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AMC12 2021 B

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AMC12 · 2021 (B)

Q1
How many integer values of $x$ satisfy $|x|<3\pi$?
有几个整数 $x$ 满足 $|x|<3\pi$？
Q2
At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts?
在一次数学竞赛中，有 $57$ 名学生穿着蓝色衬衫，另外 $75$ 名学生穿着黄色衬衫。这 $132$ 名学生被分成 $66$ 对。在这些对中恰好有 $23$ 对两个学生都穿着蓝色衬衫。那么，有几对两个学生都穿着黄色衬衫？
Q3
Suppose\[2+\frac{1}{1+\frac{1}{2+\frac{2}{3+x}}}=\frac{144}{53}.\]What is the value of $x?$
设\[2+\frac{1}{1+\frac{1}{2+\frac{2}{3+x}}}=\frac{144}{53}.\]求 $x$ 的值。
Q4
Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is $84$, and the afternoon class's mean score is $70$. The ratio of the number of students in the morning class to the number of students in the afternoon class is $\frac{3}{4}$. What is the mean of the scores of all the students?
布莱克威尔女士给两个班级的学生出了一次考试。上午班学生的成绩平均分为 $84$，下午班的平均分为 $70$。上午班学生人数与下午班学生人数之比为 $\frac{3}{4}$。所有学生的成绩平均分是多少？
Q5
The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^\circ$ around the point $(1,5)$ and then reflected about the line $y = -x$. The image of $P$ after these two transformations is at $(-6,3)$. What is $b - a ?$
平面直角坐标系中，点 $P(a,b)$ 先绕点 $(1,5)$ 逆时针旋转 $90^\circ$，然后关于直线 $y = -x$ 作反射。经过这两个变换后，$P$ 的像位于 $(-6,3)$。求 $b - a$ 的值。
Q6
An inverted cone with base radius $12 \mathrm{cm}$ and height $18 \mathrm{cm}$ is full of water. The water is poured into a tall cylinder whose horizontal base has radius of $24 \mathrm{cm}$. What is the height in centimeters of the water in the cylinder?
一个底面半径为$12 \mathrm{cm}$、高$18 \mathrm{cm}$的倒锥体装满了水。水被倒入一个高圆柱体中，该圆柱体的水平底面半径为$24 \mathrm{cm}$。圆柱体中水的液面高度是多少厘米？
Q7
Let $N = 34 \cdot 34 \cdot 63 \cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$?
设$N = 34 \cdot 34 \cdot 63 \cdot 270$。$N$的奇约数之和与偶约数之和的比值为多少？
Q8
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38,38,$ and $34$. What is the distance between two adjacent parallel lines?
三条等间距的平行线与一个圆相交，形成三条弦，长分别为$38,38,$和$34$。两条相邻平行线之间的距离是多少？
Q9
What is the value of\[\frac{\log_2 80}{\log_{40}2}-\frac{\log_2 160}{\log_{20}2}?\]
求\[\frac{\log_2 80}{\log_{40}2}-\frac{\log_2 160}{\log_{20}2}\]的值。
Q10
Two distinct numbers are selected from the set $\{1,2,3,4,\dots,36,37\}$ so that the sum of the remaining $35$ numbers is the product of these two numbers. What is the difference of these two numbers?
从集合$\{1,2,3,4,\dots,36,37\}$中选出两个不同的数，使得剩余$35$个数的和等于这两个数的乘积。这两个数的差是多少？
Q11
Triangle $ABC$ has $AB=13,BC=14$ and $AC=15$. Let $P$ be the point on $\overline{AC}$ such that $PC=10$. There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. What is the distance $DE?$
三角形 $ABC$ 有 $AB=13,BC=14$ 和 $AC=15$。让 $P$ 为 $\overline{AC}$ 上的点，使得 $PC=10$。线 $BP$ 上恰有两点 $D$ 和 $E$，使得四边形 $ABCD$ 和 $ABCE$ 是梯形。$DE$ 的距离是多少？
Q12
Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer in $S$ is also removed, then the average value of the integers remaining is $35$. If the greatest integer is then returned to the set, the average value of the integers rises to $40$. The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$. What is the average value of all the integers in the set $S$?
假设 $S$ 是一个有限的正整数集。如果从 $S$ 中移除 $S$ 中的最大整数，则剩余整数的平均值（算术平均）为 $32$。如果再移除 $S$ 中的最小整数，则剩余整数的平均值为 $35$。如果将最大整数放回集合，则整数的平均值上升到 $40$。原集合 $S$ 中的最大整数比最小整数大 $72$。集合 $S$ 中所有整数的平均值是多少？
Q13
How many values of $\theta$ in the interval $0<\theta\le 2\pi$ satisfy \[1-3\sin\theta+5\cos3\theta = 0?\]
区间 $0<\theta\le 2\pi$ 内有多少个 $\theta$ 满足 \[1-3\sin\theta+5\cos3\theta = 0?\]
Q14
Let $ABCD$ be a rectangle and let $\overline{DM}$ be a segment perpendicular to the plane of $ABCD$. Suppose that $\overline{DM}$ has integer length, and the lengths of $\overline{MA},\overline{MC},$ and $\overline{MB}$ are consecutive odd positive integers (in this order). What is the volume of pyramid $MABCD?$
设 $ABCD$ 为矩形，$\overline{DM}$ 是垂直于 $ABCD$ 平面的线段。假设 $\overline{DM}$ 长度为整数，且 $\overline{MA},\overline{MC},$ 和 $\overline{MB}$ 的长度依次为连续的奇正整数（按此顺序）。金字塔 $MABCD$ 的体积是多少？
Q15
The figure is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written as $\sqrt{m} + \sqrt{n}$, where $m$ and $n$ are positive integers. What is $m + n ?$
该图形由 $11$ 条长度均为 $2$ 的线段构成。五边形 $ABCDE$ 的面积可写为 $\sqrt{m} + \sqrt{n}$，其中 $m$ 和 $n$ 是正整数。$m + n$ 是多少？
stem
Q16
Let $g(x)$ be a polynomial with leading coefficient $1,$ whose three roots are the reciprocals of the three roots of $f(x)=x^3+ax^2+bx+c,$ where $1<a<b<c.$ What is $g(1)$ in terms of $a,b,$ and $c?$
设 $g(x)$ 是一个首项系数为 $1$ 的多项式，其三个根是 $f(x)=x^3+ax^2+bx+c$ 的三个根的倒数，其中 $1<a<b<c$。$g(1)$ 用 $a,b,c$ 表示是什么？
Q17
Let $ABCD$ be an isosceles trapezoid having parallel bases $\overline{AB}$ and $\overline{CD}$ with $AB>CD.$ Line segments from a point inside $ABCD$ to the vertices divide the trapezoid into four triangles whose areas are $2, 3, 4,$ and $5$ starting with the triangle with base $\overline{CD}$ and moving clockwise as shown in the diagram below. What is the ratio $\frac{AB}{CD}?$
设 $ABCD$ 是一个等腰梯形，具有平行底边 $\overline{AB}$ 和 $\overline{CD}$，且 $AB>CD$。从梯形内部一点到顶点的线段将梯形分成四个三角形，其面积分别为 $2, 3, 4,$ 和 $5$，从底边 $\overline{CD}$ 的三角形开始顺时针方向如图所示。$ rac{AB}{CD}$ 的比值为多少？
stem
Q18
Let $z$ be a complex number satisfying $12|z|^2=2|z+2|^2+|z^2+1|^2+31.$ What is the value of $z+\frac 6z?$
设 $z$ 是一个满足 $12|z|^2=2|z+2|^2+|z^2+1|^2+31$ 的复数。$z+\frac 6z$ 的值为多少？
Q19
Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\frac34$ of the probability of rolling a sum of $10,$ and the probability of rolling a sum of $12$ is $\frac{1}{12}$. What is the least possible number of faces on the two dice combined?
掷两个公平的骰子，每个骰子至少有 $6$ 个面。每个骰子的每个面上印有从 $1$ 到该骰子面数的不同整数。掷出和为 $7$ 的概率是掷出和为 $10$ 的概率的 $\frac34$，掷出和为 $12$ 的概率是 $\frac{1}{12}$。两骰子面数总和的最小可能值为多少？
Q20
Let $Q(z)$ and $R(z)$ be the unique polynomials such that\[z^{2021}+1=(z^2+z+1)Q(z)+R(z)\]and the degree of $R$ is less than $2.$ What is $R(z)?$
设 $Q(z)$ 和 $R(z)$ 是唯一多项式，使得\[z^{2021}+1=(z^2+z+1)Q(z)+R(z)\]且 $R$ 的度数小于 $2$。$R(z)$ 是什么？
Q21
Let $S$ be the sum of all positive real numbers $x$ for which\[x^{2^{\sqrt2}}=\sqrt2^{2^x}.\]Which of the following statements is true?
设 $S$ 是所有满足 \[x^{2^{\sqrt2}}=\sqrt2^{2^x}\] 的正实数 $x$ 的和。以下哪项陈述正确？
Q22
Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by one move: $(3,2),(2,1,2),(4),(4,1),(2,2),$ or $(1,1,2).$ Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?
Arjun 和 Beth 玩一个游戏，他们轮流从一组砖墙中移除一块砖或两块相邻的砖，移除来自同一“墙”，间隙可能创建新墙。墙高为一砖。例如，大小为 $4$ 和 $2$ 的墙组可以通过一步移动变为以下之一：$(3,2),(2,1,2),(4),(4,1),(2,2)$ 或 $(1,1,2)$。 Arjun 先手，取走最后一块砖的玩家获胜。对于以下哪种起始配置，Beth 有必胜策略？
stem
Q23
Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $i$ is $2^{-i}$ for $i=1,2,3,....$ More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is $\frac pq,$ where $p$ and $q$ are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins $3,17,$ and $10.$) What is $p+q?$
三个球被随机且独立地扔入编号为正整数的箱子中，对于每个球，扔入箱子 $i$ 的概率为 $2^{-i}$，$i=1,2,3,....$ 每个箱子允许多个球。球最终落在不同箱子中且均匀间隔的概率为 $\frac pq$，其中 $p$ 和 $q$ 互质正整数。（例如，如果球扔入箱子 $3,17$ 和 $10$，则均匀间隔。）求 $p+q$？
Q24
Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points. Suppose $PQ=6$ and $RS=8,$ and let $d$ denote the length of $\overline{BD},$ the longer diagonal of $ABCD.$ Then $d^2$ can be written in the form $m+n\sqrt p,$ where $m,n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m+n+p?$
设 $ABCD$ 是面积为 $15$ 的平行四边形。点 $P$ 和 $Q$ 分别是 $A$ 和 $C$ 在直线 $BD$ 上的投影；点 $R$ 和 $S$ 分别是 $B$ 和 $D$ 在直线 $AC$ 上的投影。见图，该图还显示了这些点的相对位置。 假设 $PQ=6$ 和 $RS=8$，设 $d$ 表示 $\overline{BD}$ 的长度，即 $ABCD$ 的较长对角线。然后 $d^2$ 可写成 $m+n\sqrt p$ 的形式，其中 $m,n,p$ 是正整数，且 $p$ 没有被任何质数的平方整除。求 $m+n+p$？
stem
Q25
Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positive integers. What is $a+b?$
设 $S$ 是坐标平面中坐标均为 $1$ 到 $30$（包含）整数的格点。恰有 $300$ 个点位于直线 $y=mx$ 上或下方。$m$ 的可能值位于长度为 $\frac ab$ 的区间中，其中 $a$ 和 $b$ 互质正整数。求 $a+b$？
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