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AMC10 2021 B

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AMC10 · 2021 (B)

Q1
How many integer values of $x$ satisfy $|x|<3\pi$?
有几个整数值 $x$ 满足 $|x|<3\pi$？
Q2
What is the value of $\sqrt{\left(3-2\sqrt{3}\right)^2}+\sqrt{\left(3+2\sqrt{3}\right)^2}$?
求 $\sqrt{\left(3-2\sqrt{3}\right)^2}+\sqrt{\left(3+2\sqrt{3}\right)^2}$ 的值。
Q3
In an after-school program for juniors and seniors, there is a debate team with an equal number of students from each class on the team. Among the $28$ students in the program, $25\%$ of the juniors as a class and $10\%$ of the seniors as a class are on the debate team. How many juniors are in the program?
在一个为高二和高三学生开设的课后项目中，辩论队中有来自每个年级的学生数量相等。该项目共有 $28$ 名学生，高二年级学生的 $25\%$ 和高三年级学生的 $10\%$ 在辩论队中。项目中有多少高二学生？
Q4
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?
在一场数学竞赛中，有 $57$ 名学生穿蓝色衬衫，另外 $75$ 名学生穿黄色衬衫。这 $132$ 名学生被分成 $66$ 对。在恰好 $23$ 对中，两名学生都穿蓝色衬衫。有多少对中两名学生都穿黄色衬衫？
Q5
The ages of Jonie's four cousins are distinct single-digit positive integers. Two of the cousins' ages multiplied together give $24$, while the other two multiply to $30$. What is the sum of the ages of Jonie's four cousins?
Jonie 的四个堂兄弟的年龄是不同的个位正整数。其中两个堂兄弟的年龄相乘得 $24$，另外两个相乘得 $30$。Jonie 四个堂兄弟年龄的总和是多少？
Q6
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?
布莱克韦尔女士给两个班级的学生出了一场考试。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？
Q7
In a plane, four circles with radii $1,3,5,$ and $7$ are tangent to line $\ell$ at the same point $A,$ but they may be on either side of $\ell$. Region $S$ consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region $S$?
在平面上一条直线$\ell$上有四个半径分别为$1,3,5,$和$7$的圆，它们在同一点$A$处与直线$\ell$相切，但可以位于$\ell$的两侧。区域$S$由位于恰好一个圆内部的所有点组成。$S$区域的最大可能面积是多少？
Q8
Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest number and the least number that appear in the second row from the top?
周先生将整数从$1$到$225$放入一个$15$乘$15$的网格中。他将$1$放在中间的方格（第八行第八列），然后顺时针逐一放置其他数字，如下图部分所示。从顶部第二行中出现的最小数和最大数的和是多少？
stem
Q9
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 $b - a ?$
平面上的点$P(a,b)$先绕点$(1,5)$逆时针旋转$90^\circ$，然后关于直线$y = -x$反射。$P$经过这两个变换后的像位于$(-6,3)$。$b - a$的值是多少？
Q10
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$12 \mathrm{cm}$、高$18 \mathrm{cm}$的倒锥体盛满水。水被倒入一个高圆柱体中，该圆柱体的底面半径为$24 \mathrm{cm}$。圆柱体中水的液面高度（厘米）是多少？
Q11
Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with straight cuts parallel to the sides of the pan. Each cut must be made entirely across the pan. Grandma wants to make the same number of interior pieces as pieces along the perimeter of the pan. What is the greatest possible number of brownies she can produce?
奶奶刚刚烤好了一大块长方形布朗尼。她计划切成相同大小和形状的长方形小块，使用平行于锅边的直切。每刀必须完全横切锅。奶奶希望内部小块的数量与锅边缘小块的数量相同。她能制作的最大布朗尼数量是多少？
Q12
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？
Q13
Let $n$ be a positive integer and $d$ be a digit such that the value of the numeral $\underline{32d}$ in base $n$ equals $263$, and the value of the numeral $\underline{324}$ in base $n$ equals the value of the numeral $\underline{11d1}$ in base six. What is $n + d ?$
设$n$为正整数，$d$为一个数字，使得$n$进制数$\underline{32d}$的值等于263，且$n$进制数$\underline{324}$的值等于6进制数$\underline{11d1}$的值。求$n + d$？
Q14
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?
三条等距平行线与一个圆相交，形成三条长度分别为38、38和34的弦。相邻两条平行线之间的距离是多少？
Q15
The real number $x$ satisfies the equation $x+\frac{1}{x} = \sqrt{5}$. What is the value of $x^{11}-7x^{7}+x^3?$
实数$x$满足方程$x+\frac{1}{x} = \sqrt{5}$。求$x^{11}-7x^{7}+x^3$的值？
Q16
Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, $1357, 89,$ and $5$ are all uphill integers, but $32, 1240,$ and $466$ are not. How many uphill integers are divisible by $15$?
称一个正整数为上坡整数，如果每个数字严格大于前一个数字。例如，$1357, 89,$ 和 $5$ 都是上坡整数，但 $32, 1240,$ 和 $466$ 不是。有多少个上坡整数能被 $15$ 整除？
Q17
Ravon, Oscar, Aditi, Tyrone, and Kim play a card game. Each person is given $2$ cards out of a set of $10$ cards numbered $1,2,3, \dots,10.$ The score of a player is the sum of the numbers of their cards. The scores of the players are as follows: Ravon--$11,$ Oscar--$4,$ Aditi--$7,$ Tyrone--$16,$ Kim--$17.$ Which of the following statements is true?
Ravon、Oscar、Aditi、Tyrone 和 Kim 玩一个纸牌游戏。每人从编号为 $1,2,3, \dots,10$ 的 $10$ 张牌中获得 $2$ 张牌。玩家的得分是他们牌上数字之和。玩家的得分为：Ravon--$11,$ Oscar--$4,$ Aditi--$7,$ Tyrone--$16,$ Kim--$17$。以下哪个陈述是正确的？
Q18
A fair $6$-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?
一个公平的 $6$ 面骰子反复掷直到出现奇数。每个偶数在第一次出现奇数之前至少出现一次的概率是多少？
Q19
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合。如果从 $S$ 中移除集合中的最大整数，则剩余整数的平均值（算术平均）是 $32$。如果再移除集合中的最小整数，则剩余整数的平均值是 $35$。如果将最大整数放回集合，则平均值上升到 $40$。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？
Q20
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 $ABCDE$ 的面积可以写成 $\sqrt{m} + \sqrt{n}$，其中 $m$ 和 $n$ 是正整数。$m + n$ 等于多少？
stem
Q21
A square piece of paper has side length $1$ and vertices $A,B,C,$ and $D$ in that order. As shown in the figure, the paper is folded so that vertex $C$ meets edge $\overline{AD}$ at point $C'$, and edge $\overline{BC}$ intersects edge $\overline{AB}$ at point $E$. Suppose that $C'D = \frac{1}{3}$. What is the perimeter of triangle $\bigtriangleup AEC' ?$
一张边长为$1$的正方形纸片，顶点依次为$A,B,C,D$。如图所示，将纸片折叠使顶点$C$与边$\overline{AD}$上的点$C'$重合，且边$\overline{BC}$与边$\overline{AB}$相交于点$E$。已知$C'D=\frac{1}{3}$。求三角形$\bigtriangleup AEC'$的周长。
stem
Q22
Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n ?$
Ang、Ben和Jasmin各有5块积木，颜色分别为红、蓝、黄、白、绿；有5个空盒子。三人独立随机地将各自的一块积木放入每个盒子中。至少有一个盒子收到3块同色积木的概率为$\frac{m}{n}$，其中$m$和$n$互质。求$m+n$？
Q23
A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\sqrt{2}$ in the center of the square, as shown in the diagram. A circular coin with diameter $1$ is dropped onto the square and lands in a random location where the coin is completely contained within the square. The probability that the coin will cover part of the black region of the square can be written as $\frac{1}{196}\left(a+b\sqrt{2}+\pi\right)$, where $a$ and $b$ are positive integers. What is $a+b$?
一个边长为$8$的正方形，除四个角各有一个腿长为$2$的黑色的等腰直角三角形区域和正方形中心一个边长为$2\sqrt{2}$的黑色菱形外，其余涂白色，如图所示。将一个直径为$1$的圆形硬币随机丢到正方形上，且硬币完全在正方形内。硬币覆盖部分黑色区域的概率可写为$\frac{1}{196}\left(a+b\sqrt{2}+\pi\right)$，其中$a,b$为正整数。求$a+b$？
stem
Q24
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，他们轮流从几堵砖墙中选择一堵墙，取走一块砖或两块相邻的砖，间隙可能产生新墙。墙高为一砖。例如，尺寸为$4$和$2$的墙组，一步可变为$(3,2),(2,1,2),(4),(4,1),(2,2)$或$(1,1,2)$。 Arjun先手，取走最后一块砖者胜。对于哪种起始配置，Beth有必胜策略？
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$S$为坐标平面中坐标均为1到30（包含）整数的格点。恰有$300$个$S$中的点位于直线$y=mx$上或下方。$m$的可能值位于长度为$\frac ab$的区间，其中$a$和$b$互质。求$a+b$？
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