/

AMC12 2021 B

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

AMC12 · 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
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 $N?$
在某一天的中午，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。
Q4
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 $\frac{n}{4}$?
Q5
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 $\frac{a}{b}$（不一定为最简形式）为特殊分数，如果 $a$ 和 $b$ 是正整数且它们的和为 $15$。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？
Q6
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？
Q7
Which of the following conditions is sufficient to guarantee that integers $x$, $y$, and $z$ satisfy the equation \[x(x-y)+y(y-z)+z(z-x) = 1?\]
以下哪一个条件足以保证整数 $x$, $y$, 和 $z$ 满足方程 \[x(x-y)+y(y-z)+z(z-x) = 1?\]
Q8
The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle's height to the base. What is the measure, in degrees, of the vertex angle of this triangle?
一个钝角等腰三角形的两条全等边的长度乘积，等于底边与该三角形对底边的高的两倍的乘积。求该三角形顶角的度数。
Q9
Triangle $ABC$ is equilateral with side length $6$. Suppose that $O$ is the center of the inscribed circle of this triangle. What is the area of the circle passing through $A$, $O$, and $C$?
等边三角形 $ABC$ 的边长为 $6$。设 $O$ 为该三角形内切圆的圆心。过 $A$、$O$、$C$ 三点的圆的面积是多少？
Q10
What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are \[(\cos 40^\circ,\sin 40^\circ), (\cos 60^\circ,\sin 60^\circ), \text{ and } (\cos t^\circ,\sin t^\circ)\] is isosceles?
在 $0$ 到 $360$ 之间，所有可能的 $t$ 的取值之和是多少，使得坐标平面上顶点为 \[(\cos 40^\circ,\sin 40^\circ), (\cos 60^\circ,\sin 60^\circ), \text{ and } (\cos t^\circ,\sin t^\circ)\] 的三角形是等腰三角形？
Q11
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？
Q12
For $n$ a positive integer, let $f(n)$ be the quotient obtained when the sum of all positive divisors of $n$ is divided by $n.$ For example, \[f(14)=(1+2+7+14)\div 14=\frac{12}{7}\] What is $f(768)-f(384)?$
对于正整数 $n$，令 $f(n)$ 为 $n$ 的所有正因数之和除以 $n$ 所得到的商。例如，\[f(14)=(1+2+7+14)\div 14=\frac{12}{7}\] 求 $f(768)-f(384)$ 的值。
Q13
Let $c = \frac{2\pi}{11}.$ What is the value of \[\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}?\]
设 $c = \frac{2\pi}{11}.$ 下式的值是多少？ \[\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}?\]
Q14
Suppose that $P(z), Q(z)$, and $R(z)$ are polynomials with real coefficients, having degrees $2$, $3$, and $6$, respectively, and constant terms $1$, $2$, and $3$, respectively. Let $N$ be the number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z)=R(z)$. What is the minimum possible value of $N$?
设 $P(z), Q(z)$ 和 $R(z)$ 是实系数多项式，次数分别为 $2$、$3$ 和 $6$，常数项分别为 $1$、$2$ 和 $3$。令 $N$ 为满足方程 $P(z) \cdot Q(z)=R(z)$ 的不同复数 $z$ 的个数。$N$ 的最小可能值是多少？
Q15
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 $a+b+c$? $(
三张边长均为 $6$ 的全等正方形纸片叠放在一起。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 $a+b+c$。 $(
stem
Q16
Suppose $a$, $b$, $c$ are positive integers such that \[a+b+c=23\] and \[\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9.\] What is the sum of all possible distinct values of $a^2+b^2+c^2$?
设 $a$, $b$, $c$ 为正整数，满足 \[a+b+c=23\] 且 \[\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9.\] 所有可能的不同的 $a^2+b^2+c^2$ 的取值之和是多少？
Q17
A bug starts at a vertex of a grid made of equilateral triangles of side length $1$. At each step the bug moves in one of the $6$ possible directions along the grid lines randomly and independently with equal probability. What is the probability that after $5$ moves the bug never will have been more than $1$ unit away from the starting position?
一只虫子从一个边长为 $1$ 的等边三角形网格的某个顶点出发。每一步虫子沿着网格线以相同概率随机且相互独立地朝 $6$ 个可能方向之一移动。问：在移动 $5$ 步之后，虫子从未距离起点超过 $1$ 个单位的概率是多少？
Q18
Set $u_0 = \frac{1}{4}$, and for $k \ge 0$ let $u_{k+1}$ be determined by the recurrence \[u_{k+1} = 2u_k - 2u_k^2.\] This sequence tends to a limit; call it $L$. What is the least value of $k$ such that \[|u_k-L| \le \frac{1}{2^{1000}}?\]
设 $u_0 = \frac{1}{4}$，并且对 $k \ge 0$，令 $u_{k+1}$ 由递推式确定：\[u_{k+1} = 2u_k - 2u_k^2.\] 该数列趋于一个极限，记为 $L$。求满足 \[|u_k-L| \le \frac{1}{2^{1000}}\] 的最小 $k$ 值。
Q19
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？ $(
Q20
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.)
用 $4$ 个白色单位小立方体和 $4$ 个蓝色单位小立方体拼成一个大立方体。用这些小立方体拼成 $2 \times 2 \times 2$ 立方体共有多少种不同的方法？（如果一种拼法可以通过旋转与另一种拼法重合，则认为它们相同。）
Q21
For real numbers $x$, let \[P(x)=1+\cos(x)+i\sin(x)-\cos(2x)-i\sin(2x)+\cos(3x)+i\sin(3x)\] where $i = \sqrt{-1}$. For how many values of $x$ with $0\leq x<2\pi$ does \[P(x)=0?\]
对实数 $x$，令 \[P(x)=1+\cos(x)+i\sin(x)-\cos(2x)-i\sin(2x)+\cos(3x)+i\sin(3x)\] 其中 $i = \sqrt{-1}$。当 $0\leq x<2\pi$ 时，有多少个 $x$ 的取值使得 \[P(x)=0?\]
Q22
Right triangle $ABC$ has side lengths $BC=6$, $AC=8$, and $AB=10$. A circle centered at $O$ is tangent to line $BC$ at $B$ and passes through $A$. A circle centered at $P$ is tangent to line $AC$ at $A$ and passes through $B$. What is $OP$?
直角三角形 $ABC$ 的边长为 $BC=6$、$AC=8$、$AB=10$。以 $O$ 为圆心的圆在 $B$ 点与直线 $BC$ 相切，并经过点 $A$。以 $P$ 为圆心的圆在 $A$ 点与直线 $AC$ 相切，并经过点 $B$。求 $OP$。
Q23
What is the average number of pairs of consecutive integers in a randomly selected subset of $5$ distinct integers chosen from the set $\{ 1, 2, 3, …, 30\}$? (For example the set $\{1, 17, 18, 19, 30\}$ has $2$ pairs of consecutive integers.)
从集合 $\{ 1, 2, 3, …, 30\}$ 中随机选取 $5$ 个互不相同的整数构成一个子集。该子集中相邻整数对的平均个数是多少？（例如集合 $\{1, 17, 18, 19, 30\}$ 有 $2$ 对相邻整数。）
Q24
Triangle $ABC$ has side lengths $AB = 11, BC=24$, and $CA = 20$. The bisector of $\angle{BAC}$ intersects $\overline{BC}$ in point $D$, and intersects the circumcircle of $\triangle{ABC}$ in point $E \ne A$. The circumcircle of $\triangle{BED}$ intersects the line $AB$ in points $F \ne B$. What is $CF$?
三角形 $ABC$ 的边长为 $AB = 11, BC=24$, 且 $CA = 20$。$\angle{BAC}$ 的角平分线与 $\overline{BC}$ 交于点 $D$，并与 $\triangle{ABC}$ 的外接圆交于点 $E \ne A$。$\triangle{BED}$ 的外接圆与直线 $AB$ 交于点 $F \ne B$。求 $CF$。
Q25
For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$
对正整数 $n$，令 $R(n)$ 为 $n$ 分别除以 $2$、$3$、$4$、$5$、$6$、$7$、$8$、$9$ 和 $10$ 时所得余数之和。例如，$R(15) = 1+0+3+0+3+1+7+6+5=26$。有多少个两位正整数 $n$ 满足 $R(n) = R(n+1)\,?$
Time Left 75:00