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AMC12 2014 B

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AMC12 · 2014 (B)

Q1
Leah has 13 coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah’s coins worth?
Leah 有 13 枚硬币，全都是便士和镍币。如果她再多一枚镍币，那么她就会有相同数量的便士和镍币。Leah 的硬币总价值多少美分？
Q2
Orvin went to the store with just enough money to buy 30 balloons. When he arrived he discovered that the store had a special sale on balloons: buy 1 balloon at the regular price and get a second at $\frac{1}{3}$ off the regular price. What is the greatest number of balloons Orvin could buy?
Orvin 带着刚好够买 30 个气球的钱去商店。到了那里，他发现商店正在搞特价活动：买一个气球按原价，第二个气球打 $\frac{1}{3}$ 折。Orvin 最多能买多少个气球？
Q3
Randy drove the first third of his trip on a gravel road, the next 20 miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy’s trip?
Randy 的行程前三分之一在碎石路上，接下来 20 英里在铺装路上，剩余五分之一在土路上。Randy 的总行程多长（英里）？
Q4
Susie pays for 4 muffins and 3 bananas. Calvin spends twice as much paying for 2 muffins and 16 bananas. A muffin is how many times as expensive as a banana?
Susie 买了 4 个松饼和 3 个香蕉。Calvin 花费是 Susie 的两倍，买了 2 个松饼和 16 个香蕉。一个松饼比一个香蕉贵多少倍？
Q5
Doug constructs a square window using 8 equal-size panes of glass, as shown. The ratio of the height to width for each pane is 5 : 2, and the borders around and between the panes are 2 inches wide. In inches, what is the side length of the square window?
Doug 使用 8 块相同大小的玻璃板构成一个正方形窗户，如图所示。每块玻璃板的宽度与高度之比为 5 : 2，玻璃板之间和周围的边框宽 2 英寸。正方形窗户的边长是多少英寸？
stem
Q6
Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is $50\%$ more than the regular. After both consume $\frac{3}{4}$ of their drinks, Ann gives Ed a third of what she has left, and 2 additional ounces. When they finish their lemonades they realize that they both drank the same amount. How many ounces of lemonade did they drink together?
Ed 和 Ann 午餐时都点了柠檬水。Ed 点了常规尺寸。Ann 点了大杯柠檬水，比常规尺寸多 $50\%$。两人各自喝掉 $\frac{3}{4}$ 的饮料后，Ann 给了 Ed 她剩下柠檬水的三分之一，外加 2 盎司。当他们喝完柠檬水时，发现两人喝的量相同。他们总共喝了多少盎司的柠檬水？
Q7
For how many positive integers $n$ is $\frac{n}{30-n}$ also a positive integer?
有且仅有几个正整数 $n$ 使得 $\frac{n}{30-n}$ 也是正整数？
Q8
In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$? \[ \begin{array}{r} \mathrm{ABBCB} \\ + \mathrm{BCADA} \\ \hline \mathrm{DBDDD} \end{array} \]
如下所示的加法中，$A$、$B$、$C$、$D$ 是不同的数字。$D$ 有多少个不同的可能值？ \[ \begin{array}{r} \mathrm{ABBCB} \\ + \mathrm{BCADA} \\ \hline \mathrm{DBDDD} \end{array} \]
Q9
Convex quadrilateral $ABCD$ has $AB = 3$, $BC = 4$, $CD = 13$, $AD = 12$, and $\angle ABC = 90^\circ$, as shown. What is the area of the quadrilateral?
凸四边形 $ABCD$ 有 $AB = 3$，$BC = 4$，$CD = 13$，$AD = 12$，且 $\angle ABC = 90^\circ$，如图所示。四边形的面积是多少？
stem
Q10
Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, $abc$ miles was displayed on the odometer, where $abc$ is a 3-digit number with $a \ge 1$ and $a + b + c \le 7$. At the end of the trip, the odometer showed $cba$ miles. What is $a^2 + b^2 + c^2$?
Danica 开着新车旅行了整数组小时，以平均 55 英里/小时的速度。旅行开始时，里程表显示 $abc$ 英里，其中 $abc$ 是三位数，$a \ge 1$ 且 $a + b + c \le 7$。旅行结束时，里程表显示 $cba$ 英里。求 $a^2 + b^2 + c^2$？
Q11
A list of 11 positive integers has a mean of 10, a median of 9, and a unique mode of 8. What is the largest possible value of an integer in the list?
一个包含11个正整数的列表，平均值为10，中位数为9，唯一众数为8。列表中整数的最大可能值是多少？
Q12
A set $S$ consists of triangles whose sides have integer lengths less than 5, and no two elements of $S$ are congruent or similar. What is the largest number of elements that $S$ can have?
集合$S$由边长小于5的整数长度的三角形组成，且$S$中没有两个元素全等或相似。$S$的最大元素个数是多少？
Q13
Real numbers $a$ and $b$ are chosen with $1 < a < b$ such that no triangle with positive area has side lengths $1, a,$ and $b$ or $\frac{1}{b}, \frac{1}{a},$ and $1$. What is the smallest possible value of $b$?
选择实数$a$和$b$，满足$1 < a < b$，使得没有正面积三角形具有边长$1, a,$和$b$或$\frac{1}{b}, \frac{1}{a},$和$1$。$b$的最小可能值是多少？
Q14
A rectangular box has a total surface area of 94 square inches. The sum of the lengths of all its edges is 48 inches. What is the sum of the lengths in inches of all of its interior diagonals?
一个长方体盒子总表面积为94平方英寸。所有棱长的和为48英寸。其所有体内对角线长度的和是多少英寸？
Q15
When $p = \sum_{k=1}^{6} k \ln k$, the number $e^{p}$ is an integer. What is the largest power of 2 that is a factor of $e^{p}$?
当$p = \sum_{k=1}^{6} k \ln k$时，数$e^{p}$是一个整数。$e^{p}$的因式中2的最大幂是多少？
Q16
Let $P$ be a cubic polynomial with $P(0) = k$, $P(1) = 2k$, and $P(-1) = 3k$. What is $P(2) + P(-2)$?
设 $P$ 是一个三次多项式，满足 $P(0) = k$，$P(1) = 2k$，且 $P(-1) = 3k$。求 $P(2) + P(-2)$。
Q17
Let $P$ be the parabola with equation $y = x^2$ and let $Q = (20, 14)$. There are real numbers $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $P$ if and only if $r < m < s$. What is $r + s$?
设 $P$ 是方程 $y = x^2$ 的抛物线，$Q = (20, 14)$。存在实数 $r$ 和 $s$，使得通过 $Q$ 且斜率为 $m$ 的直线不与 $P$ 相交当且仅当 $r < m < s$。求 $r + s$。
Q18
The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is bad if it is not true that for every $n$ from 1 to 15 one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?
将数字 1, 2, 3, 4, 5 排列成一个圆圈。如果对于从 1 到 15 的每个 $n$，不能找到圆圈上连续出现的数字子集使其和为 $n$，则该排列为坏排列。只考虑旋转或反射不同的排列为不同。坏排列有多少种？
Q19
A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the truncated cone. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?
如图所示，一个球体内接于一个右圆锥台中。圆锥台的体积是球体体积的两倍。求圆锥台底面半径与顶面半径的比。
stem
Q20
For how many positive integers $x$ is $\log_{10}(x - 40) + \log_{10}(60 - x) < 2$?
有正整数 $x$ 多少个满足 $\log_{10}(x - 40) + \log_{10}(60 - x) < 2$？
Q21
In the figure, $ABCD$ is a square of side length 1. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$?
在图中，$ABCD$是一个边长为1的正方形。矩形$JKHG$和$EBCF$全等。$BE$的长度是多少？
stem
Q22
In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad $N$, $0 < N < 10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1 - \frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad 0 it will be eaten by a patiently waiting snake. If the frog reaches pad 10 it will exit the pond, never to return. What is the probability that the frog will escape being eaten by the snake?
一个小池塘中有11个睡莲垫，依次标记为0到10。一只青蛙坐在垫1上。当青蛙在垫$N$上时，$0 < N < 10$，它以概率$\frac{N}{10}$跳到垫$N-1$，以概率$1 - \frac{N}{10}$跳到垫$N+1$。每次跳跃独立于前一次。如果青蛙到达垫0，它将被耐心等待的蛇吃掉。如果到达垫10，它将离开池塘，再不返回。青蛙逃脱被蛇吃掉的概率是多少？
Q23
The number 2017 is prime. Let $S = \sum_{k=0}^{62} \binom{2014}{k}$. What is the remainder when $S$ is divided by 2017?
数2017是素数。设$S = \sum_{k=0}^{62} \binom{2014}{k}$。$S$除以2017的余数是多少？
Q24
Let $ABCDE$ be a pentagon inscribed in a circle such that $AB = CD = 3$, $BC = DE = 10$, and $AE = 14$. The sum of the lengths of all diagonals of $ABCDE$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?
设$ABCDE$是一个内接于圆的五边形，使得$AB = CD = 3$，$BC = DE = 10$，且$AE = 14$。五边形$ABCDE$所有对角线长度之和等于$\frac{m}{n}$，其中$m$和$n$互质。求$m + n$。
Q25
What is the sum of all positive real solutions $x$ to the equation $2\cos(2x)\left(\cos(2x)-\cos\left(\frac{2014\pi^2}{x}\right)\right)=\cos(4x)-1$?
求方程 $2\cos(2x)\left(\cos(2x)-\cos\left(\frac{2014\pi^2}{x}\right)\right)=\cos(4x)-1$ 的所有正实数解 $x$ 之和是多少？
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