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AMC12 2012 A

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AMC12 · 2012 (A)

Q1
A bug crawls along a number line, starting at $-2$. It crawls to $-6$, then turns around and crawls to $5$. How many units does the bug crawl altogether?
一只虫子沿着数轴爬行，从 $-2$ 开始。它爬到 $-6$，然后转身爬到 $5$。虫子总共爬了多少单位距离？
Q2
Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes?
Cagney 每 20 秒可以给一个纸杯蛋糕上糖霜，Lacey 每 30 秒可以给一个纸杯蛋糕上糖霜。他们一起工作，5 分钟内可以给多少个纸杯蛋糕上糖霜？
Q3
A box $2$ centimeters high, $3$ centimeters wide, and $5$ centimeters long can hold $40$ grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold $n$ grams of clay. What is $n$?
一个高 $2$ 厘米、宽 $3$ 厘米、长 $5$ 厘米的盒子可以装 $40$ 克粘土。第二个盒子的高度是第一个的两倍，宽度是三倍，长度与第一个盒子相同，可以装 $n$ 克粘土。$n$ 是多少？
Q4
In a bag of marbles, $\frac{3}{5}$ of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red?
一袋弹珠中，$\frac{3}{5}$ 的弹珠是蓝色的，其余是红色的。如果红色弹珠的数量加倍而蓝色弹珠的数量保持不变，那么红色弹珠将占弹珠总数的几分之几？
Q5
A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of $280$ pieces of fruit. There are twice as many raspberries as blueberries, three times as many grapes as cherries, and four times as many cherries as raspberries. How many cherries are there in the fruit salad?
一份水果沙拉由蓝莓、覆盆子、葡萄和樱桃组成。这份水果沙拉共有 $280$ 块水果。覆盆子的数量是蓝莓的两倍，葡萄的数量是樱桃的三倍，樱桃的数量是覆盆子的四倍。水果沙拉中有多少块樱桃？
Q6
The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?
三个整数两两相加的和分别为 12、17 和 19。中间的那个数是多少？
Q7
Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?
玛丽将一个圆分成12个扇形。这些扇形的圆心角（以度为单位）都是整数，并且形成一个等差数列。最小的扇形角度数可能是多少？
Q8
An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?
对数字 1、2、3、4 和 5 进行迭代平均，方法如下：将这五个数按某种顺序排列。先求前两个数的平均数，再将该平均数与第三个数求平均，再将所得结果与第四个数求平均，最后将所得结果与第五个数求平均。用此过程可能得到的最大值与最小值之差是多少？
Q9
A year is a leap year if and only if the year number is divisible by 400 (such as 2000) or is divisible by 4 but not 100 (such as 2012). The 200th anniversary of the birth of novelist Charles Dickens was celebrated on February 7, 2012, a Tuesday. On what day of the week was Dickens born?
一年是闰年当且仅当该年份能被 400 整除（如 2000 年），或能被 4 整除但不能被 100 整除（如 2012 年）。小说家查尔斯·狄更斯诞辰 200 周年纪念日是 2012 年 2 月 7 日（星期二）。狄更斯出生那天是星期几？
Q10
A triangle has area $30$, one side of length $10$, and the median to that side of length $9$. Let $\theta$ be the acute angle formed by that side and the median. What is $\sin{\theta}$?
一个三角形的面积为 $30$，有一边长为 $10$，且到该边的中线长为 $9$。设 $\theta$ 为该边与中线所成的锐角。求 $\sin{\theta}$。
Q11
Alex, Mel, and Chelsea play a game that has $6$ rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For each round the probability that Alex wins is $\frac{1}{2}$, and Mel is twice as likely to win as Chelsea. What is the probability that Alex wins three rounds, Mel wins two rounds, and Chelsea wins one round?
Alex、Mel 和 Chelsea 玩一个有 $6$ 轮的游戏。每轮有一个获胜者，且各轮结果相互独立。每轮 Alex 获胜的概率为 $\frac{1}{2}$，并且 Mel 获胜的可能性是 Chelsea 的两倍。求 Alex 赢 $3$ 轮、Mel 赢 $2$ 轮、Chelsea 赢 $1$ 轮的概率。
Q12
A square region $ABCD$ is externally tangent to the circle with equation $x^2+y^2=1$ at the point $(0,1)$ on the side $CD$. Vertices $A$ and $B$ are on the circle with equation $x^2+y^2=4$. What is the side length of this square?
正方形区域 $ABCD$ 在边 $CD$ 上的点 $(0,1)$ 处与圆 $x^2+y^2=1$ 外切。顶点 $A$ 和 $B$ 在圆 $x^2+y^2=4$ 上。求该正方形的边长。
Q13
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two helpers painted only 24% of the house and quit at 2:12 PM. On Wednesday Paula worked by herself and finished the house by working until 7:12 P.M. How long, in minutes, was each day's lunch break?
画家 Paula 和她的两个助手各自以恒定但不同的速率作画。他们总是在上午 8:00 开始，而且三个人每天吃午饭所用的时间都相同。周一三人画完了一栋房子的 50%，并在下午 4:00 停工。周二 Paula 不在时，两位助手只画了房子的 24%，并在下午 2:12 停工。周三 Paula 独自工作，一直干到晚上 7:12 才完成整栋房子。问每天的午饭休息时间是多少分钟？
Q14
The closed curve in the figure is made up of 9 congruent circular arcs each of length $\frac{2\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side 2. What is the area enclosed by the curve?
图中的闭合曲线由 9 段全等的圆弧组成，每段圆弧的长度为 $\frac{2\pi}{3}$，且这些圆弧所对应圆的圆心都位于边长为 2 的正六边形的顶点中。求该曲线所围成的面积。
stem
Q15
A $3 \times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90\,^{\circ}$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability the grid is now entirely black?
一个 $3 \times 3$ 的正方形被分割成 $9$ 个单位正方形。每个单位正方形被涂成白色或黑色，两种颜色等可能，且彼此独立、随机选择。然后将整个正方形绕其中心顺时针旋转 $90\,^{\circ}$，并把所有处在“原先由黑色正方形占据的位置”上的白色正方形涂成黑色。其余正方形颜色保持不变。问此时整个网格全为黑色的概率是多少？
Q16
Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. What is the radius of circle $C_1$?
圆 $C_1$ 的圆心 $O$ 位于圆 $C_2$ 上。两圆相交于点 $X$ 和 $Y$。点 $Z$ 在 $C_1$ 的外部，且位于圆 $C_2$ 上，并且 $XZ=13$，$OZ=11$，$YZ=7$。圆 $C_1$ 的半径是多少？
Q17
Let $S$ be a subset of $\{1,2,3,\dots,30\}$ with the property that no pair of distinct elements in $S$ has a sum divisible by $5$. What is the largest possible size of $S$?
设 $S$ 是集合 $\{1,2,3,\dots,30\}$ 的子集，且 $S$ 中任意一对不同元素之和都不能被 $5$ 整除。$S$ 的最大可能大小是多少？
Q18
Triangle $ABC$ has $AB=27$, $AC=26$, and $BC=25$. Let $I$ be the intersection of the internal angle bisectors of $\triangle ABC$. What is $BI$?
三角形 $ABC$ 满足 $AB=27$，$AC=26$，$BC=25$。设 $I$ 为 $\triangle ABC$ 的内部角平分线的交点。求 $BI$。
Q19
Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?
Adam、Benin、Chiang、Deshawn、Esther 和 Fiona 有互联网账户。他们之中有些（但不是全部）彼此是互联网朋友，并且他们都没有这个组外的互联网朋友。他们每个人拥有相同数量的互联网朋友。这种情况可以有多少种不同的方式发生？
Q20
Consider the polynomial \[P(x)=\prod_{k=0}^{10}(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots (x^{1024}+1024)\] The coefficient of $x^{2012}$ is equal to $2^a$. What is $a$? \[
考虑多项式 \[P(x)=\prod_{k=0}^{10}(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots (x^{1024}+1024)\] $x^{2012}$ 的系数等于 $2^a$。$a$ 是多少？ \[
Q21
Let $a$, $b$, and $c$ be positive integers with $a\ge$ $b\ge$ $c$ such that $a^2-b^2-c^2+ab=2011$ and $a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$. What is $a$?
设 $a$、$b$ 和 $c$ 是正整数且 $a\ge$ $b\ge$ $c$，使得 $a^2-b^2-c^2+ab=2011$ 且 $a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$。 求 $a$。
Q22
Distinct planes $p_1,p_2,....,p_k$ intersect the interior of a cube $Q$. Let $S$ be the union of the faces of $Q$ and let $P =\bigcup_{j=1}^{k}p_{j}$. The intersection of $P$ and $S$ consists of the union of all segments joining the midpoints of every pair of edges belonging to the same face of $Q$. What is the difference between the maximum and minimum possible values of $k$?
不同的平面 $p_1,p_2,....,p_k$ 与立方体 $Q$ 的内部相交。设 $S$ 为 $Q$ 的各个面的并集，且 $P =\bigcup_{j=1}^{k}p_{j}$。$P$ 与 $S$ 的交集由所有线段的并集组成，这些线段连接 $Q$ 的同一面上任意一对边的中点。$k$ 的最大可能值与最小可能值之差是多少？
Q23
Let $S$ be the square one of whose diagonals has endpoints $(1/10,7/10)$ and $(-1/10,-7/10)$. A point $v=(x,y)$ is chosen uniformly at random over all pairs of real numbers $x$ and $y$ such that $0 \le x \le 2012$ and $0\le y\le 2012$. Let $T(v)$ be a translated copy of $S$ centered at $v$. What is the probability that the square region determined by $T(v)$ contains exactly two points with integer coefficients in its interior?
设 $S$ 是一个正方形，其一条对角线的端点为 $(1/10,7/10)$ 和 $(-1/10,-7/10)$。点 $v=(x,y)$ 在所有满足 $0 \le x \le 2012$ 且 $0\le y\le 2012$ 的实数对 $(x,y)$ 中均匀随机选取。设 $T(v)$ 为以 $v$ 为中心的 $S$ 的平移拷贝。由 $T(v)$ 确定的正方形区域在其内部恰好包含两个整数坐标点的概率是多少？
Q24
Let $\{a_k\}_{k=1}^{2011}$ be the sequence of real numbers defined by $a_1=0.201,$ $a_2=(0.2011)^{a_1},$ $a_3=(0.20101)^{a_2},$ $a_4=(0.201011)^{a_3}$, and in general, \[a_k=\begin{cases}(0.\underbrace{20101\cdots 0101}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is odd,}\\(0.\underbrace{20101\cdots 01011}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is even.}\end{cases}\] Rearranging the numbers in the sequence $\{a_k\}_{k=1}^{2011}$ in decreasing order produces a new sequence $\{b_k\}_{k=1}^{2011}$. What is the sum of all integers $k$, $1\le k \le 2011$, such that $a_k=b_k?$
设 $\{a_k\}_{k=1}^{2011}$ 为实数序列，定义为 $a_1=0.201,$ $a_2=(0.2011)^{a_1},$ $a_3=(0.20101)^{a_2},$ $a_4=(0.201011)^{a_3}$，一般地， \[a_k=\begin{cases}(0.\underbrace{20101\cdots 0101}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is odd,}\\(0.\underbrace{20101\cdots 01011}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is even.}\end{cases}\] 将序列 $\{a_k\}_{k=1}^{2011}$ 中的数按降序重排得到新序列 $\{b_k\}_{k=1}^{2011}$。所有满足 $a_k=b_k$ 的整数 $k$（$1\le k \le 2011$）之和是多少？
Q25
Let $f(x)=|2\{x\}-1|$ where $\{x\}$ denotes the fractional part of $x$. The number $n$ is the smallest positive integer such that the equation \[nf(xf(x))=x\] has at least $2012$ real solutions. What is $n$? Note: the fractional part of $x$ is a real number $y=\{x\}$ such that $0\le y<1$ and $x-y$ is an integer.
设 $f(x)=|2\{x\}-1|$，其中 $\{x\}$ 表示 $x$ 的小数部分。$n$ 是使得方程 \[nf(xf(x))=x\] 至少有 $2012$ 个实数解的最小正整数。求 $n$。注：$x$ 的小数部分是实数 $y=\{x\}$，满足 $0\le y<1$ 且 $x-y$ 是整数。
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