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AMC12 2017 B

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AMC12 · 2017 (B)

Q1
Kymbrea's comic book collection currently has 30 comic books in it, and she is adding to her collection at the rate of 2 comic books per month. LaShawn's collection currently has 10 comic books in it, and he is adding to his collection at the rate of 6 comic books per month. After how many months will LaShawn's collection have twice as many comic books as Kymbrea's?
Kymbrea 的漫画书收藏目前有 30 本漫画书，她每月以 2 本漫画书的速率增加收藏。LaShawn 的收藏目前有 10 本漫画书，他每月以 6 本漫画书的速率增加收藏。几个月后 LaShawn 的收藏将有 Kymbrea 收藏的两倍那么多？
Q2
Real numbers $x, y,$ and $z$ satisfy the inequalities $0 < x < 1, \quad -1 < y < 0, \quad \text{and} \quad 1 < z < 2$. Which of the following numbers is necessarily positive?
实数 $x, y,$ 和 $z$ 满足不等式 $0 < x < 1, \quad -1 < y < 0, \quad \text{and} \quad 1 < z < 2$。以下哪个数一定是正的？
Q3
Suppose that $x$ and $y$ are nonzero real numbers such that $\frac{3x + y}{x - 3y} = -2$. What is the value of $\frac{x + 3y}{3x - y}$?
假设 $x$ 和 $y$ 是非零实数，使得 $\frac{3x + y}{x - 3y} = -2$。求 $\frac{x + 3y}{3x - y}$ 的值？
Q4
Samia set off on her bicycle to visit her friend, traveling at an average speed of 17 kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at 5 kilometers per hour. In all it took her 44 minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?
Samia 骑自行车去朋友家，平均速度为 17 公里/小时。当她走了一半距离时，轮胎爆了，她以 5 公里/小时的速度步行剩余路程。总共用了 44 分钟到达朋友家。四舍五入到十分位，Samia 步行了多少公里？
Q5
The data set $[6, 19, 33, 33, 39, 41, 41, 43, 51, 57]$ has median $Q_2 = 40$, first quartile $Q_1 = 33$, and third quartile $Q_3 = 43$. An outlier in a data set is a value that is more than 1.5 times the interquartile range below the first quartile ($Q_1$) or more than 1.5 times the interquartile range above the third quartile ($Q_3$), where the interquartile range is defined as $Q_3 - Q_1$. How many outliers does this data set have?
数据集 $[6, 19, 33, 33, 39, 41, 41, 43, 51, 57]$ 的中位数 $Q_2 = 40$，第一四分位数 $Q_1 = 33$，第三四分位数 $Q_3 = 43$。数据集中离群值是低于第一四分位数 ($Q_1$) 1.5 倍四分位距的值，或高于第三四分位数 ($Q_3$) 1.5 倍四分位距的值，其中四分位距定义为 $Q_3 - Q_1$。这个数据集有多少个离群值？
Q6
The circle having (0, 0) and (8, 6) as the endpoints of a diameter intersects the x-axis at a second point. What is the x-coordinate of this point?
以点 (0, 0) 和 (8, 6) 为直径端点的圆与 x 轴相交于另一个点。这个点的 x 坐标是多少？
Q7
The functions $\sin(x)$ and $\cos(x)$ are periodic with least period $2\pi$. What is the least period of the function $\cos(\sin(x))$?
函数 $\sin(x)$ 和 $\cos(x)$ 的最小周期为 $2\pi$。函数 $\cos(\sin(x))$ 的最小周期是多少？
Q8
The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle?
某个矩形的短边与长边的比等于长边与对角线的比。这个矩形短边与长边的比的平方是多少？
Q9
A circle has center (−10, −4) and radius 13. Another circle has center (3, 9) and radius $\sqrt{65}$. The line passing through the two points of intersection of the two circles has equation $x + y = c$. What is $c$?
一个圆的圆心为 (−10, −4)，半径 13。另一个圆的圆心为 (3, 9)，半径 $\sqrt{65}$。通过两个圆交点连线方程为 $x + y = c$。$c$ 是多少？
Q10
At Typico High School, 60% of the students like dancing, and the rest dislike it. Of those who like dancing, 80% say that they like it, and the rest say that they dislike it. Of those who dislike dancing, 90% say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it?
在 Typico 高中，60% 的学生喜欢跳舞，其余的不喜欢。喜欢跳舞的学生中，80% 说他们喜欢，其余说不喜欢。不喜欢跳舞的学生中，90% 说他们不喜欢，其余说喜欢。说不喜欢跳舞的学生中，实际喜欢跳舞的学生分数是多少？
Q11
Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, 3, 23578, and 987620 are monotonous, but 88, 7434, and 23557 are not. How many monotonous positive integers are there?
称一个正整数为单调的，如果它是一位数，或者其数字从左到右阅读时形成严格递增或严格递减的序列。例如，3、23578 和 987620 是单调的，但 88、7434 和 23557 不是。有多少个单调正整数？
Q12
What is the sum of the roots of $z^{12} = 64$ that have a positive real part?
$z^{12} = 64$ 中，具有正实部的根的和是多少？
Q13
In the figure below, 3 of the 6 disks are to be painted blue, 2 are to be painted red, and 1 is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?
在下面的图形中，要将 6 个圆盘中的 3 个涂成蓝色，2 个涂成红色，1 个涂成绿色。通过整个图形的旋转或反射可以相互得到的两种涂法视为相同。有多少种不同的涂法可能？
stem
Q14
An ice-cream novelty item consists of a cup in the shape of a 4-inch-tall frustum of a right circular cone, with a 2-inch-diameter base at the bottom and a 4-inch-diameter base at the top, packed solid with ice cream, together with a solid cone of ice cream of height 4 inches, whose base, at the bottom, is the top base of the frustum. What is the total volume of the ice cream, in cubic inches?
一个冰激凌新奇物品由一个高 4 英寸的圆锥台形状杯子组成，底部基底直径 2 英寸，顶部基底直径 4 英寸，里面塞满冰激凌，再加上一个高 4 英寸的实心冰激凌圆锥，其底部基底即圆锥台的顶部基底。冰激凌总体积是多少立方英寸？
Q15
Let ABC be an equilateral triangle. Extend side AB beyond B to a point B' so that BB' = 3AB. Similarly, extend side BC beyond C to a point C' so that CC' = 3BC, and extend side CA beyond A to a point A' so that AA' = 3CA. What is the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$?
设 ABC 为正三角形。将边 AB 在 B 外延至点 B'，使 BB' = 3AB。类似地，将边 BC 在 C 外延至点 C'，使 CC' = 3BC，将边 CA 在 A 外延至点 A'，使 AA' = 3CA。$ riangle A'B'C'$ 与 $ riangle ABC$ 的面积比是多少？
Q16
The number $21! = 51{,}090{,}942{,}171{,}709{,}440{,}000$ has over $60{,}000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd?
数 $21! = 51{,}090{,}942{,}171{,}709{,}440{,}000$ 有超过 $60{,}000$ 个正整数除数。其中一个被随机选中。它是奇数的概率是多少？
Q17
A coin is biased in such a way that on each toss the probability of heads is $\frac{2}{3}$ and the probability of tails is $\frac{1}{3}$. The outcomes of the tosses are independent. A player has the choice of playing Game A or Game B. In Game A she tosses the coin three times and wins if all three outcomes are the same. In Game B she tosses the coin four times and wins if both the outcomes of the first and second tosses are the same and the outcomes of the third and fourth tosses are the same. How do the chances of winning Game A compare to the chances of winning Game B?
一枚硬币是偏倚的，每次抛掷正面概率为 $\frac{2}{3}$，反面概率为 $\frac{1}{3}$。抛掷结果相互独立。玩家可以选择玩游戏 A 或游戏 B。在游戏 A 中，她抛三次硬币，如果三次结果都相同则获胜。在游戏 B 中，她抛四次硬币，如果第一次和第二次结果相同且第三次和第四次结果相同则获胜。游戏 A 的获胜机会与游戏 B 的获胜机会相比如何？
Q18
The diameter AB of a circle of radius 2 is extended to a point D outside the circle so that BD = 3. Point E is chosen so that ED = 5 and line ED is perpendicular to line AD. Segment AE intersects the circle at a point C between A and E. What is the area of $\triangle ABC$?
半径为 2 的圆的直径 AB 被延长到圆外一点 D，使得 BD = 3。选择点 E 使得 ED = 5 且直线 ED 垂直于直线 AD。线段 AE 与圆相交于 A 和 E 之间的点 C。求 $\triangle ABC$ 的面积。
Q19
Let $N = 123456789101112 \dots 4344$ be the 79-digit number that is formed by writing the integers from 1 to 44 in order, one after the other. What is the remainder when $N$ is divided by 45?
令 $N = 123456789101112 \dots 4344$ 为由 1 到 44 的整数依次写成的 79 位数。$N$ 除以 45 的余数是多少？
Q20
Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $(0, 1)$. What is the probability that $\lfloor \log_2 x \rfloor = \lfloor \log_2 y \rfloor$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to the real number $r$?
实数 $x$ 和 $y$ 从区间 $(0, 1)$ 中独立均匀随机选择。求 $\lfloor \log_2 x \rfloor = \lfloor \log_2 y \rfloor$ 的概率，其中 $\lfloor r \rfloor$ 表示不大于实数 $r$ 的最大整数。
Q21
Last year Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was 95. What was her score on the sixth test?
去年Isabella参加了7次数学测验，得到了7个不同的分数，每个分数都是91到100之间的整数。每次测验后，她注意到她的测验平均分都是整数。她的第七次测验分数是95。她的第六次测验分数是多少？
Q22
Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four rounds. In each round, four balls are placed in an urn—one green, one red, and two white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one coin to whoever gets the red ball. What is the probability that, at the end of the fourth round, each of the players has four coins?
Abby、Bernardo、Carl和Debra玩一个游戏，每人起始有4个硬币。游戏有4轮。每轮中，将4个球放入一个瓮中——一个绿色、一个红色、两个白色。玩家依次无放回抽取球。抽到绿色球的人给抽到红色球的人一个硬币。第四轮结束后，每位玩家都有4个硬币的概率是多少？
Q23
The graph of $y = f(x)$, where $f(x)$ is a polynomial of degree 3, contains points $A(2, 4)$, $B(3, 9)$, and $C(4, 16)$. Lines $AB$, $AC$, and $BC$ intersect the graph again at points $D$, $E$, and $F$, respectively, and the sum of the x-coordinates of $D$, $E$, and $F$ is 24. What is $f(0)$?
$y = f(x)$的图像，其中$f(x)$是三次多项式，包含点$A(2, 4)$、$B(3, 9)$和$C(4, 16)$。直线$AB$、$AC$和$BC$分别再次与图像相交于点$D$、$E$和$F$，且$D$、$E$、$F$的$x$坐标之和为24。求$f(0)$？
Q24
Quadrilateral ABCD has right angles at B and C, $\triangle ABC \sim \triangle BCD$, and AB > BC. There is a point E in the interior of ABCD such that $\triangle ABC \sim \triangle CEB$ and the area of $\triangle AED$ is 17 times the area of $\triangle CEB$. What is $\frac{AB}{BC}$?
四边形ABCD在B和C处有直角，$\triangle ABC \sim \triangle BCD$，且AB > BC。存在点E在ABCD内部，使得$\triangle ABC \sim \triangle CEB$，且$\triangle AED$的面积是$\triangle CEB$面积的17倍。求$\frac{AB}{BC}$？
Q25
A set of n people participate in an online video basketball tournament. Each person may be a member of any number of 5-player teams, but no two teams may have exactly the same 5 members. The site statistics show a curious fact: The average, over all subsets of size 9 of the set of n participants, of the number of complete teams whose members are among those 9 people is equal to the reciprocal of the average, over all subsets of size 8 of the set of n participants, of the number of complete teams whose members are among those 8 people. How many values n, $9 \leq n \leq 2017$, can be the number of participants?
有n个人参加在线视频篮球锦标赛。每人可加入任意多个5人团队，但无两个团队有完全相同的5名成员。网站统计显示一个奇特事实：对n名参与者所有大小为9的子集，完整团队（成员都在这9人中）的数量的平均值，等于对所有大小为8的子集的相同平均值的倒数。有多少个n（$9 \leq n \leq 2017$）可以是参与者人数？
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