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AMC12 2023 A

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AMC12 · 2023 (A)

Q1
Cities $A$ and $B$ are $45$ miles apart. Alicia lives in $A$ and Beth lives in $B$. Alicia bikes towards $B$ at 18 miles per hour. Leaving at the same time, Beth bikes toward $A$ at 12 miles per hour. How many miles from City $A$ will they be when they meet?
城市 $A$ 和 $B$ 相距 $45$ 英里。Alicia 住在 $A$，Beth 住在 $B$。Alicia 以每小时 $18$ 英里的速度向 $B$ 骑行。与此同时，Beth 以每小时 $12$ 英里的速度向 $A$ 骑行。他们相遇时距离城市 $A$ 有多少英里？
Q2
The weight of $\frac{1}{3}$ of a large pizza together with $3 \frac{1}{2}$ cups of orange slices is the same as the weight of $\frac{3}{4}$ of a large pizza together with $\frac{1}{2}$ cup of orange slices. A cup of orange slices weighs $\frac{1}{4}$ of a pound. What is the weight, in pounds, of a large pizza?
一大块披萨的 $\frac{1}{3}$ 连同 $3 \frac{1}{2}$ 杯橙子片的总重量，等于一大块披萨的 $\frac{3}{4}$ 连同 $\frac{1}{2}$ 杯橙子片的总重量。一杯橙子片重 $\frac{1}{4}$ 磅。一大块披萨的重量是多少磅？
Q3
How many positive perfect squares less than $2023$ are divisible by $5$?
小于 $2023$ 的正完全平方数中，能被 $5$ 整除的有多少个？
Q4
How many digits are in the base-ten representation of $8^5 \cdot 5^{10} \cdot 15^5$?
$8^5 \cdot 5^{10} \cdot 15^5$ 的十进制表示中有多少位数字？
Q5
Janet rolls a standard $6$-sided die $4$ times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal $3$?
Janet 掷一次标准的 $6$ 面骰子 $4$ 次，并保持掷骰数字的累加总和。她的累加总和某时刻等于 $3$ 的概率是多少？
Q6
Points $A$ and $B$ lie on the graph of $y=\log_{2}x$. The midpoint of $\overline{AB}$ is $(6, 2)$. What is the positive difference between the $x$-coordinates of $A$ and $B$?
点 $A$ 和 $B$ 位于 $y=\log_{2}x$ 的图像上。线段 $\overline{AB}$ 的中点为 $(6, 2)$。$A$ 和 $B$ 的 $x$ 坐标的正差是多少？
Q7
A digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-digit month, followed by a $2$-digit date within the month. For example, Arbor Day this year is displayed as $20230428.$ For how many dates in $2023$ does each digit appear an even number of times in the $8$-digital display for that date?
一个数字显示屏将当前日期显示为一个由 $8$ 位数字组成的整数，包括 $4$ 位年份、紧接着的 $2$ 位月份、以及月份内的 $2$ 位日期。例如，今年树日显示为 $20230428$。在 $2023$ 年的多少个日期中，$8$ 位数字显示中每个数字出现的次数都是偶数？
Q8
Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an $11$ on the next quiz, her mean will increase by $1$. If she scores an $11$ on each of the next three quizzes, her mean will increase by $2$. What is the mean of her quiz scores currently?
莫琳正在记录本学期测验成绩的平均分。如果莫琳在下一次测验中得 $11$ 分，她的平均分将增加 $1$ 分。如果她在接下来的三场测验中每场都得 $11$ 分，她的平均分将增加 $2$ 分。目前她的测验成绩平均分是多少？
Q9
A square of area $2$ is inscribed in a square of area $3$, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?
一个面积为 $2$ 的正方形内接在一个面积为 $3$ 的正方形中，形成了四个全等的三角形，如下图所示。阴影直角三角形的短腿与长腿的比是多少？
stem
Q10
Positive real numbers $x$ and $y$ satisfy $y^3=x^2$ and $(y-x)^2=4y^2$. What is $x+y$?
正实数 $x$ 和 $y$ 满足 $y^3=x^2$ 和 $(y-x)^2=4y^2$。$x+y$ 等于多少？
Q11
What is the degree measure of the acute angle formed by lines with slopes $2$ and $\frac{1}{3}$?
斜率为 $2$ 和 $\frac{1}{3}$ 的两条直线所形成的锐角的度量是多少度？
Q12
What is the value of \[2^3 - 1^3 + 4^3 - 3^3 + 6^3 - 5^3 + \dots + 18^3 - 17^3?\]
求 $2^3 - 1^3 + 4^3 - 3^3 + 6^3 - 5^3 + \dots + 18^3 - 17^3$ 的值。
Q13
In a table tennis tournament, every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was $40\%$ more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?
在一场乒乓球锦标赛中，每位选手与其他每位选手恰好对战一次。虽然右手选手是左手选手的两倍，但左手选手赢得的比赛数比右手选手多 $40\%$。（没有平局，也没有双手选手。）总共进行了多少场比赛？
Q14
How many complex numbers satisfy the equation $z^5=\overline{z}$, where $\overline{z}$ is the conjugate of the complex number $z$?
有几个复数满足方程 $z^5=\overline{z}$，其中 $\overline{z}$ 是复数 $z$ 的共轭？
Q15
Usain is walking for exercise by zigzagging across a $100$-meter by $30$-meter rectangular field, beginning at point $A$ and ending on the segment $\overline{BC}$. He wants to increase the distance walked by zigzagging as shown in the figure below $(APQRS)$. What angle $\theta = \angle PAB=\angle QPC=\angle RQB=\cdots$ will produce a length that is $120$ meters? (This figure is not drawn to scale. Do not assume that he zigzag path has exactly four segments as shown; there could be more or fewer.)
Usain 通过在 $100$ 米长 $30$ 米宽的矩形场地内之字形行走锻炼，从点 $A$ 开始，结束在线段 $\overline{BC}$ 上。他想通过如图所示（$APQRS$）之字形路径增加行走距离。什么角度 $\theta = \angle PAB=\angle QPC=\angle RQB=\cdots$ 会使路径长度为 $120$ 米？（图未按比例绘制，不要假设之字形路径恰好有四段；可能更多或更少。）
stem
Q16
Consider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can be written in the form $\tfrac{\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
考虑满足 $|1+z+z^{2}|=4$ 的复数集 $z$。$z$ 的虚部的最大值可以写成形式 $\tfrac{\sqrt{m}}{n}$，其中 $m$ 和 $n$ 是互质的正整数。求 $m+n$？
Q17
Flora the frog starts at 0 on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\frac{1}{2^m}$. What is the probability that Flora will eventually land at 10?
青蛙 Flora 从数轴上的 0 开始，向右进行一系列跳跃。在任意一次跳跃中，与之前的跳跃无关，Flora 以概率 $\frac{1}{2^m}$ 跳跃正整数距离 $m$。 Flora 最终落在 10 上的概率是多少？
Q18
Circle $C_1$ and $C_2$ each have radius $1$, and the distance between their centers is $\frac{1}{2}$. Circle $C_3$ is the largest circle internally tangent to both $C_1$ and $C_2$. Circle $C_4$ is internally tangent to both $C_1$ and $C_2$ and externally tangent to $C_3$. What is the radius of $C_4$?
圆 $C_1$ 和 $C_2$ 半径均为 1，其圆心间距离为 $\frac{1}{2}$。圆 $C_3$ 是与 $C_1$ 和 $C_2$ 都内切的最大的圆。圆 $C_4$ 与 $C_1$ 和 $C_2$ 都内切，并且与 $C_3$ 外切。$C_4$ 的半径是多少？
stem
Q19
What is the product of all solutions to the equation \[\log_{7x}2023\cdot \log_{289x}2023=\log_{2023x}2023\]
方程 \[\log_{7x}2023\cdot \log_{289x}2023=\log_{2023x}2023\] 的所有解的乘积是多少？
Q20
Rows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below. Each row after the first row is formed by placing a 1 at each end of the row, and each interior entry is 1 greater than the sum of the two numbers diagonally above it in the previous row. What is the units digits of the sum of the 2023 numbers in the 2023rd row?
一个整数三角形的第 1、2、3、4 和 5 行如下所示。 每行之后的行在行首尾放置 1，内部每个条目比前一行对角线上方的两个数的和大 1。求第 2023 行 2023 个数的和的个位数是多少？
stem
Q21
If $A$ and $B$ are vertices of a polyhedron, define the distance $d(A,B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B$. For example, if $\overline{AB}$ is an edge of the polyhedron, then $d(A, B) = 1$, but if $\overline{AC}$ and $\overline{CB}$ are edges and $\overline{AB}$ is not an edge, then $d(A, B) = 2$. Let $Q$, $R$, and $S$ be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles). What is the probability that $d(Q, R) > d(R, S)$?
如果 $A$ 和 $B$ 是多面体的顶点，定义距离 $d(A,B)$ 为连接 $A$ 和 $B$ 所需穿越的多面体边的最小数量。例如，如果 $\overline{AB}$ 是多面体的边，则 $d(A, B) = 1$，但如果 $\overline{AC}$ 和 $\overline{CB}$ 是边而 $\overline{AB}$ 不是边，则 $d(A, B) = 2$。让 $Q$、$R$ 和 $S$ 是从正二十面体（由 20 个等边三角形组成的正多面体）中随机选择的不同的顶点。$d(Q, R) > d(R, S)$ 的概率是多少？
Q22
Let $f$ be the unique function defined on the positive integers such that \[\sum_{d\mid n}d\cdot f\left(\frac{n}{d}\right)=1\] for all positive integers $n$. What is $f(2023)$?
设 $f$ 是定义在正整数上的唯一函数，使得对所有正整数 $n$，\[\sum_{d\mid n}d\cdot f\left(\frac{n}{d}\right)=1\]。求 $f(2023)$。
Q23
How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation \[(1+2a)(2+2b)(2a+b) = 32ab?\]
有几个正实数有序对 $(a,b)$ 满足方程 \[(1+2a)(2+2b)(2a+b) = 32ab？\]
Q24
Let $K$ be the number of sequences $A_1$, $A_2$, $\dots$, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of $\{1, 2, 3, \dots, 10\}$, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, $\{\}$, $\{5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 6, 7, 9\}$ is one such sequence, with $n = 5$.What is the remainder when $K$ is divided by $10$?
设 $K$ 为满足以下条件的序列 $A_1$、$A_2$、…、$A_n$ 的数量，其中 $n$ 是小于或等于 10 的正整数，每个 $A_i$ 是集合 $\{1, 2, 3, \dots, 10\}$ 的子集，且对每个 $i$ 从 2 到 $n$（包含），$A_{i-1}$ 是 $A_i$ 的子集。例如，$\{\}$、$\{5, 7\}$、$\{2, 5, 7\}$、$\{2, 5, 7\}$、$\{2, 5, 6, 7, 9\}$ 是一个这样的序列，$n = 5$。$K$ 除以 10 的余数是多少？
Q25
There is a unique sequence of integers $a_1, a_2, \cdots a_{2023}$ such that \[\tan2023x = \frac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + \cdots + a_{2023} \tan^{2023} x}{1 + a_2 \tan^2 x + a_4 \tan^4 x \cdots + a_{2022} \tan^{2022} x}\]whenever $\tan 2023x$ is defined. What is $a_{2023}?$
存在唯一的整数序列 $a_1, a_2, \cdots a_{2023}$，使得只要 $\tan 2023x$ 定义，则 \[\tan2023x = \frac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + \cdots + a_{2023} \tan^{2023} x}{1 + a_2 \tan^2 x + a_4 \tan^4 x \cdots + a_{2022} \tan^{2022} x}\]。求 $a_{2023}$？
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