1997 AIME Problems/Problem 14

Problem

Let $\displaystyle v$ and $\displaystyle w$ be distinct, randomly chosen roots of the equation $\displaystyle z^{1997}-1=0$ . Let $\displaystyle \frac{m}{n}$ be the probability that $\displaystyle\sqrt{2+\sqrt{3}}\le\left|v+w\right|$ , where $\displaystyle m$ and $\displaystyle n$ are relatively prime positive integers. Find $\displaystyle m+n$ .

Solution

The solution requires the use of Euler's formula:

$\displaystyle e^{i\theta}=\cos(\theta)+i\sin(\theta)$

If $\displaystyle \theta=2\pi k$ , where k is any constant, the equation reduces to:

$\displaystyle e^{2\pi ik}=\cos(2\pi k)+i\sin(2\pi k)$

$\displaystyle =1+0i$

$\displaystyle =1+0$

$\displaystyle =1$

Now, substitute this into the equation:

$\displaystyle z^{1997}-1=0$

$\displaystyle z^{1997}=1$

$\displaystyle z^{1997}=e^{2\pi ik}$

$\displaystyle z=e^{\frac{2\pi ik}{1997}}$

$\displaystyle z=\cos\left(\frac{2\pi k}{1997}\right)+i\sin\left(\frac{2\pi k}{1997}\right)$

Now, let $\displaystyle v$ be the root corresponding to $\displaystyle \theta=\frac{2\pi m}{1997}$ , and let $\displaystyle w$ be the root corresponding to $\displaystyle \theta=\frac{2\pi n}{1997}$ . The magnitude of $\displaystyle v+w$ is therefore:

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1997 AIME Problems/Problem 14

Problem

Solution

See also