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AMC12 2011 B

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AMC12 · 2011 (B)

Q1
What is $\frac{2+4+6}{1+3+5} - \frac{1+3+5}{2+4+6}?$
$\frac{2+4+6}{1+3+5} - \frac{1+3+5}{2+4+6}$ 的值是多少？
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 到目前为止的测试分数是 $90, 80, 70, 60,$ 和 $85$。她的目标是用下一次测试将平均分提高至少 $3$ 分。她需要的最低测试分数是多少？
Q3
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。在这一周中，他们各自支付了各种共同开销，如汽油费和租车费。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 多少美元才能使他们平分费用？
Q4
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 $b$?
在将两个正整数 $a$ 和 $b$ 相乘时，Ron 把两位数 $a$ 的数字顺序颠倒了。他得到的错误乘积是 $161$。$a$ 与 $b$ 的正确乘积是多少？
Q5
Let $N$ be the second smallest positive integer that is divisible by every positive integer less than $7$. What is the sum of the digits of $N$?
设 $N$ 是能被所有小于 $7$ 的正整数整除的第二小的正整数。$N$ 的各位数字之和是多少？
Q6
Two tangents to a circle are drawn from a point $A$. The points of contact $B$ and $C$ divide the circle into arcs with lengths in the ratio $2 : 3$. What is the degree measure of $\angle{BAC}$?
从一点 $A$ 引出两条圆的切线。切点 $B$ 和 $C$ 将圆分成弧，其长度比为 $2:3$。$\angle{BAC}$ 的度量是多少度？
Q7
Let $x$ and $y$ be two-digit positive integers with mean $60$. What is the maximum value of the ratio $\frac{x}{y}$?
设 $x$ 和 $y$ 是两位正整数，其平均数为 $60$。比值 $\frac{x}{y}$ 的最大值是多少？
Q8
Keiko walks once around a track at the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of $6$ 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 每天以相同的恒定速度绕跑道走一圈。跑道两侧为直线，两端为半圆。跑道宽度为 $6$ 米，她绕外侧边缘走一圈比绕内侧边缘多花 $36$ 秒。Keiko 的速度是多少米每秒？
stem
Q9
Two real numbers are selected independently and 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。这两个数的乘积大于零的概率是多少？
Q10
Rectangle $ABCD$ has $AB=6$ and $BC=3$. Point $M$ is chosen on side $AB$ so that $\angle AMD=\angle CMD$. What is the degree measure of $\angle AMD$?
矩形 $ABCD$ 有 $AB=6$ 且 $BC=3$。在边 $AB$ 上选择点 $M$，使得 $\angle AMD=\angle CMD$。$\angle AMD$ 的度量是多少度？
Q11
A frog located at $(x,y)$, with both $x$ and $y$ integers, makes successive jumps of length $5$ and always lands on points with integer coordinates. Suppose that the frog starts at $(0,0)$ and ends at $(1,0)$. What is the smallest possible number of jumps the frog makes?
一只青蛙位于 $(x,y)$，其中 $x$ 和 $y$ 都是整数。它连续进行长度为 $5$ 的跳跃，并且总是落在整数坐标点上。假设青蛙从 $(0,0)$ 出发，最终到达 $(1,0)$。青蛙最少需要跳跃多少次？
Q12
A dart board is a regular octagon divided into regions as shown below. 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?
一个飞镖盘是如下图所示的正八边形，并被分成若干区域。假设飞镖投向飞镖盘时，落在盘内任意位置的可能性相同。飞镖落在中心正方形内的概率是多少？
stem
Q13
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 of $w$?
Brian 写下四个整数 $w > x > y > z$，它们的和为 $44$。这四个数两两之间的正差分别为 $1, 3, 4, 5, 6$ 和 $9$。所有可能的 $w$ 的取值之和是多少？
Q14
A segment through the focus $F$ of a parabola with vertex $V$ is perpendicular to $\overline{FV}$ and intersects the parabola in points $A$ and $B$. What is $\cos\left(\angle AVB\right)$?
一条经过抛物线焦点 $F$ 的线段与 $\overline{FV}$ 垂直（$V$ 为顶点），并与抛物线交于点 $A$ 和 $B$。求 $\cos\left(\angle AVB\right)$。
Q15
How many positive two-digit integers are factors of $2^{24}-1$?
$2^{24}-1$ 有多少个正的两位整数因数？
Q16
Rhombus $ABCD$ has side length $2$ and $\angle B = 120$°. 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 $ABCD$ 的边长为 $2$，且 $\angle B = 120$°。区域 $R$ 由菱形内部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？
Q17
Let $f(x) = 10^{10x}, g(x) = \log_{10}\left(\frac{x}{10}\right), h_1(x) = g(f(x))$, and $h_n(x) = h_1(h_{n-1}(x))$ for integers $n \geq 2$. What is the sum of the digits of $h_{2011}(1)$?
设 $f(x) = 10^{10x}, g(x) = \log_{10}\left(\frac{x}{10}\right), h_1(x) = g(f(x))$，且对整数 $n \geq 2$，$h_n(x) = h_1(h_{n-1}(x))$。$h_{2011}(1)$ 的各位数字之和是多少？
Q18
A pyramid has a square base with side 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，且侧面为等边三角形。在金字塔内放置一个立方体，使其中一个面在金字塔底面上，其对面的所有边都在金字塔的侧面上。求此立方体的体积。
Q19
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 $0 < x \leq 100$ for all $m$ such that $\frac{1}{2} < m < a$. What is the maximum possible value of $a$?
在 $xy$ 坐标系中，晶格点是指 $(x, y)$ 中 $x$ 和 $y$ 都为整数的点。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？
Q20
Triangle $ABC$ has $AB = 13, BC = 14$, and $AC = 15$. The points $D, E$, and $F$ are the midpoints of $\overline{AB}, \overline{BC}$, and $\overline{AC}$ respectively. Let $X \neq E$ be the intersection of the circumcircles of $\triangle BDE$ and $\triangle CEF$. What is $XA + XB + XC$?
三角形 $ABC$ 有 $AB = 13, BC = 14$，且 $AC = 15$。点 $D, E$, 和 $F$ 分别为 $\overline{AB}, \overline{BC}$, 和 $\overline{AC}$ 的中点。设 $X \neq E$ 为 $\triangle BDE$ 和 $\triangle CEF$ 的外接圆交点。$XA + XB + XC$ 是多少？
Q21
The arithmetic mean of two distinct positive integers $x$ and $y$ is a two-digit integer. The geometric mean of $x$ and $y$ is obtained by reversing the digits of the arithmetic mean. What is $|x - y|$?
两个不同的正整数 $x$ 和 $y$ 的算术平均数是一个两位整数。$x$ 和 $y$ 的几何平均数是通过反转算术平均数的数字得到的。求 $|x - y|$？
Q22
Let $T_1$ be a triangle with side lengths $2011$, $2012$, and $2013$. For $n \geq 1$, 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 $\left(T_n\right)$?
设 $T_1$ 是一个边长为 $2011$, $2012$, $2013$ 的三角形。对 $n \geq 1$，若 $T_n = \triangle ABC$，且 $D, E, F$ 分别是 $\triangle ABC$ 的内切圆与边 $AB$, $BC$, $AC$ 的切点，则（若存在）$T_{n+1}$ 是一边长为 $AD, BE, CF$ 的三角形。序列 $\left(T_n\right)$ 中最后一个三角形的周长是多少？
Q23
A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$-axis or $y$-axis. Let $A = (-3, 2)$ and $B = (3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most $20$. How many points with integer coordinates lie on at least one of these paths?
一只虫子在坐标平面中移动，只能沿与 $x$ 轴或 $y$ 轴平行的直线移动。设 $A = (-3, 2)$，$B = (3, -2)$。考虑从 $A$ 到 $B$ 的所有长度不超过 $20$ 的可能路径。至少位于其中一条路径上的整数坐标点共有多少个？
Q24
Let $P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)$. What is the minimum perimeter among all the $8$-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$?
设 $P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)$。在复平面中，所有顶点恰为 $P(z)$ 的零点的 8 边形中，最小周长是多少？
Q25
For every $m$ and $k$ integers with $k$ odd, denote by $\left[\frac{m}{k}\right]$ the integer closest to $\frac{m}{k}$. For every odd integer $k$, let $P(k)$ be the probability that \[\left[\frac{n}{k}\right] + \left[\frac{100 - n}{k}\right] = \left[\frac{100}{k}\right]\] for an integer $n$ randomly chosen from the interval $1 \leq n \leq 99!$. What is the minimum possible value of $P(k)$ over the odd integers $k$ in the interval $1 \leq k \leq 99$?
对任意整数 $m$ 和 $k$（其中 $k$ 为奇数），用 $\left[\frac{m}{k}\right]$ 表示最接近 $\frac{m}{k}$ 的整数。对每个奇整数 $k$，令 $P(k)$ 为如下事件发生的概率： \[\left[\frac{n}{k}\right] + \left[\frac{100 - n}{k}\right] = \left[\frac{100}{k}\right]\] 其中整数 $n$ 从区间 $1 \leq n \leq 99!$ 中随机选取。对区间 $1 \leq k \leq 99$ 内的奇整数 $k$，$P(k)$ 的最小可能值是多少？
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