 ounces of soda. What is the minimum number of cans needed to provide a gallon ($128$ ounces) of soda?
一罐装12盎司苏打水。提供一加仑（128盎司）苏打水需要的最少罐数是多少？
Q2
Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could not be the total value of the four coins, in cents?
从一个装有便士、镍币、角币和25美分硬币的存钱罐中取出4枚硬币。以下哪个不可能是这4枚硬币的总价值（以美分为单位）？
Q3
Which of the following is equal to $1 + \frac{1}{1 + \frac{1}{1+1}}$?
以下哪个等于 $1 + \frac{1}{1 + \frac{1}{1+1}}$？
Q4
Eric plans to compete in a triathlon. He can average 2 miles per hour in the 1-mile swim and 6 miles per hour in the 3-mile run. His goal is to finish the triathlon in 2 hours. To accomplish his goal what must his average speed, in miles per hour, be for the 15-mile bicycle ride?
Eric计划参加一场铁人三项比赛。他在1英里游泳中平均速度为2英里每小时，在3英里跑步中平均速度为6英里每小时。他的目标是在2小时内完成比赛。为了实现目标，他在15英里自行车骑行中的平均速度必须是多少英里每小时？
Q5
What is the sum of the digits of the square of 111,111,111?
111,111,111的平方数的各位数字之和是多少？
Q6
A circle of radius 2 is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is shaded. What fraction of the semicircle's area is shaded?
一个半径为2的圆内切于一个半圆中，如图所示。半圆内圆外阴影部分的面积占半圆面积的几分之几？
stem
Q7
A carton contains milk that is 2% fat, an amount that is 40% less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk?
一盒牛奶脂肪含量为2%，这是全脂牛奶同体积脂肪含量的40%少。 全脂牛奶的脂肪百分比是多少？
Q8
Three generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a 50% discount as children. The two members of the oldest generation receive a 25% discount as senior citizens. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs $6.00, is paying for everyone. How many dollars must he pay?
温家三代人每代两人去电影院。最年轻一代两人作为儿童享受50%折扣。最年长一代两人作为老人享受25%折扣。中间一代两人无折扣。温爷爷的老人票价为6.00美元，他为所有人付款。他需要支付多少美元？
Q9
Positive integers $a$, $b$, and 2009, with $a < b < 2009$, form a geometric sequence with an integer ratio. What is $a$?
正整数 $a$、$b$ 和 2009 构成公比为整数的等比数列，且 $a < b < 2009$。$a$ 等于多少？
Q10
Triangle $ABC$ has a right angle at $B$. Point $D$ is the foot of the altitude from $B$, $AD = 3$, and $DC = 4$. What is the area of $\triangle ABC$?
三角形 $ABC$ 在 $B$ 处为直角。$D$ 是 $B$ 垂足，$AD = 3$，$DC = 4$。$ riangle ABC$ 的面积是多少？
stem
Q11
One dimension of a cube is increased by 1, another is decreased by 1, and the third is left unchanged. The volume of the new rectangular solid is 5 less than that of the cube. What was the volume of the cube?
一个立方体的边长有一个维度增加1，另一个维度减少1，第三个维度保持不变。新矩形体的体积比立方体的小5。立方体的体积是多少？
Q12
In quadrilateral ABCD, AB = 5, BC = 17, CD = 5, DA = 9, and BD is an integer. What is BD?
在四边形ABCD中，AB = 5, BC = 17, CD = 5, DA = 9，且BD是整数。BD是多少？
stem
Q13
Suppose that P = 2^m and Q = 3^n. Which of the following is equal to 12^{mn} for every pair of integers (m, n)?
设P = $2^m$且Q = $3^n$。以下哪个等价于$12^{mn}$，对任意整数对(m, n)成立？
Q14
Four congruent rectangles are placed as shown. The area of the outer square is 4 times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?
如图所示放置了四个全等的矩形。外正方形的面积是内正方形的4倍。每个矩形长边与短边的比是多少？
stem
Q15
The figures F₁, F₂, F₃ and F₄ shown are the first in a sequence of figures. For n ≥ 3, Fₙ is constructed from Fₙ₋₁ by surrounding it with a square and placing one more diamond on each side of the new square than Fₙ₋₁ had on each side of its outside square. For example, figure F₃ has 13 diamonds. How many diamonds are there in figure F₂₀?
所示的图形F₁, F₂, F₃和F₄是图形序列中的前几个。对于n ≥ 3，Fₙ由Fₙ₋₁构成，通过在其周围围上一个正方形，并在新正方形的每边放置比Fₙ₋₁外正方形每边多一个菱形。例如，图形F₃有13个菱形。图形F₂₀有多少个菱形？
stem
Q16
Let $a, b, c,$ and $d$ be real numbers with $|a - b| = 2$, $|b - c| = 3$, and $|c - d| = 4$. What is the sum of all possible values of $|a - d|$?
设 $a, b, c,$ 和 $d$ 是实数，且 $|a - b| = 2$，$|b - c| = 3$，$|c - d| = 4$。所有可能的 $|a - d|$ 的值之和是多少？
Q17
Rectangle $ABCD$ has $AB = 4$ and $BC = 3$. Segment $EF$ is constructed through $B$ so that $EF \perp DB$, and $A$ and $C$ lie on $DE$ and $DF$, respectively. What is $EF$?
矩形 $ABCD$ 有 $AB = 4$ 和 $BC = 3$。通过 $B$ 构造线段 $EF$ 使得 $EF \perp DB$，且 $A$ 和 $C$ 分别位于 $DE$ 和 $DF$ 上。$EF$ 等于多少？
Q18
At Jefferson Summer Camp, 60% of the children play soccer, 30% of the children swim, and 40% of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer?
在杰斐逊夏令营，60%的孩子踢足球，30%的孩子游泳，且40%的踢足球的孩子游泳。不游泳的孩子中有大约百分之多少踢足球？（结果四舍五入到最接近的整百分比）
Q19
Circle $A$ has radius 100. Circle $B$ has an integer radius $r < 100$ and remains internally tangent to circle $A$ as it rolls once around the circumference of circle $A$. The two circles have the same points of tangency at the beginning and end of circle $B$'s trip. How many possible values can $r$ have?
圆 $A$ 的半径为100。圆 $B$ 的整数半径 $r < 100$，它在绕圆 $A$ 的周长滚一圈时始终与圆 $A$ 内部相切。在 $B$ 滚完一圈的开始和结束时，两圆有相同的切点。$r$ 有多少个可能值？
Q20
Andrea and Lauren are 20 kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of 1 kilometer per minute. After 5 minutes, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from the time they started to bike does Lauren reach Andrea?
Andrea 和 Lauren 相距20千米。她们相向骑车，Andrea 的速度是 Lauren 的三倍，且它们之间的距离以每分钟1千米的速度减小。5分钟后，Andrea 因为爆胎停止骑车并等待 Lauren。从她们开始骑车起，经过多少分钟 Lauren 到达 Andrea 处？
Q21
Many Gothic cathedrals have windows with portions containing a ring of congruent circles that are circumscribed by a larger circle. In the figure shown, the number of smaller circles is four. What is the ratio of the sum of the areas of the four smaller circles to the area of the larger circle?
许多哥特式大教堂的窗户中有部分包含一个由全等圆组成的环，这些圆被一个更大的圆外切。在图示中，小圆的数量是四个。小圆面积之和与大圆面积的比率为多少？
stem
Q22
Two cubical dice each have removable numbers 1 through 6. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probability that the sum is 7?
两个立方体骰子各有可移除的1至6数字。两个骰子上的十二个数字被移除，放入袋中，然后逐一随机抽出并重新贴回骰子面上，每个面一个数字。然后掷骰子，两个顶面数字相加。和为7的概率是多少？
Q23
Convex quadrilateral ABCD has AB = 9 and CD = 12. Diagonals AC and BD intersect at E, AC = 14, and ΔAED and ΔBEC have equal areas. What is AE?
凸四边形ABCD有$AB=9$且$CD=12$。对角线AC和BD交于E，$AC=14$，且$\triangle AED$与$\triangle BEC$面积相等。$AE$是多少？
Q24
Three distinct vertices of a cube are chosen at random. What is the probability that the plane determined by these three vertices contains points inside the cube?
从立方体的三个不同顶点中随机选择。确定的平面包含立方体内点的概率是多少？
Q25
For $k > 0$, let $I_k = 10\dots064$, where there are $k$ zeros between the 1 and the 6. Let $N(k)$ be the number of factors of 2 in the prime factorization of $I_k$. What is the maximum value of $N(k)$?
对于$k>0$，令$I_k=10\dots064$，其中1和6之间有$k$个零。$N(k)$是$I_k$质因数分解中因子2的个数。$N(k)$的最大值是多少？
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