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AMC12 2018 A

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AMC12 · 2018 (A)

Q1
A large urn contains 100 balls, of which 36% are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be 72%? (No red balls are to be removed.)
一个大瓮中有100个球，其中36%是红色的，其余是蓝色的。需要移除多少个蓝球，使得瓮中红球的百分比变为72%？（不得移除红球。）
Q2
While exploring a cave, Carl comes across a collection of 5-pound rocks worth \$14 each, 4-pound rocks worth \$11 each, and 1-pound rocks worth \$2 each. There are at least 20 of each size. He can carry at most 18 pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?
卡尔在探洞时发现了一些5磅重的石头每块价值14美元、4磅重的石头每块价值11美元、1磅重的石头每块价值2美元。每种石头至少有20个。他最多能携带18磅。问他能带出洞的最大价值（美元）是多少？
Q3
How many ways can a student schedule 3 mathematics courses—algebra, geometry, and number theory—in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)
学生如何在6节课的一天中安排3门数学课——代数、几何和数论，使得没有两门数学课在连续节次？（其他3节课上什么课无关。）有多少种方式？
Q4
Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, “We are at least 6 miles away,” Bob replied, “We are at most 5 miles away.” Charlie then remarked, “Actually the nearest town is at most 4 miles away.” It turned out that none of the three statements was true. Let $d$ be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of $d$?
爱丽丝、鲍勃和查理徒步时想知道最近的城镇有多远。爱丽丝说：“我们至少6英里远。”鲍勃回答：“我们至多5英里远。”查理说：“其实最近的城镇至多4英里远。”结果三人的陈述都不真。设$d$为到最近城镇的英里距离。以下哪个区间是$d$的所有可能值的集合？
Q5
What is the sum of all possible values of $k$ for which the polynomials $x^{2} -3x + 2$ and $x^{2} -5x + k$ have a root in common?
对于哪些$k$的值，多项式$x^{2} -3x + 2$和$x^{2} -5x + k$有公共根？所有可能$k$之和是多少？
Q6
For positive integers $m$ and $n$ such that $m + 10 < n + 1$, both the mean and the median of the set $\{m, m + 4, m + 10, n + 1, n + 2, 2n\}$ are equal to $n$. What is $m + n$?
对于正整数 $m$ 和 $n$，满足 $m + 10 < n + 1$，集合 $\{m, m + 4, m + 10, n + 1, n + 2, 2n\}$ 的平均数和中位数均为 $n$。求 $m + n$ 的值。
Q7
For how many (not necessarily positive) integer values of $n$ is the value of $4000 \cdot \left(\frac{2}{5}\right)^n$ an integer?
$4000 \cdot \left(\frac{2}{5}\right)^n$ 的值为整数的整数 $n$（不一定是正整数）有多少个？
Q8
All of the triangles in the diagram below are similar to isosceles triangle $ABC$, in which $AB = AC$. Each of the 7 smallest triangles has area 1, and $\triangle ABC$ has area 40. What is the area of trapezoid $DBCE$?
图中所有的三角形都与等腰三角形 $ABC$ 相似，其中 $AB = AC$。7 个最小三角形的面积均为 1，$\triangle ABC$ 的面积为 40。梯形 $DBCE$ 的面积是多少？
stem
Q9
Which of the following describes the largest subset of values of $y$ within the closed interval $[0, \pi]$ for which $\sin(x + y) \le \sin(x) + \sin(y)$ for every $x$ between 0 and $\pi$, inclusive?
以下哪项描述了在闭区间 $[0, \pi]$ 内最大的 $y$ 值子集，使得对于所有 $x \in [0, \pi]$，有 $\sin(x + y) \le \sin(x) + \sin(y)$？
Q10
How many ordered pairs of real numbers $(x, y)$ satisfy the following system of equations? $$x + 3y = 3$$$$|x| - |y| = 1$$
下列方程组有多少个实数有序对 $(x, y)$ 满足？ $$x + 3y = 3$$$$|x| - |y| = 1$$
Q11
A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point A falls on point B. What is the length in inches of the crease?
一个纸质三角形，边长分别为3、4和5英寸，如图所示，将其折叠使得点A落在点B上。折痕的长度是多少英寸？
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Q12
Let $S$ be a set of 6 integers taken from $\{1, 2, \dots, 12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a < b$, then $b$ is not a multiple of $a$. What is the least possible value of an element of $S$?
设$S$是从$\{1, 2, \dots, 12\}$中取的6个整数的集合，具有如下性质：如果$a$和$b$是$S$的元素且$a < b$，则$b$不是$a$的倍数。$S$的一个元素的最小可能值为多少？
Q13
How many nonnegative integers can be written in the form $a_7 \cdot 3^7 + a_6 \cdot 3^6 + a_5 \cdot 3^5 + a_4 \cdot 3^4 + a_3 \cdot 3^3 + a_2 \cdot 3^2 + a_1 \cdot 3^1 + a_0 \cdot 3^0$, where $a_i \in \{-1, 0, 1\}$ for $0 \le i \le 7$?
有多少个非负整数可以表示为 $a_7 \cdot 3^7 + a_6 \cdot 3^6 + a_5 \cdot 3^5 + a_4 \cdot 3^4 + a_3 \cdot 3^3 + a_2 \cdot 3^2 + a_1 \cdot 3^1 + a_0 \cdot 3^0$， 其中$a_i \in \{-1, 0, 1\}$对于$0 \le i \le 7$？
Q14
The solution to the equation $\log_{3x} 4 = \log_{2x} 8$, where $x$ is a positive real number other than $\frac{1}{3}$ or $\frac{1}{2}$, can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p + q$?
方程$\log_{3x} 4 = \log_{2x} 8$的解（其中$x$是除$\frac{1}{3}$或$\frac{1}{2}$外的正实数）可以写成$\frac{p}{q}$，其中$p$和$q$是互质的正整数。$p + q$等于多少？
Q15
A scanning code consists of a $7 \times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of 49 squares. A scanning code is called symmetric if its look does not change when the entire square is rotated by a multiple of 90° counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?
一个扫描码由$7 \times 7$的方格网格组成，其中一些方格涂黑，其余涂白。该49个方格中必须至少有一个方格是每种颜色。扫描码被称为对称的，如果整个方格绕中心逆时针旋转90°的倍数时外观不变，也不改变当反射穿过连接对角线的线或连接对边中点的线时。可能的对称扫描码总数是多少？
Q16
Which of the following describes the set of values of $a$ for which the curves $x^2 + y^2 = a^2$ and $y = x^2 - a$ in the real $xy$-plane intersect at exactly 3 points?
以下哪个描述了曲线 $x^2 + y^2 = a^2$ 和 $y = x^2 - a$ 在实数 $xy$ 平面上相交恰好 3 个点的 $a$ 的取值集合？
Q17
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths of 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square $S$ so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from $S$ to the hypotenuse is 2 units. What fraction of the field is planted?
毕达哥拉斯农夫有一个直角三角形田地，直角三角形的两条直角边长分别为 3 和 4 单位。在两条直角边相交的直角角上，他留出一个小的未种植正方形 $S$，从空中看像直角符号。田地的其余部分都种植了。从 $S$ 到斜边的最近距离为 2 单位。田地中有多少分数被种植了？
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Q18
Triangle $ABC$ with $AB = 50$ and $AC = 10$ has area 120. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$?
三角形 $ABC$ 有 $AB = 50$ 和 $AC = 10$，面积为 120。$D$ 为 $\overline{AB}$ 中点，$E$ 为 $\overline{AC}$ 中点。$\angle BAC$ 的角平分线分别与 $\overline{DE}$ 和 $\overline{BC}$ 相交于 $F$ 和 $G$。四边形 $FDBG$ 的面积是多少？
Q19
Let $A$ be the set of positive integers that have no prime factors other than 2, 3, or 5. The infinite sum $$ \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \dots $$ of the reciprocals of all the elements of $A$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
令 $A$ 为仅由质因数 2、3 或 5 构成的正整数集合。无限和 $$ \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \dots $$ 是 $A$ 中所有元素的倒数之和，可表示为 $\frac{m}{n}$，其中 $m$ 和 $n$ 互质正整数。求 $m+n$？
Q20
Triangle ABC is an isosceles right triangle with AB = AC = 3. Let M be the midpoint of hypotenuse BC. Points I and E lie on sides AC and AB, respectively, so that AI > AE and AIME is a cyclic quadrilateral. Given that triangle EMI has area 2, the length CI can be written as $a-\frac{\sqrt{b}}{c}$, where a, b, and c are positive integers and b is square-free. What is the value of a + b + c?
三角形 ABC 是等腰直角三角形，$AB = AC = 3$。$M$ 为斜边 BC 中点。点 I 和 E 在边 AC 和 AB 上，且 $AI > AE$，使得 AIME 是圆内接四边形。已知三角形 EMI 面积为 2，$CI$ 的长度可写为 $a-\frac{\sqrt{b}}{c}$，其中 $a, b, c$ 为正整数且 $b$ 无平方因子。求 $a + b + c$ 的值？
Q21
Which of the following polynomials has the greatest real root?
下列多项式中，具有最大实根的多项式是哪一个？
Q22
The solutions to the equations $z^2 = 4 + 4\sqrt{15}i$ and $z^2 = 2 + 2\sqrt{3}i$, where $i = \sqrt{-1}$, form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form $p\sqrt{q} - r\sqrt{s}$, where $p, q, r,$ and $s$ are positive integers and neither $q$ nor $s$ is divisible by the square of any prime number. What is $p + q + r + s$?
方程 $z^2 = 4 + 4\sqrt{15}i$ 和 $z^2 = 2 + 2\sqrt{3}i$ 的解（其中 $i = \sqrt{-1}$）在复平面中形成一个平行四边形的顶点。这个平行四边形的面积可以写成 $p\sqrt{q} - r\sqrt{s}$ 的形式，其中 $p, q, r, s$ 是正整数，且 $q$ 和 $s$ 都不被任何质数的平方整除。求 $p + q + r + s$ 的值。
Q23
In $\triangle PAT$, $\angle P = 36^\circ$, $\angle A = 56^\circ$, and $PA = 10$. Points $U$ and $G$ lie on sides $TP$ and $TA$, respectively, so that $PU = AG = 1$. Let $M$ and $N$ be the midpoints of segments $PA$ and $UG$, respectively. What is the degree measure of the acute angle formed by lines $MN$ and $PA$?
在 $\triangle PAT$ 中，$\angle P = 36^\circ$，$\angle A = 56^\circ$，$PA = 10$。点 $U$ 和 $G$ 分别在边 $TP$ 和 $TA$ 上，使得 $PU = AG = 1$。设 $M$ 和 $N$ 分别是线段 $PA$ 和 $UG$ 的中点。求直线 $MN$ 和 $PA$ 形成的锐角的度数。
Q24
Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between 0 and 1, and Bob announces that he will choose his number uniformly at random from all the numbers between $\frac{1}{2}$ and $\frac{2}{3}$. Armed with this information, what number should Carol choose to maximize her chance of winning?
Alice、Bob 和 Carol 玩一个游戏，每人选择 0 到 1 之间的实数。获胜者是其数位于其他两人所选数之间的那个人。Alice 宣布她将从 0 到 1 的所有数中均匀随机选择她的数，Bob 宣布他将从 $\frac{1}{2}$ 到 $\frac{2}{3}$ 的所有数中均匀随机选择他的数。有了这些信息，Carol 应该选择什么数来最大化她的获胜概率？
Q25
For a positive integer $n$ and nonzero digits $a, b,$ and $c$, let $A_n$ be the $n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $n$-digit integer each of whose digits is equal to $b$; and let $C_n$ be the $2n$-digit integer each of whose digits is equal to $c$. What is the greatest possible value of $a + b + c$ for which there are at least two values of $n$ such that $C_n - B_n = A_n^2$?
对于正整数 $n$ 和非零数字 $a, b, c$，设 $A_n$ 是每个数字均为 $a$ 的 $n$ 位整数；$B_n$ 是每个数字均为 $b$ 的 $n$ 位整数；$C_n$ 是每个数字均为 $c$ 的 $2n$ 位整数。求存在至少两个 $n$ 值使得 $C_n - B_n = A_n^2$ 的最大可能 $a + b + c$ 值。
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