/

AMC10 2022 B

You are not logged in. After submit, your report may not be available on other devices. Login

AMC10 · 2022 (B)

Q1
Define $x\diamond y$ to be $|x-y|$ for all real numbers $x$ and $y.$ What is the value of \[(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?\]
定义对于所有实数$x$和$y$,$x\diamond y=|x-y|$。$(1\diamond(2\diamond3))-((1\diamond2)\diamond3)$的值是多少?
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?
stem
Q3
How many three-digit positive integers have an odd number of even digits?
有多少个三位正整数具有奇数个偶数数字?
Q4
A donkey suffers an attack of hiccups and the first hiccup happens at $4:00$ one afternoon. Suppose that the donkey hiccups regularly every $5$ seconds. At what time does the donkey’s $700$th hiccup occur?
一头驴打嗝,第一声嗝发生在下午4:00。假设驴每5秒规律地打一次嗝。驴的第700次嗝发生在几点?
Q5
What is the value of \[\frac{\left(1+\frac13\right)\left(1+\frac15\right)\left(1+\frac17\right)}{\sqrt{\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{5^2}\right)\left(1-\frac{1}{7^2}\right)}}?\]
计算\[\frac{\left(1+\frac13\right)\left(1+\frac15\right)\left(1+\frac17\right)}{\sqrt{\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{5^2}\right)\left(1-\frac{1}{7^2}\right)}}\]的值?
Q6
How many of the first ten numbers of the sequence $121, 11211, 1112111, \ldots$ are prime numbers?
数列 $121, 11211, 1112111, \ldots$ 的前十项中有多少项是质数?
Q7
For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?
常数 $k$ 有多少个取值,使得多项式 $x^{2}+kx+36$ 有两个不同的整数根?
Q8
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*} 其中有多少个集合恰好包含两个 $7$ 的倍数?
Q9
The sum \[\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\cdots+\frac{2021}{2022!}\] can be expressed as $a-\frac{1}{b!}$, where $a$ and $b$ are positive integers. What is $a+b$?
和 \[\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\cdots+\frac{2021}{2022!}\] 可以表示为 $a-\frac{1}{b!}$,其中 $a$ 和 $b$ 是正整数。求 $a+b$?
Q10
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?
Camila 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 $2$。众数的最小可能值为多少?
Q11
All the high schools in a large school district are involved in a fundraiser selling T-shirts. Which of the choices below is logically equivalent to the statement "No school bigger than Euclid HS sold more T-shirts than Euclid HS"?
一个大型学区内所有高中都参与了销售T恤的筹款活动。以下哪个选项与陈述“没有比Euclid高中更大的学校卖出的T恤比Euclid高中多”在逻辑上是等价的?
Q12
A pair of fair $6$-sided dice is rolled $n$ times. What is the least value of $n$ such that the probability that the sum of the numbers face up on a roll equals $7$ at least once is greater than $\frac{1}{2}$?
一对公平的$6$面骰子掷$n$次。求最小的$n$,使得至少一次掷骰和为$7$的概率大于$\frac{1}{2}$。
Q13
The positive difference between a pair of primes is equal to $2$, and the positive difference between the cubes of the two primes is $31106$. What is the sum of the digits of the least prime that is greater than those two primes?
一对质数的正差等于$2$,这两个质数的立方正差等于$31106$。这两个质数之后的最小质数的各位数字之和是多少?
Q14
Suppose that $S$ is a subset of $\left\{ 1, 2, 3, \ldots , 25 \right\}$ such that the sum of any two (not necessarily distinct) elements of $S$ is never an element of $S.$ What is the maximum number of elements $S$ may contain?
设$S$是$\left\{ 1, 2, 3, \ldots , 25 \right\}$的子集,使得$S$中任意两个(不一定不同)元素的和都不在$S$中。$S$最多能包含多少个元素?
Q15
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence that has a common difference of $2$. The quotient $\frac{S_{3n}}{S_n}$ does not depend on $n$. What is $S_{20}$?
设$S_n$是一个公差为$2$的等差数列的前$n$项和。商$\frac{S_{3n}}{S_n}$不依赖于$n$。求$S_{20}$。
Q16
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分别为$4$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。平方与矩形交集区域的面积是多少?
stem
Q17
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?
Q18
Consider systems of three linear equations with unknowns $x$, $y$, and $z$, \begin{align*} a_1 x + b_1 y + c_1 z & = 0 \\ a_2 x + b_2 y + c_2 z & = 0 \\ a_3 x + b_3 y + c_3 z & = 0 \end{align*} where each of the coefficients is either $0$ or $1$ and the system has a solution other than $x=y=z=0$. For example, one such system is \[\{ 1x + 1y + 0z = 0, 0x + 1y + 1z = 0, 0x + 0y + 0z = 0 \}\] with a nonzero solution of $\{x,y,z\} = \{1, -1, 1\}$. How many such systems of equations are there? (The equations in a system need not be distinct, and two systems containing the same equations in a different order are considered different.)
考虑含有未知数 $x$、$y$ 和 $z$ 的三个线性方程组, \begin{align*} a_1 x + b_1 y + c_1 z & = 0 \\ a_2 x + b_2 y + c_2 z & = 0 \\ a_3 x + b_3 y + c_3 z & = 0 \end{align*} 其中每个系数要么是 $0$ 要么是 $1$,且该方程组有非平凡解(即非 $x=y=z=0$ 的解)。 例如,一个这样的方程组是 \[\{ 1x + 1y + 0z = 0, 0x + 1y + 1z = 0, 0x + 0y + 0z = 0 \}\]\n它有一个非零解 $\{x,y,z\} = \{1, -1, 1\}$。有多少这样的方程组? (方程组中的方程不必互异,且含有相同方程但顺序不同的两个方程组被视为不同。)
Q19
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为空,每个方格最多有八个相邻邻居方格,其中相邻方格共享一条边或一个角。该网格按照以下规则变换: 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构成的边框,包围着一个 $3 \times 3$ 子网格。经过一次变换后,有多少种初始配置会得到变换网格只有一个中心填充方格?(同一配置的旋转和反射被视为不同。)
stem stem
Q20
Let $ABCD$ be a rhombus with $\angle ADC = 46^\circ$. Let $E$ be the midpoint of $\overline{CD}$, and let $F$ be the point on $\overline{BE}$ such that $\overline{AF}$ is perpendicular to $\overline{BE}$. What is the degree measure of $\angle BFC$?
设 $ABCD$ 是一个菱形,$\angle ADC = 46^\circ$。令 $E$ 为 $\overline{CD}$ 的中点,$F$ 为 $\overline{BE}$ 上的点,使得 $\overline{AF}$ 与 $\overline{BE}$ 垂直。求 $\angle BFC$ 的度数。
Q21
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,使得当 $P(x)$ 被多项式 $x^2 + x + 1$ 除时,余数是 $x+2$,当 $P(x)$ 被多项式 $x^2+1$ 除时,余数是 $2x+1$。存在一个唯一的最低次数的多项式具有这两个性质。该多项式的系数平方和是多少?
Q22
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 $S$ 为坐标平面中与圆 $x^{2}+y^{2}=4$、$x^{2}+y^{2}=64$ 和 $(x-5)^{2}+y^{2}=3$ 各相切的圆的集合。$S$ 中所有圆面积的和是多少?
Q23
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 $0$ 开始,按照以下方式爬行。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 $x_n$。在过程的第 $n$ 步中,Amelia 正向移动 $x_n$ 个单位,使用 $t_n$ 分钟。如果总经过时间在第 $n$ 步期间超过 $1$ 分钟,她在那一步结束时停止;否则,她继续下一步,总共最多 $3$ 步。Amelia 停止时位置大于 $1$ 的概率是多少?
Q24
Consider functions $f$ that satisfy \[|f(x)-f(y)|\leq \frac{1}{2}|x-y|\] for all real numbers $x$ and $y$. Of all such functions that also satisfy the equation $f(300) = f(900)$, what is the greatest possible value of \[f(f(800))-f(f(400))?\]
考虑满足 \[|f(x)-f(y)|\leq \frac{1}{2}|x-y|\] 对所有实数 $x$ 和 $y$ 成立的函数 $f$。在所有也满足方程 $f(300) = f(900)$ 的此类函数中,\[f(f(800))-f(f(400))?\] 的最大可能值是多少?
Q25
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 $x_k$ 要么是 $0$ 要么是 $1$。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。
Time Left 75:00