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1997 AIME Problems/Problem 14

Problem

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

Solution

Solution 1

$z^{1997}=1=1(\cos 0 + i \sin 0)$

By De Moivre's Theorem, we find that ($k \in \{0,1,\ldots,1996\}$)

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

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

$\sqrt{\left(\cos\left(\frac{2\pi m}{1997}\right) + \cos\left(\frac{2\pi n}{1997}\right)\right)^2 + \left(\sin\left(\frac{2\pi m}{1997}\right) + \sin\left(\frac{2\pi n}{1997}\right)\right)^2}$
$=\sqrt{2 + 2\cos\left(\frac{2\pi m}{1997}\right)\cos\left(\frac{2\pi n}{1997}\right) + 2\sin\left(\frac{2\pi m}{1997}\right)\sin\left(\frac{2\pi n}{1997}\right)}$

We need $\cos \left(\frac{2\pi m}{1997}\right)\cos \left(\frac{2\pi n}{1997}\right) + \sin \left(\frac{2\pi m}{1997}\right)\sin \left(\frac{2\pi n}{1997}\right) \ge \frac{\sqrt{3}}{2}$. The cosine difference identity simplifies that to $\cos\left(\frac{2\pi m}{1997} - \frac{2\pi n}{1997}\right) \ge \frac{\sqrt{3}}{2}$. Thus, $|m - n| \le \frac{\pi}{6} \cdot \frac{1997}{2 \pi} = \lfloor \frac{1997}{12} \rfloor =166$.

Therefore, $m$ and $n$ cannot be more than $166$ away from each other. This means that for a given value of $m$, there are $332$ values for $n$ that satisfy the inequality; $166$ of them $> m$, and $166$ of them $< m$. Since $m$ and $n$ must be distinct, $n$ can have $1996$ possible values. Therefore, the probability is $\frac{332}{1996}=\frac{83}{499}$. The answer is then $499+83=\boxed{582}$.

Solution 2

The solutions of the equation $z^{1997} = 1$ are the $1997$th roots of unity and are equal to $\cos\left(\frac {2\pi k}{1997}\right) + i\sin\left(\frac {2\pi k}{1997}\right)$ for $k = 0,1,\ldots,1996.$ They are also located at the vertices of a regular $1997$-gon that is centered at the origin in the complex plane.

Without loss of generality, let $v = 1.$ Then \begin{eqnarray*} |v + w|^2 & = & |\cos\left(\frac {2\pi k}{1997}\right) + i\sin\left(\frac {2\pi k}{1997}\right) + 1|^2 \\ & = & \left|\left[\cos\left(\frac {2\pi k}{1997}\right) + 1\right] + i\sin\left(\frac {2\pi k}{1997}\right)\right|^2 \\ & = & \cos^2\left(\frac {2\pi k}{1997}\right) + 2\cos\left(\frac {2\pi k}{1997}\right) + 1 + \sin^2\left(\frac {2\pi k}{1997}\right) \\ & = & 2 + 2\cos\left(\frac {2\pi k}{1997}\right) \end{eqnarray*}

We want $|v + w|^2\ge 2 + \sqrt {3}.$ From what we just obtained, this is equivalent to $\cos\left(\frac {2\pi k}{1997}\right)\ge \frac {\sqrt {3}}2.$ This occurs when $\frac {\pi}6\ge \frac {2\pi k}{1997}\ge - \frac {\pi}6$ which is satisfied by $k = 166,165,\ldots, - 165, - 166$ (we don't include 0 because that corresponds to $v$). So out of the $1996$ possible $k$, $332$ work. Thus, $m/n = 332/1996 = 83/499.$ So our answer is $83 + 499 = \boxed{582}.$

See also

1997 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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