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AMC10 2011 B

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AMC10 · 2011 (B)

Q1
What is $\frac{2 + 4 + 6}{1 + 3 + 5 - 1 + 3 + 5} - \frac{2 + 4 + 6}{?}$
什么是 $\frac{2 + 4 + 6}{1 + 3 + 5 - 1 + 3 + 5} - \frac{2 + 4 + 6}{?}$
Q2
Josanna’s test scores to date are 90, 80, 70, 60, and 85. Her goal is to raise her test average at least 3 points with her next test. What is the minimum test score she would need to accomplish this goal?
Josanna 目前的测试分数是 90、80、70、60 和 85。她的目标是用下一次测试将平均分至少提高 3 分。她需要的最低测试分数是多少？
Q3
At a store, when a length is reported as x inches that means the length is at least x − 0.5 inches and at most x + 0.5 inches. Suppose the dimensions of a rectangular tile are reported as 2 inches by 3 inches. In square inches, what is the minimum area for the rectangle?
在一家商店，当长度报告为 x 英寸时，意味着长度至少为 x − 0.5 英寸，至多为 x + 0.5 英寸。假设一块矩形瓷砖的尺寸报告为 2 英寸乘 3 英寸。矩形的最小面积是多少平方英寸？
Q4
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid A dollars and Bernardo had paid B dollars, where A < B. How many dollars must LeRoy give to Bernardo so that they share the costs equally?
LeRoy 和 Bernardo 一起进行了一次为期一周的旅行，并同意平分费用。在一周内，他们各自支付了各种共同费用，如汽油和租车费用。旅行结束时，LeRoy 支付了 A 美元，Bernardo 支付了 B 美元，其中 A < B。LeRoy 必须给 Bernardo 多少美元才能平分费用？
Q5
In multiplying two positive integers a and b, Ron reversed the digits of the two-digit number a. His erroneous product was 161. What is the correct value of the product of a and b ?
在乘两个正整数 a 和 b 时，Ron 把两位数 a 的数字颠倒了。他的错误积是 161。a 和 b 的正确积是多少？
Q6
On Halloween Casper ate $\frac{1}{3}$ of his candies and then gave 2 candies to his brother. The next day he ate $\frac{1}{3}$ of his remaining candies and then gave 4 candies to his sister. On the third day he ate his final 8 candies. How many candies did Casper have at the beginning?
在万圣节，Casper 吃了他的糖果的 $\frac{1}{3}$，然后给了兄弟 2 颗糖果。第二天，他吃了剩余糖果的 $\frac{1}{3}$，然后给了姐姐 4 颗糖果。第三天，他吃掉了最后的 8 颗糖果。Casper 一开始有多少糖果？
Q7
The sum of two angles of a triangle is $\frac{6}{5}$ of a right angle, and one of these two angles is 30° larger than the other. What is the degree measure of the largest angle in the triangle?
一个三角形的两个角之和是直角的 $\frac{6}{5}$，这两个角之一比另一个大 30°。该三角形最大角的度数是多少？
Q8
At a certain beach if it is at least 80°F and sunny, then the beach will be crowded. On June 10 the beach was not crowded. What can be concluded about the weather conditions on June 10?
在某个海滩，如果温度至少 80°F 且晴天，则海滩会很拥挤。6 月 10 日海滩不拥挤。关于 6 月 10 日的天气条件可以得出什么结论？
Q9
The area of △EBD is one third of the area of 3 – 4 – 5 △ABC. Segment DE is perpendicular to segment AB. What is BD ?
△EBD 的面积是 3 – 4 – 5 △ABC 面积的三分之一。线段 DE 垂直于线段 AB。BD 是多少？
stem
Q10
Consider the set of numbers $\{1, 10, 10^2, 10^3, \dots, 10^{10}\}$. The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer?
考虑集合 $\{1, 10, 10^2, 10^3, \dots, 10^{10}\}$。该集合中最大元素与其它十个元素之和的比值最接近哪个整数？
Q11
There are 52 people in a room. What is the largest value of n such that the statement “At least n people in this room have birthdays falling in the same month” is always true?
房间里有 52 个人。最大的 n 值是多少，使得“房间里至少有 n 个人生日在同一个月”这个陈述总是成立？
Q12
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width 6 meters, and it takes her 36 seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko’s speed in meters per second?
Keiko 每天以完全相同的恒定速度绕跑道走一圈。跑道两侧是直线，两端是半圆。跑道宽度 6 米，她绕外侧边缘走一圈比内侧边缘多花 36 秒。Keiko 的速度是多少米/秒？
stem
Q13
Two real numbers are selected independently at random from the interval [−20, 10]. What is the probability that the product of those numbers is greater than zero?
从区间 [-20, 10] 中独立随机选取两个实数。它们的乘积大于 0 的概率是多少？
Q14
A rectangular parking lot has a diagonal of 25 meters and an area of 168 square meters. In meters, what is the perimeter of the parking lot?
一个矩形停车场对角线长 25 米，面积 168 平方米。停车场的周长是多少米？
Q15
Let @ denote the “averaged with” operation: a @ b = $\frac{a+b}{2} $. Which of the following distributive laws hold for all numbers x, y, and z ? I. x @ (y + z) = (x @ y) + (x @ z) II. x + (y @ z) = (x + y) @ (x + z) III. x @ (y @ z) = (x @ y) @ (x @ z)
设 @ 表示“平均”运算：a @ b = $\frac{a+b}{2}$。以下哪些分配律对所有数 x, y, z 成立？ I. x @ (y + z) = (x @ y) + (x @ z) II. x + (y @ z) = (x + y) @ (x + z) III. x @ (y @ z) = (x @ y) @ (x @ z)
Q16
A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?
飞镖盘是一个正八边形，按照图示划分成区域。假设飞镖投向飞镖盘时，均匀随机落在飞镖盘的任何位置。飞镖落在中心正方形内的概率是多少？
stem
Q17
In the given circle, the diameter EB is parallel to DC, and AB is parallel to ED. The angles AEB and ABE are in the ratio 4 : 5. What is the degree measure of angle BCD ?
在给定的圆中，直径 EB 平行于 DC，且 AB 平行于 ED。角 AEB 和 ABE 的比值为 4 : 5。角 BCD 的度数是多少？
stem
Q18
Rectangle ABCD has AB = 6 and BC = 3. Point M is chosen on side AB so that ∠AMD = ∠CMD. What is the degree measure of ∠AMD ?
矩形 ABCD 有 AB = 6，BC = 3。在边 AB 上选择点 M，使得 ∠AMD = ∠CMD。∠AMD 的度数是多少？
Q19
What is the product of all the roots of the equation $\sqrt{5|x| + 8} = \sqrt{x^2 −16}$.
方程 $\sqrt{5|x| + 8} = \sqrt{x^2 −16}$ 的所有根的乘积是多少。
Q20
Rhombus ABCD has side length 2 and ∠B = 120°. Region R consists of all points inside the rhombus that are closer to vertex B than any of the other three vertices. What is the area of R ?
菱形 ABCD 边长为 2，∠B = 120°。区域 R 由菱形内所有比其他三个顶点更靠近顶点 B 的点组成。R 的面积是多少？
Q21
Brian writes down four integers w > x > y > z whose sum is 44. The pairwise positive differences of these numbers are 1, 3, 4, 5, 6, and 9. What is the sum of the possible values for w ?
Brian 写下四个整数 $w > x > y > z$，它们的和为 44。这些数的成对正差为 1、3、4、5、6 和 9。$w$ 的可能值的和是多少？
Q22
A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
一个金字塔具有边长为 1 的正方形底面，其侧面为等边三角形。在金字塔内放置一个立方体，使一面在金字塔底面上，其相对面所有的边都在金字塔的侧面上。这个立方体的体积是多少？
Q23
What is the hundreds digit of 2011^{2011} ?
$2011^{2011}$ 的百位数字是多少？
Q24
A lattice point in an xy-coordinate system is any point (x, y) where both x and y are integers. The graph of y = mx + 2 passes through no lattice point with 0 < x ≤100 for all m such that $\frac{1}{2}$ < m < a. What is the maximum possible value of a ?
在 $xy$ 坐标系中，晶格点是任意点 $(x, y)$ 其中 $x$ 和 $y$ 均为整数。对于所有满足 $\frac{1}{2} < m < a$ 的 $m$，直线 $y = mx + 2$ 在 $0 < x \leq 100$ 时不经过任何晶格点。$a$ 的最大可能值是多少？
Q25
Let $T_1$ be a triangle with sides $2011$, $2012$, and $2013$. For $n \ge 1$, if $T_n=\triangle ABC$ and $D$, $E$, and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB$, $BC$, and $AC$, respectively, then $T_{n+1}$ is a triangle with side lengths $AD$, $BE$, and $CF$, if it exists. What is the perimeter of the last triangle in the sequence $(T_n)$?
设 $T_1$ 为一边长分别为 $2011$、$2012$、$2013$ 的三角形。对任意 $n \ge 1$，若 $T_n=\triangle ABC$，且 $D,E,F$ 分别为 $\triangle ABC$ 的内切圆与边 $AB,BC,AC$ 的切点，则（若存在）$T_{n+1}$ 定义为一三角形，其三边长分别为 $AD,BE,CF$。求序列 $(T_n)$ 中最后一个三角形的周长。
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