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AMC12 2020 A

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AMC12 · 2020 (A)

Q1
Carlos took 70% of a whole pie. Maria took one third of the remainder. What portion of the original pie was left?
Carlos 拿走了整个饼的 70%。Maria 拿走了剩余部分的 1/3。原饼还剩下多少？
Q2
The acronym AMC is shown in the rectangular grid below with grid lines spaced 1 unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC?
下面的矩形网格中显示了缩写 AMC，网格线间距为 1 个单位。形成缩写 AMC 的线段长度总和是多少单位？
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Q3
A driver travels for 2 hours at 60 miles per hour, during which her car gets 30 miles per gallon of gasoline. She is paid $0.50 per mile, and her only expense is gasoline at $2.00 per gallon. What is her net rate of pay, in dollars per hour, after this expense?
一位司机以 60 英里/小时的速度行驶 2 小时，其间她的车耗油 30 英里/加仑汽油。她每英里工资 0.50 美元，唯一开支是每加仑 2.00 美元的汽油。扣除此开支后，她的净时薪是多少美元/小时？
Q4
How many 4-digit positive integers (that is, integers between 1000 and 9999, inclusive) having only even digits are divisible by 5?
有多少个 4 位正整数（即 1000 到 9999 之间，包括两端）仅由偶数数字组成且能被 5 整除？
Q5
The 25 integers from −10 to 14, inclusive, can be arranged to form a 5-by-5 square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?
从 −10 到 14 的 25 个整数（包括两端）可以排列成一个 5×5 方阵，使得每行之和、每列之和以及两条主对角线之和都相同。这个公共和的值是多少？
Q6
In the plane figure shown below, 3 of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry?
在下面的平面图形中，有 3 个单位正方形被涂黑。 必须额外涂黑的最少单位正方形数量是多少，使得最终图形具有两条对称轴？
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Q7
Seven cubes, whose volumes are 1, 8, 27, 64, 125, 216, and 343 cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?
有七个立方体，它们的体积分别是 1、8、27、64、125、216 和 343 立方单位。这些立方体垂直堆叠成一座塔，体积从底部到顶部递减。除了最底部的立方体，每个立方体的底面完全位于下方立方体的顶面上。塔的总表面积（包括底部）有多少平方单位？
Q8
What is the median of the following list of 4040 numbers? $1, 2, 3, \dots , 2020, 1^2, 2^2, 3^2, \dots , 2020^2$
以下 4040 个数的.median 是多少？ $1, 2, 3, \dots , 2020, 1^2, 2^2, 3^2, \dots , 2020^2$
Q9
How many solutions does the equation $\tan(2x) = \cos\left(\frac{x}{2}\right)$ have on the interval $[0, 2\pi]$?
方程 $\tan(2x) = \cos\left(\frac{x}{2}\right)$ 在区间 $[0, 2\pi]$ 上有多少个解？
Q10
There is a unique positive integer $n$ such that $\log_2(\log_{16} n) = \log_4(\log_4 n)$. What is the sum of the digits of $n$?
存在唯一的正整数 $n$ 使得 $\log_2(\log_{16} n) = \log_4(\log_4 n)$。 $n$ 的各位数字之和是多少？
Q11
A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length 1, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0, 0)$, $(0, 4)$, $(4, 4)$, and $(4, 0)$. What is the probability that the sequence of jumps ends on a vertical side of the square?
一只青蛙坐在点 $(1, 2)$ 开始一系列跳跃，每一次跳跃平行于坐标轴且长度为 1，每次跳跃的方向（上、下、右或左）独立随机选择。序列在青蛙到达顶点为 $(0, 0)$、$(0, 4)$、$(4, 4)$ 和 $(4, 0)$ 的正方形的一条边时结束。序列结束在正方形垂直边上的概率是多少？
Q12
Line $\ell$ in the coordinate plane has equation $3x-5y+40=0$. This line is rotated $45^\circ$ counterclockwise about the point $(20,20)$ to obtain line $k$. What is the $x$-coordinate of the $x$-intercept of line $k$?
坐标平面中的直线 $\ell$ 的方程为 $3x-5y+40=0$。这条直线绕点 $(20,20)$ 逆时针旋转 $45^\circ$ 得到直线 $k$。直线 $k$ 的 $x$ 截距的 $x$ 坐标是多少？
Q13
There are integers $a$, $b$, and $c$, each greater than 1, such that $\sqrt[a]{N}\sqrt[b]{N}\sqrt[c]{N}=\sqrt[36]{N^{25}}$ for all $N>1$. What is $b$?
存在整数 $a$、$b$ 和 $c$，每个都大于 1，使得 $\sqrt[a]{N}\sqrt[b]{N}\sqrt[c]{N}=\sqrt[36]{N^{25}}$ 对所有 $N>1$ 成立。$b$ 是多少？
Q14
Regular octagon $ABCDEFGH$ has area $n$. Let $m$ be the area of quadrilateral $ACEG$. What is $\frac{m}{n}$?
正八边形 $ABCDEFGH$ 的面积为 $n$。设 $m$ 为四边形 $ACEG$ 的面积。$\frac{m}{n}$ 是多少？
Q15
In the complex plane, let $A$ be the set of solutions to $z^3-8=0$ and let $B$ be the set of solutions to $z^3-8z^2-8z+64=0$. What is the greatest distance between a point of $A$ and a point of $B$?
在复平面中，设 $A$ 为 $z^3-8=0$ 的解集，$B$ 为 $z^3-8z^2-8z+64=0$ 的解集。$A$ 中一点与 $B$ 中一点之间的最大距离是多少？
Q16
A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0)$, $(2020, 0)$, $(2020, 2020)$, and $(0, 2020)$. The probability that the point lies within $d$ units of a lattice point is $\frac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest tenth?
在坐标平面内的一个边长为2020的正方形中随机选择一点，该点到最近格点距离不超过$d$的概率为$\frac{1}{2}$。（点$(x, y)$是格点当且仅当$x$和$y$均为整数。）$d$的最接近的十分位数是多少？
Q17
The vertices of a quadrilateral lie on the graph of $y = \ln x$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln \frac{91}{90}$. What is the $x$-coordinate of the leftmost vertex?
一个四边形的顶点位于$y = \ln x$的图像上，其$x$坐标为四个连续的正整数。该四边形的面积为$\ln \frac{91}{90}$。最左顶点的$x$坐标是多少？
Q18
Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^\circ$, $AC = 20$, and $CD = 30$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E$, and $AE = 5$. What is the area of quadrilateral $ABCD$?
四边形$ABCD$满足$\angle ABC = \angle ACD = 90^\circ$，$AC = 20$，$CD = 30$。对角线$\overline{AC}$和$\overline{BD}$相交于点$E$，且$AE = 5$。四边形$ABCD$的面积是多少？
Q19
There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < \cdots < a_k$ such that $$\frac{2^{289} + 1}{2^{17} + 1} = 2^{a_1} + 2^{a_2} + \cdots + 2^{a_k}.$$ What is $k$?
存在唯一的严格递增的非负整数序列$a_1 < a_2 < \cdots < a_k$，使得$$\frac{2^{289} + 1}{2^{17} + 1} = 2^{a_1} + 2^{a_2} + \cdots + 2^{a_k}.$$$k$是多少？
Q20
Let $T$ be the triangle in the coordinate plane with vertices $(0, 0)$, $(4, 0)$, and $(0, 3)$. Consider the following five isometries of the plane: rotations of $90^\circ$, $180^\circ$, and $270^\circ$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position?
设$T$为坐标平面上的三角形，顶点为$(0, 0)$、$(4, 0)$、$(0, 3)$。考虑平面的以下五种等距变换：绕原点逆时针旋转$90^\circ$、$180^\circ$、$270^\circ$，关于$x$轴反射，关于$y$轴反射。在这125种由三个（不一定不同）这些变换组成的序列中，有多少种将$T$变回到原位置？
Q21
How many positive integers $n$ are there such that $n$ is a multiple of 5, and the least common multiple of $5!$ and $n$ equals 5 times the greatest common divisor of $10!$ and $n$?
有有多少个正整数$n$，使得$n$是5的倍数，并且$5!$和$n$的最小公倍数等于10！和$n$的最大公因数的5倍？
Q22
Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that $(2 + i)^n = a_n + b_n i$ for all integers $n \geq 0$, where $i = \sqrt{-1}$. What is $\sum_{n=0}^\infty \frac{a_n b_n}{7^n}$?
设$(a_n)$和$(b_n)$是实数序列，使得$(2 + i)^n = a_n + b_n i$对所有整数$n \geq 0$成立，其中$i = \sqrt{-1}$。求$\sum_{n=0}^\infty \frac{a_n b_n}{7^n}$？
Q23
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly 7. Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
Jason掷三个公平的六面骰子。然后他查看掷出的点数，并选择一部分骰子（可能为空，可能全部三个）重新掷。重新掷后，他赢得比赛当且仅当三个骰子上面朝上的点数之和恰好为7。Jason总是为了优化获胜几率而玩。求他选择重新掷恰好两个骰子的概率？
Q24
Suppose that $\triangle ABC$ is an equilateral triangle of side length $s$, with the property that there is a unique point $P$ inside the triangle such that $AP = 1$, $BP = \sqrt{3}$, and $CP = 2$. What is $s$?
假设$\triangle ABC$是边长为$s$的等边三角形，具有唯一一点$P$在三角形内部使得$AP = 1$，$BP = \sqrt{3}$，$CP = 2$。求$s$？
Q25
The number $a = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying $\lfloor x \rfloor \cdot \{x\} = a \cdot x^2$ is 420, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$. What is $p + q$?
数$a = \frac{p}{q}$，其中$p$和$q$互质正整数，具有所有满足$\lfloor x \rfloor \cdot \{x\} = a \cdot x^2$的实数$x$之和为420，其中$\lfloor x \rfloor$表示小于等于$x$的最大整数，$\{x\} = x - \lfloor x \rfloor$表示$x$的分数部分。求$p + q$？
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