amcdrill | AMC8 AMC10 AMC12 Practice

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？

(A) {-}2 {-}2 (B) {-}1 {-}1 (C) 0 0 (D) 1 1 (E) 2 2

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？

(A) 3\sqrt{5} 3\sqrt{5} (B) 10 10 (C) 6\sqrt{5} 6\sqrt{5} (D) 20 20 (E) 25 25

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？

(A) 150 150 (B) 250 250 (C) 350 350 (D) 450 450 (E) 550 550

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？

(A) 15 \text{ seconds after } 4:58 15 \text{ seconds after } 4:58 (B) 20 \text{ seconds after } 4:58 20 \text{ seconds after } 4:58 (C) 25 \text{ seconds after } 4:58 25 \text{ seconds after } 4:58 (D) 30 \text{ seconds after } 4:58 30 \text{ seconds after } 4:58 (E) 35 \text{ seconds after } 4:58 35 \text{ seconds after } 4:58

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？

(A) \sqrt3 \sqrt3 (B) 2 2 (C) \sqrt{15} \sqrt{15} (D) 4 4 (E) \sqrt{105} \sqrt{105}

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？

(A) 0 0 (B) 1 1 (C) 2 2 (D) 3 3 (E) 4 4

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？

(A) 6 6 (B) 8 8 (C) 9 9 (D) 14 14 (E) 16 16

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？

(A) 40 40 (B) 42 42 (C) 43 43 (D) 49 49 (E) 50 50

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？

(A) 5 5 (B) 7 7 (C) 9 9 (D) 11 11 (E) 13 13

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？

(A) All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS. All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS. (B) No school that sold more T-shirts than Euclid HS is bigger than Euclid HS. No school that sold more T-shirts than Euclid HS is bigger than Euclid HS. (C) All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS. All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS. (D) All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS. All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS. (E) All schools smaller than Euclid HS sold more T-shirts than Euclid HS. All schools smaller than Euclid HS sold more T-shirts than Euclid HS.

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$\frac{1}{2}$。

(A) 2 2 (B) 3 3 (C) 4 4 (D) 5 5 (E) 6 6

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？

(A) 8 8 (B) 10 10 (C) 11 11 (D) 13 13 (E) 16 16

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？

(A) 12 12 (B) 13 13 (C) 14 14 (D) 15 15 (E) 16 16

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$S_{20}$。

(A) 340 340 (B) 360 360 (C) 380 380 (D) 400 400 (E) 420 420

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？

(A) 15\dfrac{1}{8} 15\dfrac{1}{8} (B) 15\dfrac{3}{8} 15\dfrac{3}{8} (C) 15\dfrac{1}{2} 15\dfrac{1}{2} (D) 15\dfrac{5}{8} 15\dfrac{5}{8} (E) 15\dfrac{7}{8} 15\dfrac{7}{8}

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？

(A) 2^{606}-1 2^{606}-1 (B) 2^{606}+1 2^{606}+1 (C) 2^{607}-1 2^{607}-1 (D) 2^{607}+1 2^{607}+1 (E) 2^{607}+3^{607} 2^{607}+3^{607}

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。）

(A) 302 302 (B) 338 338 (C) 340 340 (D) 343 343 (E) 344 344

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。）

(A) 14 14 (B) 18 18 (C) 22 22 (D) 26 26 (E) 30 30

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。

(A) 110 110 (B) 111 111 (C) 112 112 (D) 113 113 (E) 114 114

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？

(A) 10 10 (B) 13 13 (C) 19 19 (D) 20 20 (E) 23 23

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？

(A) 48\pi 48\pi (B) 68\pi 68\pi (C) 96\pi 96\pi (D) 102\pi 102\pi (E) 136\pi\qquad 136\pi\qquad

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？

(A) \frac{1}{3} \frac{1}{3} (B) \frac{1}{2} \frac{1}{2} (C) \frac{2}{3} \frac{2}{3} (D) \frac{3}{4} \frac{3}{4} (E) \frac{5}{6} \frac{5}{6}

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？

(A) 25 25 (B) 50 50 (C) 100 100 (D) 150 150 (E) 200 200

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。

(A) 6 6 (B) 7 7 (C) 12 12 (D) 14 14 (E) 15 15

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