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AMC12 2012 B

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AMC12 · 2012 (B)

Q1
Each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. How many more students than rabbits are there in all $4$ of the third-grade classrooms?
Pearl Creek 小学的每个三年级教室有 $18$ 名学生和 $2$ 只宠物兔子。所有 $4$ 个三年级教室中，学生比兔子多多少？
Q2
A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?
如图，一个半径为 $5$ 的圆内接于一个矩形中。矩形的长与宽之比为 $2:1$。这个矩形的面积是多少？
stem
Q3
For a science project, Sammy observed a chipmunk and squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?
为了一个科学项目，Sammy 观察到一只花栗鼠和一只松鼠把橡子藏在洞里。花栗鼠在它挖的每个洞里藏 $3$ 个橡子。松鼠在它挖的每个洞里藏 $4$ 个橡子。它们藏的橡子总数相同，但松鼠所需的洞比花栗鼠少 $4$ 个。花栗鼠藏了多少个橡子？
Q4
Suppose that the euro is worth 1.3 dollars. If Diana has 500 dollars and Etienne has 400 euros, by what percent is the value of Etienne's money greater that the value of Diana's money?
假设欧元的价值为 1.3 美元。如果 Diana 有 500 美元而 Etienne 有 400 欧元，那么 Etienne 的钱的价值比 Diana 的钱的价值多百分之多少？
Q5
Two integers have a sum of $26$. when two more integers are added to the first two, the sum is $41$. Finally, when two more integers are added to the sum of the previous $4$ integers, the sum is $57$. What is the minimum number of even integers among the $6$ integers?
两个整数的和为 $26$。当再把两个整数加到前两个整数上时，和为 $41$。最后，当再把两个整数加到前面 $4$ 个整数的和上时，和为 $57$。这 $6$ 个整数中偶数的最小个数是多少？
Q6
In order to estimate the value of $x-y$ where $x$ and $y$ are real numbers with $x>y>0$, Xiaoli rounded $x$ up by a small amount, rounded $y$ down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?
为了估计 $x-y$ 的值，其中 $x$ 和 $y$ 是实数且 $x>y>0$，小丽将 $x$ 向上取整一个很小的量，将 $y$ 向下取整相同的量，然后用取整后的值相减。以下哪个陈述一定正确？
Q7
Small lights are hung on a string $6$ inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of $2$ red lights followed by $3$ green lights. How many feet separate the 3rd red light and the 21st red light? Note: $1$ foot is equal to $12$ inches.
小灯挂在一根绳子上，相邻两盏灯相距 $6$ 英寸，按顺序为红、红、绿、绿、绿、红、红、绿、绿、绿，依此类推，持续重复“$2$ 盏红灯后接 $3$ 盏绿灯”的模式。第 $3$ 盏红灯与第 $21$ 盏红灯相隔多少英尺？ 注：$1$ 英尺等于 $12$ 英寸。
Q8
A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?
一位甜点师从星期日开始为一周的每天准备甜点。每天的甜点要么是蛋糕、派、冰淇淋或布丁。相同的甜点不能连续两天供应。由于生日，星期五必须供应蛋糕。一周可能有多少种不同的甜点菜单？
Q9
It takes Clea 60 seconds to walk down an escalator when it is not moving, and 24 seconds when it is moving. How many seconds would it take Clea to ride the escalator down when she is not walking?
当自动扶梯不运行时，Clea 走下去需要 60 秒；当自动扶梯运行时，她走下去需要 24 秒。如果 Clea 不走路，只是站在自动扶梯上乘下去，需要多少秒？
Q10
What is the area of the polygon whose vertices are the points of intersection of the curves $x^2 + y^2 =25$ and $(x-4)^2 + 9y^2 = 81 ?$
由曲线 $x^2 + y^2 =25$ 与 $(x-4)^2 + 9y^2 = 81$ 的交点作为顶点所形成的多边形的面积是多少？
Q11
In the equation below, $A$ and $B$ are consecutive positive integers, and $A$, $B$, and $A+B$ represent number bases: \[132_A+43_B=69_{A+B}.\] What is $A+B$?
在下面的方程中，$A$ 和 $B$ 是连续的正整数，且 $A$、$B$ 和 $A+B$ 表示进制：\[132_A+43_B=69_{A+B}.\] $A+B$ 是多少？
Q12
How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both?
长度为 20 的由 0 和 1 组成的序列中，有多少个序列满足：所有的 0 都是连续的，或所有的 1 都是连续的，或两者皆是？
Q13
Two parabolas have equations $y= x^2 + ax +b$ and $y= x^2 + cx +d$, where $a, b, c,$ and $d$ are integers, each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas will have at least one point in common?
两条抛物线的方程为 $y= x^2 + ax +b$ 和 $y= x^2 + cx +d$，其中 $a, b, c,$ 和 $d$ 为整数，且各自独立地通过掷一枚公平的六面骰子确定。两条抛物线至少有一个公共点的概率是多少？
Q14
Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds $50$ to it and passes the result to Bernardo. The winner is the last person who produces a number less than $1000$. Let $N$ be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of $N$?
Bernardo 和 Silvia 玩如下游戏。从 $0$ 到 $999$（含）之间选取一个整数并给 Bernardo。每当 Bernardo 收到一个数，他将其加倍并把结果传给 Silvia。每当 Silvia 收到一个数，她将其加 $50$ 并把结果传给 Bernardo。最后一个产生小于 $1000$ 的数的人获胜。设 $N$ 为使 Bernardo 获胜的最小初始数。$N$ 的各位数字之和是多少？
Q15
Jesse cuts a circular disk of radius 12, along 2 radii to form 2 sectors, one with a central angle of 120. He makes two circular cones using each sector to form the lateral surface of each cone. What is the ratio of the volume of the smaller cone to the larger cone?
Jesse 将一个半径为 12 的圆盘沿两条半径切开，得到两个扇形，其中一个的圆心角为 120。他用每个扇形分别制作一个圆锥，使扇形成为圆锥的侧面。较小圆锥的体积与较大圆锥的体积之比是多少？
Q16
Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible?
Amy、Beth 和 Jo 听了四首不同的歌曲，并讨论她们喜欢哪些歌。没有一首歌是三人都喜欢的。此外，对于女孩的三对中的每一对，都至少有一首歌是这两女孩喜欢的但第三人不喜欢的。这种情况可能有多少种不同的方式？
Q17
Square $PQRS$ lies in the first quadrant. Points $(3,0), (5,0), (7,0),$ and $(13,0)$ lie on lines $SP, RQ, PQ$, and $SR$, respectively. What is the sum of the coordinates of the center of the square $PQRS$?
正方形 $PQRS$ 位于第一象限。点 $(3,0)$、$(5,0)$、$(7,0)$ 和 $(13,0)$ 分别位于直线 $SP$、$RQ$、$PQ$ 和 $SR$ 上。正方形 $PQRS$ 的中心坐标之和是多少？
Q18
Let $(a_1,a_2, \dots ,a_{10})$ be a list of the first 10 positive integers such that for each $2 \le i \le 10$ either $a_i+1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list. How many such lists are there?
设 $(a_1, a_2, \dots, a_{10})$ 是前 10 个正整数的一个排列，使得对于每个 $2 \le i \le 10$，$a_i + 1$ 或 $a_i -1$ 或两者都在 $a_i$ 之前的位置出现过。有多少这样的排列？
Q19
A unit cube has vertices $P_1,P_2,P_3,P_4,P_1',P_2',P_3',$ and $P_4'$. Vertices $P_2$, $P_3$, and $P_4$ are adjacent to $P_1$, and for $1\le i\le 4,$ vertices $P_i$ and $P_i'$ are opposite to each other. A regular octahedron has one vertex in each of the segments $P_1P_2$, $P_1P_3$, $P_1P_4$, $P_1'P_2'$, $P_1'P_3'$, and $P_1'P_4'$. What is the octahedron's side length?
一个单位立方体有顶点 $P_1,P_2,P_3,P_4,P_1',P_2',P_3',$ 和 $P_4'$。顶点 $P_2$, $P_3$, 和 $P_4$ 与 $P_1$ 相邻，且对于 $1\le i\le 4,$ 顶点 $P_i$ 和 $P_i'$ 互为对顶点。正八面体在每一段 $P_1P_2$, $P_1P_3$, $P_1P_4$, $P_1'P_2'$, $P_1'P_3'$, 和 $P_1'P_4'$ 上各有一个顶点。这个正八面体的边长是多少？
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Q20
A trapezoid has side lengths 3, 5, 7, and 11. The sum of all the possible areas of the trapezoid can be written in the form of $r_1\sqrt{n_1}+r_2\sqrt{n_2}+r_3$, where $r_1$, $r_2$, and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of any prime. What is the greatest integer less than or equal to $r_1+r_2+r_3+n_1+n_2$?
一个梯形边长为 3、5、7 和 11。所有可能梯形面积之和可写成 $r_1\sqrt{n_1}+r_2\sqrt{n_2}+r_3$ 的形式，其中 $r_1$, $r_2,$ 和 $r_3$ 是有理数，$n_1$ 和 $n_2$ 是不能被质数的平方整除的正整数。$r_1+r_2+r_3+n_1+n_2$ 的最大整数小于或等于该值的是多少？
Q21
Square $AXYZ$ is inscribed in equiangular hexagon $ABCDEF$ with $X$ on $\overline{BC}$, $Y$ on $\overline{DE}$, and $Z$ on $\overline{EF}$. Suppose that $AB=40$, and $EF=41(\sqrt{3}-1)$. What is the side-length of the square? (diagram by djmathman)
正方形 $AXYZ$ 内接于等角六边形 $ABCDEF$ 中，$X$ 在 $\overline{BC}$ 上，$Y$ 在 $\overline{DE}$ 上，$Z$ 在 $\overline{EF}$ 上。已知 $AB=40$，且 $EF=41(\sqrt{3}-1)$。求该正方形的边长。 (diagram by djmathman)
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Q22
A bug travels from A to B along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there?
一只虫子沿着下图所示六角形格点中的线段从 A 移动到 B。标有箭头的线段只能沿箭头方向行走，且虫子不会重复走同一线段。有多少条不同的路径？
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Q23
Consider all polynomials of a complex variable, $P(z)=4z^4+az^3+bz^2+cz+d$, where $a,b,c,$ and $d$ are integers, $0\le d\le c\le b\le a\le 4$, and the polynomial has a zero $z_0$ with $|z_0|=1.$ What is the sum of all values $P(1)$ over all the polynomials with these properties?
考虑所有复变量多项式 $P(z)=4z^4+az^3+bz^2+cz+d$，其中 $a,b,c,$ 和 $d$ 为整数，$0\le d\le c\le b\le a\le 4$，且多项式有一个零点 $z_0$ 满足 $|z_0|=1$。求所有满足这些性质的多项式的 $P(1)$ 值之和。
Q24
Define the function $f_1$ on the positive integers by setting $f_1(1)=1$ and if $n=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is the prime factorization of $n>1$, then \[f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\cdots (p_k+1)^{e_k-1}.\] For every $m\ge 2$, let $f_m(n)=f_1(f_{m-1}(n))$. For how many $N$s in the range $1\le N\le 400$ is the sequence $(f_1(N),f_2(N),f_3(N),\dots )$ unbounded? Note: A sequence of positive numbers is unbounded if for every integer $B$, there is a member of the sequence greater than $B$.
定义函数 $f_1$ 于正整数，令 $f_1(1)=1$，若 $n=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ 是 $n>1$ 的质因数分解，则 \[f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\cdots (p_k+1)^{e_k-1}.\] 对每个 $m\ge 2$，令 $f_m(n)=f_1(f_{m-1}(n))$。在范围 $1\le N\le 400$ 中，有多少个 $N$ 使得序列 $(f_1(N),f_2(N),f_3(N),\dots )$ 无界？ 注：若对每个整数 $B$，序列中存在大于 $B$ 的项，则称正数序列无界。
Q25
Let $S=\{(x,y) : x\in \{0,1,2,3,4\}, y\in \{0,1,2,3,4,5\},\text{ and } (x,y)\ne (0,0)\}$. Let $T$ be the set of all right triangles whose vertices are in $S$. For every right triangle $t=\triangle{ABC}$ with vertices $A$, $B$, and $C$ in counter-clockwise order and right angle at $A$, let $f(t)=\tan(\angle{CBA})$. What is \[\prod_{t\in T} f(t)?\]
令 $S=\{(x,y) : x\in \{0,1,2,3,4\}, y\in \{0,1,2,3,4,5\},\text{ and } (x,y)\ne (0,0)\}$。 令 $T$ 为所有顶点在 $S$ 中的直角三角形的集合。对每个直角三角形 $t=\triangle{ABC}$（顶点 $A$、$B$、$C$ 按逆时针顺序排列，且直角在 $A$），令 $f(t)=\tan(\angle{CBA})$。求 \[\prod_{t\in T} f(t)?\]
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