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AMC12 2013 A

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AMC12 · 2013 (A)

Q1
Square $ABCD$ has side length 10. Point $E$ is on $\overline{BC}$, and the area of $\triangle ABE$ is 40. What is $BE$?
正方形 $ABCD$ 的边长为 10。点 $E$ 在 $\overline{BC}$ 上，$\triangle ABE$ 的面积为 40。$BE$ 等于多少？
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Q2
A softball team played ten games, scoring 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 runs. They lost by one run in exactly five games. In each of their other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?
一个垒球队打了十场比赛，得分为 1, 2, 3, 4, 5, 6, 7, 8, 9, 和 10 分。他们恰好有五场比赛以一分之差输掉。在其他比赛中，他们的得分是对手的两倍。对手总共得了多少分？
Q3
A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?
一个花束包含粉色玫瑰、红色玫瑰、粉色康乃馨和红色康乃馨。粉色花的三分之一是玫瑰，红色花的四分之三是康乃馨，花束中六成是粉色花。花束中有百分之多少是康乃馨？
Q4
What is the value of $\dfrac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}$?
$\dfrac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}$ 的值是多少？
Q5
Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid \$105, Dorothy paid \$125, and Sammy paid \$175. In order to share the costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$?
汤姆、多萝西和萨米一起去度假，并同意平均分摊费用。在旅途中，汤姆支付了 \$105，多萝西支付了 \$125，萨米支付了 \$175。为了使费用平均分摊，汤姆给了萨米 $t$ 美元，多萝西给了萨米 $d$ 美元。求 $t-d$。
Q6
In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on 20% of her three-point shots and 30% of her two-point shots. Shenille attempted 30 shots. How many points did she score?
在最近的一场篮球比赛中，Shenille 只尝试了三分球和两分球。她三分球命中率为 20%，两分球命中率为 30%。Shenille 总共尝试了 30 次投篮。她总共得了多少分？
Q7
The sequence $S_1, S_2, S_3, \dots, S_{10}$ has the property that every term beginning with the third is the sum of the previous two. That is, $S_n = S_{n-2} + S_{n-1}$ for $n \ge 3$. Suppose that $S_9 = 110$ and $S_7 = 42$. What is $S_4$?
数列 $S_1, S_2, S_3, \dots, S_{10}$ 具有从第三项开始，每一项是前两项之和的性质。即 $S_n = S_{n-2} + S_{n-1}$，$n \ge 3$。已知 $S_9 = 110$ 和 $S_7 = 42$。$S_4$ 是多少？
Q8
Given that $x$ and $y$ are distinct nonzero real numbers such that $x + \frac{2}{x} = y + \frac{2}{y}$, what is $xy$?
已知 $x$ 和 $y$ 是不同的非零实数，且 $x + \frac{2}{x} = y + \frac{2}{y}$，求 $xy$？
Q9
In $\triangle ABC$, $AB = AC = 28$ and $BC = 20$. Points $D, E, F$ are on sides $\overline{AB}, \overline{BC},$ and $\overline{AC}$, respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. What is the perimeter of parallelogram $ADEF$?
在 $\triangle ABC$ 中，$AB = AC = 28$，$BC = 20$。点 $D, E, F$ 分别在边 $\overline{AB}, \overline{BC}, \overline{AC}$ 上，使得 $\overline{DE} \parallel \overline{AC}$，$\overline{EF} \parallel \overline{AB}$。平行四边形 $ADEF$ 的周长是多少？
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Q10
Let $S$ be the set of positive integers $n$ for which $\frac{1}{n}$ has the repeating decimal representation $0.\overline{ab}=0.ababab\ldots$, with $a$ and $b$ different digits. What is the sum of the elements of $S$?
设$S$为满足如下条件的正整数$n$的集合：$\frac{1}{n}$的循环小数表示为$0.\overline{ab}=0.ababab\ldots$，其中$a$与$b$是不同的数字。求集合$S$中所有元素的和是多少？
Q11
Triangle $ABC$ is equilateral with $AB=1$. Points $E$ and $G$ are on $AC$ and points $D$ and $F$ are on $AB$ such that both $DE$ and $FG$ are parallel to $BC$. Furthermore, triangle $ADE$ and trapezoids $DFGE$ and $FBCG$ all have the same perimeter. What is $DE+FG$?
等边三角形 $ABC$ 满足 $AB=1$。点 $E$ 和 $G$ 在 $AC$ 上，点 $D$ 和 $F$ 在 $AB$ 上，并且 $DE$ 与 $FG$ 都平行于 $BC$。此外，三角形 $ADE$ 与梯形 $DFGE$、$FBCG$ 的周长都相同。求 $DE+FG$。
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Q12
The angles in a particular triangle are in arithmetic progression, and the side lengths are 4, 5, and $x$. The sum of the possible values of $x$ equals $a+\sqrt{b}+\sqrt{c}$, where $a$, $b$, and $c$ are positive integers. What is $a+b+c$?
某个三角形的三个角成等差数列，三边长分别为 4、5 和 $x$。所有可能的 $x$ 的取值之和等于 $a+\sqrt{b}+\sqrt{c}$，其中 $a$、$b$、$c$ 为正整数。求 $a+b+c$。
Q13
Let points $A=(0,0)$, $B=(1,2)$, $C=(3,3)$, and $D=(4,0)$. Quadrilateral $ABCD$ is cut into equal area pieces by a line passing through $A$. This line intersects $\overline{CD}$ at point $\left(\frac{p}{q},\frac{r}{s}\right)$, where these fractions are in lowest terms. What is $p+q+r+s$?
已知点 $A=(0,0)$，$B=(1,2)$，$C=(3,3)$，$D=(4,0)$。过点 $A$ 作一直线把四边形 $ABCD$ 分成面积相等的两部分。这条直线与线段 $\overline{CD}$ 交于点 $\left(\frac{p}{q},\frac{r}{s}\right)$，其中分数均为最简形式。求 $p+q+r+s$。
Q14
The sequence $ \log_{12} 162,\ \log_{12} x,\ \log_{12} y,\ \log_{12} z,\ \log_{12} 1250 $ is an arithmetic progression. What is $x$?
数列 $ \log_{12} 162,\ \log_{12} x,\ \log_{12} y,\ \log_{12} z,\ \log_{12} 1250 $ 是等差数列。求 $x$。
Q15
Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cottontail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done?
兔子 Peter 和 Pauline 有三个后代——Flopsie、Mopsie 和 Cottontail。这五只兔子要分送到四个不同的宠物店，使得没有商店同时得到父母和子女。不要求每个商店都得到兔子。可以有多少种不同的方式？
Q16
A, B, and C are three piles of rocks. The mean weight of the rocks in A is 40 pounds, the mean weight of the rocks in B is 50 pounds, the mean weight of the rocks in the combined piles A and B is 43 pounds, and the mean weight of the rocks in the combined piles A and C is 44 pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles B and C?
A、B 和 C 是三堆石头。A 中石头的平均重量是 40 磅，B 中石头的平均重量是 50 磅，A 和 B 合并后的平均重量是 43 磅，A 和 C 合并后的平均重量是 44 磅。B 和 C 合并后石头的平均重量最大的可能整数值是多少磅？
Q17
A group of 12 pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k$th pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the 12th pirate receive?
一共有 12 个海盗，他们同意按照以下方式分配一箱金币。第 $k$ 个海盗取份时，取走箱中剩余金币的 $\frac{k}{12}$。箱子最初的金币数是最小使得这种分配方式每个海盗都能得到正整数枚金币的数。第 12 个海盗得到多少枚金币？
Q18
Six spheres of radius 1 are positioned so that their centers are at the vertices of a regular hexagon of side length 2. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?
六个半径为 1 的球体，其中心位于边长为 2 的正六边形的顶点处。这六个球体都与一个更大的球体内切，该大球的中心是六边形的中心。还有一个第八个球体，与六个小球体外切，并与大球体内切。这个第八个球体的半径是多少？
Q19
In $\triangle ABC$, $AB=86$, and $AC=97$. A circle with center $A$ and radius $AB$ intersects $BC$ at points $B$ and $X$. Moreover, $BX$ and $CX$ have integer lengths. What is $BC$?
在 $\triangle ABC$ 中，$AB=86$，且 $AC=97$。以 $A$ 为圆心、$AB$ 为半径的圆与 $BC$ 相交于点 $B$ 和 $X$。此外，$BX$ 和 $CX$ 的长度都是整数。求 $BC$ 的长度是多少？
Q20
Let $S$ be the set $\{1,2,3,\ldots,19\}$. For $a,b\in S$, define $a\succ b$ to mean that either $0<a-b\le 9$ or $b-a>9$. How many ordered triples $(x,y,z)$ of elements of $S$ have the property that $x\succ y$, $y\succ z$, and $z\succ x$?
设 $S=\{1,2,3,\ldots,19\}$。对任意 $a,b\in S$，定义 $a\succ b$ 表示满足以下两种情况之一：$0<a-b\le 9$ 或 $b-a>9$。问：$S$ 中有多少个有序三元组 $(x,y,z)$ 满足 $x\succ y$、$y\succ z$ 且 $z\succ x$？
Q21
Consider $A = \log \left(2013 + \log \left(2012 + \log \left(2011 + \log(\cdots + \log (3 + \log 2) \cdots)\right)\right)\right)$. Which of the following intervals contains $A$?
考虑 $A = \log \left(2013 + \log \left(2012 + \log \left(2011 + \log(\cdots + \log (3 + \log 2) \cdots)\right)\right)\right)$。以下哪个区间包含 $A$？
Q22
A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome $n$ is chosen uniformly at random. What is the probability that $\frac{n}{11}$ is also a palindrome?
回文数是一个在十进制下无前导零的前后读相同的非负整数。均匀随机选择一个六位回文数 $n$。$\frac{n}{11}$ 也是回文数的概率是多少？
Q23
ABCD is a square of side length $\sqrt{3+1}$. Point P is on AC such that AP = $\sqrt{2}$. The square region bounded by ABCD is rotated 90$^\circ$ counterclockwise with center P, sweeping out a region whose area is $\frac{1}{c}(a\pi + b)$, where a, b, and c are positive integers and gcd(a, b, c) = 1. What is a + b + c ?
ABCD 是一个边长为 $\sqrt{3+1}$ 的正方形。点 P 在 AC 上使得 AP = $\sqrt{2}$。以 P 为中心将 ABCD 围成的正方形区域逆时针旋转 90$^\circ$，扫过的区域面积为 $\frac{1}{c}(a\pi + b)$，其中 a, b, c 为正整数且 gcd(a, b, c) = 1。求 a + b + c ?
Q24
Three distinct segments are chosen at random among the segments whose endpoints are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?
在正十二边形的顶点为端点的线段中随机选择三个不同的线段。这些三个线段的长度能构成一个有正面积三角形的概率是多少？
Q25
Let $f:\mathbb{C}\to\mathbb{C}$ be defined by $f(z)=z^2+iz+1$. How many complex numbers $z$ are there such that $\operatorname{Im}(z)>0$ and both the real and the imaginary parts of $f(z)$ are integers with absolute value at most $10$?
设 $f:\mathbb{C}\to\mathbb{C}$ 定义为 $f(z)=z^2+iz+1$。满足 $\operatorname{Im}(z)>0$ 且 $f(z)$ 的实部与虚部均为整数、并且它们的绝对值都不超过 $10$ 的复数 $z$ 有多少个？
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