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AMC10 2021 B

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AMC10 · 2021 (B)

Q1
What is the value of $1234 + 2341 + 3412 + 4123$
$1234 + 2341 + 3412 + 4123$ 的值是多少？
Q2
What is the area of the shaded figure shown below?
下图所示阴影图形的面积是多少？
stem
Q3
The expression $\frac{2021}{2020} - \frac{2020}{2021}$ is equal to the fraction $\frac{p}{q}$ in which $p$ and $q$ are positive integers whose greatest common divisor is ${ }1$. What is $p?$
表达式 $\frac{2021}{2020} - \frac{2020}{2021}$ 等于分数 $\frac{p}{q}$，其中 $p$ 和 $q$ 是正整数，且它们的最大公因数为 ${ }1$。$p$ 是多少？
Q4
At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values of $N?$
在某天中午，明尼阿波利斯比圣路易斯高 $N$ 华氏度。到 $4{:}00$ 时，明尼阿波利斯的气温下降了 $5$ 度，而圣路易斯的气温上升了 $3$ 度，此时两座城市的气温相差 $2$ 度。求 $N$ 的所有可能取值的乘积。
Q5
Let $n=8^{2022}$. Which of the following is equal to $\frac{n}{4}?$
设 $n=8^{2022}$. 以下哪一项等于 $\frac{n}{4}$?
Q6
The least positive integer with exactly $2021$ distinct positive divisors can be written in the form $m \cdot 6^k$, where $m$ and $k$ are integers and $6$ is not a divisor of $m$. What is $m+k?$
恰好有 $2021$ 个不同正因数的最小正整数可以写成 $m \cdot 6^k$ 的形式，其中 $m$ 和 $k$ 为整数，且 $6$ 不是 $m$ 的因数。求 $m+k$。
Q7
Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?
称分数 $\frac{a}{b}$（不一定是最简形式）为特殊分数，如果 $a$ 和 $b$ 是和为 $15$ 的正整数。有多少个不同的整数可以表示为两个（不一定不同的）特殊分数之和？
Q8
The greatest prime number that is a divisor of $16{,}384$ is $2$ because $16{,}384 = 2^{14}$. What is the sum of the digits of the greatest prime number that is a divisor of $16{,}383$?
$16{,}384$ 的最大素因数是 $2$，因为 $16{,}384 = 2^{14}$。$16{,}383$ 的最大素因数的各位数字之和是多少？
Q9
The knights in a certain kingdom come in two colors. $\frac{2}{7}$ of them are red, and the rest are blue. Furthermore, $\frac{1}{6}$ of the knights are magical, and the fraction of red knights who are magical is $2$ times the fraction of blue knights who are magical. What fraction of red knights are magical?
某个王国的骑士有两种颜色。它们中有 $\frac{2}{7}$ 是红色，其余是蓝色。此外，有 $\frac{1}{6}$ 的骑士是魔法骑士，并且红色骑士中是魔法骑士的比例是蓝色骑士中是魔法骑士的比例的 $2$ 倍。红色骑士中有多少比例是魔法骑士？
Q10
Forty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, "I can't tell who has the larger number." Then Bob says, "I know who has the larger number." Alice says, "You do? Is your number prime?" Bob replies, "Yes." Alice says, "In that case, if I multiply your number by $100$ and add my number, the result is a perfect square. " What is the sum of the two numbers drawn from the hat?
将编号为 $1$ 到 $40$ 的四十张纸条放入一顶帽子中。Alice 和 Bob 各自从帽子里不放回地抽取一个号码，并且彼此对自己的号码保密。Alice 说：“我无法判断谁的号码更大。” 然后 Bob 说：“我知道谁的号码更大。” Alice 说：“你知道？你的号码是质数吗？” Bob 回答：“是。” Alice 说：“在这种情况下，如果我把你的号码乘以 $100$ 再加上我的号码，结果是一个完全平方数。” 从帽子里抽出的两个号码之和是多少？
Q11
A regular hexagon of side length $1$ is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the region bounded by these $6$ reflected arcs?
边长为 $1$ 的正六边形内接于一个圆。由六边形的一条边所确定的圆的每一段小弧都关于该边作对称反射。由这 $6$ 条反射后的弧所围成的区域面积是多少？
Q12
Which of the following conditions is sufficient to guarantee that integers $x$, $y$, and $z$ satisfy the equation \[x(x-y)+y(y-z)+z(z-x) = 1?\]
以下哪个条件足以保证整数 $x$, $y$, 和 $z$ 满足方程 \[x(x-y)+y(y-z)+z(z-x) = 1?\]
Q13
A square with side length $3$ is inscribed in an isosceles triangle with one side of the square along the base of the triangle. A square with side length $2$ has two vertices on the other square and the other two on sides of the triangle, as shown. What is the area of the triangle?
边长为 $3$ 的正方形内接于一个等腰三角形中，且正方形的一条边与三角形的底边重合。边长为 $2$ 的正方形有两个顶点在另一个正方形上，另外两个顶点在三角形的两条边上，如图所示。这个三角形的面积是多少？
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Q14
Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6{ }$ numbers obtained. What is the probability that the product is divisible by $4?$
Una 同时掷 $6$ 个标准的 $6$ 面骰子，并计算得到的 $6$ 个数的乘积。这个乘积能被 $4$ 整除的概率是多少？
Q15
I love Bayes ThereomIn square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$, respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at right angles at $R$, with $BR = 6$ and $PR = 7$. What is the area of the square?
我爱贝叶斯定理 在正方形 $ABCD$ 中，点 $P$ 和 $Q$ 分别位于 $\overline{AD}$ 和 $\overline{AB}$ 上。线段 $\overline{BP}$ 与 $\overline{CQ}$ 在点 $R$ 处相交且成直角，并且 $BR = 6$、$PR = 7$。这个正方形的面积是多少？
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Q16
Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original positions after these two successive transpositions?
五个球围成一圈排列。Chris 随机选择两个相邻的球并交换它们的位置。然后 Silva 也做同样的操作，她选择要交换的相邻球对与 Chris 的选择相互独立。经过这两次连续的交换后，仍占据其原来位置的球的期望个数是多少？
Q17
Distinct lines $\ell$ and $m$ lie in the $xy$-plane. They intersect at the origin. Point $P(-1, 4)$ is reflected about line $\ell$ to point $P'$, and then $P'$ is reflected about line $m$ to point $P''$. The equation of line $\ell$ is $5x - y = 0$, and the coordinates of $P''$ are $(4,1)$. What is the equation of line $m?$
不同的直线 $\ell$ 和 $m$ 位于 $xy$ 平面内。它们在原点相交。点 $P(-1, 4)$ 关于直线 $\ell$ 反射到点 $P'$，然后 $P'$ 再关于直线 $m$ 反射到点 $P''$。直线 $\ell$ 的方程为 $5x - y = 0$，且 $P''$ 的坐标为 $(4,1)$。求直线 $m$ 的方程？
Q18
Three identical square sheets of paper each with side length $6{ }$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is $a+b+c?$
三张完全相同的正方形纸片，每张边长均为 $6{ }$，叠放在一起。中间那张以其中心为旋转中心顺时针旋转 $30^\circ$，最上面那张以其中心为旋转中心顺时针旋转 $60^\circ$，从而得到下图所示的 $24$ 边形。该多边形的面积可以表示为 $a-b\sqrt{c}$ 的形式，其中 $a$、$b$、$c$ 为正整数，且 $c$ 不被任何素数的平方整除。求 $a+b+c$ 的值是多少？
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Q19
Let $N$ be the positive integer $7777\ldots777$, a $313$-digit number where each digit is a $7$. Let $f(r)$ be the leading digit of the $r{ }$th root of $N$. What is\[f(2) + f(3) + f(4) + f(5)+ f(6)?\]
设 $N$ 为正整数 $7777\ldots777$，这是一个 $313$ 位数且每一位都是 $7$。令 $f(r)$ 为 $N$ 的第 $r{ }$ 次方根的首位数字。求\[f(2) + f(3) + f(4) + f(5)+ f(6)?\]
Q20
In a particular game, each of $4$ players rolls a standard $6{ }$-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again and this process will continue until one player wins. Hugo is one of the players in this game. What is the probability that Hugo's first roll was a $5,$ given that he won the game?
在某个游戏中，$4$ 名玩家各掷一次标准的 $6{ }$ 面骰子。掷出点数最高的玩家获胜。如果最高点数出现并列，则参与并列的玩家再次掷骰，如此重复直到有一名玩家获胜。Hugo 是该游戏的玩家之一。已知 Hugo 赢得了游戏，求 Hugo 第一次掷出的点数是 $5,$ 的概率。
Q21
Regular polygons with $5,6,7,$ and $8$ sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?
边数分别为 $5,6,7,$ 和 $8$ 的正多边形内接于同一个圆。任意两个多边形不共用顶点，并且它们的任意三条边不在同一点相交。在圆内有多少个点是两条边的交点？
Q22
For each integer $n\geq 2$, let $S_n$ be the sum of all products $jk$, where $j$ and $k$ are integers and $1\leq j<k\leq n$. What is the sum of the 10 least values of $n$ such that $S_n$ is divisible by $3$?
对每个整数 $n\geq 2$，令 $S_n$ 为所有乘积 $jk$ 的和，其中 $j$ 和 $k$ 为整数且 $1\leq j<k\leq n$。满足 $S_n$ 能被 $3$ 整除的 $n$ 的最小的 10 个值之和是多少？
Q23
Each of the $5{ }$ sides and the $5{ }$ diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides have the same color?
正五边形的 $5{ }$ 条边和 $5{ }$ 条对角线分别以相同概率随机且相互独立地被染成红色或蓝色。存在一个三角形，其顶点取自五边形的顶点，并且该三角形的三条边颜色都相同的概率是多少？
Q24
A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)
一个立方体由 $4$ 个白色单位立方体和 $4$ 个蓝色单位立方体构成。用这些小立方体搭建 $2 \times 2 \times 2$ 立方体共有多少种不同的方法？（如果一种搭建方式可以通过旋转与另一种重合，则认为它们相同。）
Q25
A rectangle with side lengths $1{ }$ and $3,$ a square with side length $1,$ and a rectangle $R$ are inscribed inside a larger square as shown. The sum of all possible values for the area of $R$ can be written in the form $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n?$
如图所示，一个边长为 $1{ }$ 和 $3,$ 的长方形、一个边长为 $1,$ 的正方形以及一个长方形 $R$ 内接于一个更大的正方形中。$R$ 的面积所有可能取值之和可以写成 $\tfrac mn$ 的形式，其中 $m$ 和 $n$ 是互质的正整数。求 $m+n$。
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