Question

从区间 $[0, 2)$ 内所有可以写成分数 $\dfrac{n}{d}$ 的有理数中随机选择有理数 $a$ 和 $b$ ，其中 $n$ 和 $d$ 是整数且 $1 \le d \le 5$ 。表达式是实数的概率是多少？ $(\cos(a\pi) + i\sin(b\pi))^4$

Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0, 2)$ that can be written as fractions $\dfrac{n}{d}$ where $n$ and $d$ are integers with $1 \le d \le 5.$ What is the probability that $(\cos(a\pi) + i\sin(b\pi))^4$ is a real number?

$\dfrac{3}{50}$

$\dfrac{4}{25}$

$\dfrac{41}{200}$

$\dfrac{6}{25}$

$\dfrac{13}{50}$

答案：D

知识点：复数棣莫弗定理基本计数

难度评级：2520

解答：

各有 $20$ 个可能的 $a$ 和 $b.$ 写成最简分数时，分母 $1,2,3,4,5$ 分别贡献 $2,2,4,4,8$ 个位于 $[0,2),$ 中的值，总计 $20.$

令 $x = \cos(a\pi)$ 且 $y = \sin(b\pi),$ 四次方 $(x + iy)^4$ 为实数，当且仅当 $x = 0,$ $y = 0,$ 或 $x = \pm y.$

情形 $x = 0$ 意味着 $a \in \left\{\dfrac12, \dfrac32\right\},$ 给出 $2\cdot 20 = 40$ 对；情形 $y = 0$ 意味着 $b \in \{0, 1\},$ 再给出 $40$ 对，其中 $4$ 对已经计数。

对于剩余条件 $\cos(a\pi) = \pm\sin(b\pi),$ 且两边都非零，允许的 $b$ 值为 $\dfrac14,$ $\dfrac12,$ $\dfrac34,$ $\dfrac54,$ $\dfrac32,$ 和 $\dfrac74.$ 对应的 $a$ 值个数依次为 $4,2,4,4,2,4.$ 例如，当 $b = \dfrac14,$ 时，各值为 $a = \dfrac14,$ $\dfrac34,$ $\dfrac54,$ 和 $\dfrac74;$ 其他各行同样由四分之一周的恒等式得到。因此此条件再贡献 $4+2+4+4+2+4=20$ 对。

总共有 $40 + 40 - 4 + 20 = 96$ 对有效组合，全部组合为 $400,$ 对，所以概率为 $\dfrac{96}{400} = \dfrac{6}{25}.$

因此，正确答案是 D。

There are $20$ possible values for each of $a$ and $b.$ In reduced form, denominators $1,2,3,4,5$ contribute respectively $2,2,4,4,8$ values in $[0,2),$ for a total of $20.$

Writing $x = \cos(a\pi)$ and $y = \sin(b\pi),$ the fourth power $(x + iy)^4$ is real if and only if $x = 0,$ $y = 0,$ or $x = \pm y.$

The case $x = 0$ means $a \in \left\{\dfrac12, \dfrac32\right\},$ giving $2\cdot 20 = 40$ pairs; the case $y = 0$ means $b \in \{0, 1\},$ giving another $40$ pairs, of which $4$ were already counted.

For the remaining condition $\cos(a\pi) = \pm\sin(b\pi),$ with neither side zero, the allowed values of $b$ are $\dfrac14,$ $\dfrac12,$ $\dfrac34,$ $\dfrac54,$ $\dfrac32,$ and $\dfrac74.$ The corresponding numbers of allowed values of $a$ are respectively $4,2,4,4,2,4.$ For example, when $b = \dfrac14,$ the values are $a = \dfrac14,$ $\dfrac34,$ $\dfrac54,$ and $\dfrac74;$ the other rows follow from the same quarter-turn identities. Hence this condition contributes $4+2+4+4+2+4=20$ more pairs.

In all there are $40 + 40 - 4 + 20 = 96$ valid pairs out of $400,$ so the probability is $\dfrac{96}{400} = \dfrac{6}{25}.$

Thus, the correct answer is D.

2015 AMC 12A 第 24 题

答案：D

难度评级：2520

解答：

其他年份的第 24 题