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AMC12 2022 B

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AMC12 · 2022 (B)

Q1
Define $x\diamond y$ to be $|x-y|$ for all real numbers $x$ and $y.$ What is the value of \[(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?\]
定义 $x\diamond y$ 为所有实数 $x$ 和 $y$ 的 $|x-y|$。$(1\diamond(2\diamond3))-((1\diamond2)\diamond3)$ 的值为多少？
Q2
In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ so that $\overline{BP} \perp \overline{AD}$, $AP = 3$, and $PD = 2$. What is the area of rhombus $ABCD$?
在菱形 $ABCD$ 中，点 $P$ 在线段 $\overline{AD}$ 上，使得 $\overline{BP} \perp \overline{AD}$，$AP = 3$，$PD = 2$。菱形 $ABCD$ 的面积是多少？
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Q3
How many of the first ten numbers of the sequence $121, 11211, 1112111, \ldots$ are prime numbers?
数列 $121, 11211, 1112111, \ldots$ 的前十项中有多少是质数？
Q4
For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?
常数 $k$ 有多少个值，使得多项式 $x^{2}+kx+36$ 有两个不同的整数根？
Q5
The point $(-1, -2)$ is rotated $270^{\circ}$ counterclockwise about the point $(3, 1)$. What are the coordinates of its new position?
点 $(-1, -2)$ 绕点 $(3, 1)$ 逆时针旋转 $270^{\circ}$。其新位置的坐标是多少？
Q6
Consider the following $100$ sets of $10$ elements each: \begin{align*} &\{1,2,3,\ldots,10\}, \\ &\{11,12,13,\ldots,20\},\\ &\{21,22,23,\ldots,30\},\\ &\vdots\\ &\{991,992,993,\ldots,1000\}. \end{align*} How many of these sets contain exactly two multiples of $7$?
考虑以下100个每个包含10个元素的集合： \begin{align*} &\{1,2,3,\ldots,10\}, \\ &\{11,12,13,\ldots,20\},\\ &\{21,22,23,\ldots,30\},\\ &\vdots\\ &\{991,992,993,\ldots,1000\}. \end{align*} 这些集合中有多少个包含恰好两个7的倍数？
Q7
Camila writes down five positive integers. The unique mode of these integers is $2$ greater than their median, and the median is $2$ greater than their arithmetic mean. What is the least possible value for the mode?
卡米拉写下五个正整数。这些整数的唯一众数比它们的中位数大2，中位数比它们的算术平均数大2。众数的最小可能值是多少？
Q8
What is the graph of $y^4+1=x^4+2y^2$ in the coordinate plane?
坐标平面中$y^4+1=x^4+2y^2$的图像是什么？
Q9
The sequence $a_0,a_1,a_2,\cdots$ is a strictly increasing arithmetic sequence of positive integers such that \[2^{a_7}=2^{27} \cdot a_7.\] What is the minimum possible value of $a_2$?
数列$a_0,a_1,a_2,\cdots$是一个严格递增的正整数等差数列，满足 \[2^{a_7}=2^{27} \cdot a_7.\] $a_2$的最小可能值是多少？
Q10
Regular hexagon $ABCDEF$ has side length $2$. Let $G$ be the midpoint of $\overline{AB}$, and let $H$ be the midpoint of $\overline{DE}$. What is the perimeter of $GCHF$?
正六边形$ABCDEF$边长为$2$。设$G$为$\overline{AB}$的中点，$H$为$\overline{DE}$的中点。$GCHF$的周长是多少？
Q11
Let $f(n) = \left( \frac{-1+i\sqrt{3}}{2} \right)^n + \left( \frac{-1-i\sqrt{3}}{2} \right)^n$, where $i = \sqrt{-1}$. What is $f(2022)$?
设 $f(n) = \left( \frac{-1+i\sqrt{3}}{2} \right)^n + \left( \frac{-1-i\sqrt{3}}{2} \right)^n$，其中 $i = \sqrt{-1}$。$f(2022)$ 等于多少？
Q12
Kayla rolls four fair $6$-sided dice. What is the probability that at least one of the numbers Kayla rolls is greater than $4$ and at least two of the numbers she rolls are greater than $2$?
凯拉掷四个公平的 $6$ 面骰子。掷出至少一个数字大于 $4$ 且至少两个数字大于 $2$ 的概率是多少？
Q13
The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?
下图显示了一个边长为 $4$ 和 $8$ 的矩形和一个边长为 $5$ 的正方形。正方形的三个顶点位于矩形的三个不同边上，如图所示。平方与矩形重叠区域的面积是多少？
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Q14
The graph of $y=x^2+2x-15$ intersects the $x$-axis at points $A$ and $C$ and the $y$-axis at point $B$. What is $\tan(\angle ABC)$?
$y=x^2+2x-15$ 的图像与 $x$ 轴相交于点 $A$ 和 $C$，与 $y$ 轴相交于点 $B$。$\tan(\angle ABC)$ 是多少？
Q15
One of the following numbers is not divisible by any prime number less than $10.$ Which is it?
以下数字中有一个不能被小于 $10$ 的任何质数整除。是哪一个？
Q16
Suppose $x$ and $y$ are positive real numbers such that \[x^y=2^{64}\text{ and }(\log_2{x})^{\log_2{y}}=2^{7}.\] What is the greatest possible value of $\log_2{y}$?
假设 $x$ 和 $y$ 是正实数，使得 \[x^y=2^{64}\text{ 且 }(\log_2{x})^{\log_2{y}}=2^{7}.\] $\log_2{y}$ 的最大可能值为多少？
Q17
How many $4 \times 4$ arrays whose entries are $0$s and $1$s are there such that the row sums (the sum of the entries in each row) are $1, 2, 3,$ and $4,$ in some order, and the column sums (the sum of the entries in each column) are also $1, 2, 3,$ and $4,$ in some order? For example, the array \[\left[ \begin{array}{cccc} 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ \end{array} \right]\] satisfies the condition.
有且仅有 $1,2,3,4$（顺序任意）的行和，以及列和为 $1,2,3,4$（顺序任意）的 $4 \times 4$ 由 $0$ 和 $1$ 组成的阵列有多少个？例如阵列 \[\left[ \begin{array}{cccc} 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ \end{array} \right]\] 满足条件。
Q18
Each square in a $5 \times 5$ grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules: A sample transformation is shown in the figure below. Suppose the $5 \times 5$ grid has a border of empty squares surrounding a $3 \times 3$ subgrid. How many initial configurations will lead to a transformed grid consisting of a single filled square in the center after a single transformation? (Rotations and reflections of the same configuration are considered different.)
一个 $5 \times 5$ 网格中的每个方格要么填充要么空，且每个方格最多有八个相邻邻居，相邻方格共享边或角。网格按以下规则变换： 下图显示了一个变换示例。 假设 $5 \times 5$ 网格有一个由空方格包围的 $3 \times 3$ 子网格。经过一次变换后，得到仅中心有一个填充方格的变换网格的初始配置有多少个？（同一配置的旋转和反射视为不同。）
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Q19
In $\triangle ABC$ medians $\overline{AD}$ and $\overline{BE}$ intersect at $G$ and $\triangle AGE$ is equilateral. Then $\cos(C)$ can be written as $\dfrac{m\sqrt{p}}{n}$, where $m$ and $n$ are relatively prime positive integers and $p$ is a positive integer not divisible by the square of any prime. What is $m+n+p$?
在$\triangle ABC$中，中线$\overline{AD}$与$\overline{BE}$相交于$G$，且$\triangle AGE$是等边三角形。则$\cos(C)$可表示为$\dfrac{m\sqrt{p}}{n}$，其中$m,n$为互质的正整数，$p$为不被任何素数的平方整除的正整数。求$m+n+p$。
Q20
Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1$, the remainder is $x+2$, and when $P(x)$ is divided by the polynomial $x^2+1$, the remainder is $2x+1$. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial?
设 $P(x)$ 是一个具有有理系数的多项式，当 $P(x)$ 被多项式 $x^2 + x + 1$ 除时余数为 $x+2$，被 $x^2+1$ 除时余数为 $2x+1$。具有这两个性质的最低次数多项式唯一。求该多项式系数的平方和。
Q21
Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$, $x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. What is the sum of the areas of all circles in $S$?
设 $S$ 为坐标平面中与以下三个圆相切的圆的集合：$x^{2}+y^{2}=4$，$x^{2}+y^{2}=64$，和 $(x-5)^{2}+y^{2}=3$。$S$ 中所有圆面积之和是多少？
Q22
Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n=1,2,3,$ Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1).$ During the $n$th step of the process, Amelia moves $x_n$ units in the positive direction, using up $t_n$ minutes. If the total elapsed time has exceeded $1$ minute during the $n$th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most $3$ steps in all. What is the probability that Amelia’s position when she stops will be greater than $1$?
蚂蚁 Amelia 从数轴上的 $0$ 点开始，以以下方式爬行。对于 $n=1,2,3$，Amelia 独立均匀随机地从区间 $(0,1)$ 中选择时间持续时间 $t_n$ 和增量 $x_n$。在第 $n$ 步过程中，Amelia 向正方向移动 $x_n$ 个单位，耗时 $t_n$ 分钟。如果在第 $n$ 步过程中总经过时间超过 $1$ 分钟，她在该步结束时停止；否则，她继续下一步，总共最多 $3$ 步。她停止时的位置大于 $1$ 的概率是多少？
Q23
Let $x_0,x_1,x_2,\dotsc$ be a sequence of numbers, where each $x_k$ is either $0$ or $1$. For each positive integer $n$, define \[S_n = \sum_{k=0}^{n-1} x_k 2^k\] Suppose $7S_n \equiv 1 \pmod{2^n}$ for all $n \geq 1$. What is the value of the sum \[x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}?\]
设 $x_0,x_1,x_2,\dotsc$ 是一个数列，其中每个 $x_k$ 要么为 $0$ 要么为 $1$。对于每个正整数 $n$，定义 \[S_n = \sum_{k=0}^{n-1} x_k 2^k\] 假设对所有 $n \geq 1$，有 $7S_n \equiv 1 \pmod{2^n}$。求和 \[x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}\] 的值。
Q24
The figure below depicts a regular $7$-gon inscribed in a unit circle. What is the sum of the $4$th powers of the lengths of all $21$ of its edges and diagonals?
下面的图形描绘了一个内接于单位圆的正 $7$ 边形。 所有 $21$ 条边和对角线的长度 $4$ 次幂之和是多少？
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Q25
Four regular hexagons surround a square with side length 1, each one sharing an edge with the square, as shown in the figure below. The area of the resulting 12-sided outer nonconvex polygon can be written as $m \sqrt{n} + p$, where $m$, $n$, and $p$ are integers and $n$ is not divisible by the square of any prime. What is $m+n+p$?
四个正六边形围绕着一个边长为 $1$ 的正方形，每个六边形与正方形共享一条边，如下图所示。所得 $12$ 边外部非凸多边形的面积可写成 $m \sqrt{n} + p$，其中 $m$、$n$、$p$ 为整数，且 $n$ 任何质数的平方都不整除 $n$。 $m+n+p$ 是多少？
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