Difference between revisions of "1997 AIME Problems/Problem 14"

Revision as of 20:12, 7 March 2007

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 $\displaystyle \theta=2\pi ik$ , where k is any constant, the equation reduces to: $e^{2\pi ik}=\cos(2\pi k)+i\sin(2\pi k)\\ =1+0i\\ =1+0\\ =1\\ z^{1997}-1=0\\ z^{1997}=1\\ z^{1997}=e^{2\pi ik}\\ z=e^{\frac{2\pi ik}{1997}}<\math>

== See also ==

[[1997 AIME Problems]]$ (Error compiling LaTeX. Unknown error_msg)

@@ Line 8: / Line 8: @@
 If <math>\displaystyle \theta=2\pi ik</math>, where k is any constant, the equation reduces to:
-<math>\begin{eqnarray*}
+<math>
-e^{2\pi ik}&=&\cos(2\pi k)+i\sin(2\pi k)\\
+e^{2\pi ik}=\cos(2\pi k)+i\sin(2\pi k)\\
-&=&1+0i\\
+=1+0i\\
-&=&1+0\\
+=1+0\\
-&=&1\\
+=1\\
-z^{1997}-1&=&0\\
+z^{1997}-1=0\\
-z^{1997}&=&1\\
+z^{1997}=1\\
-z^{1997}&=&e^{2\pi ik}\\
+z^{1997}=e^{2\pi ik}\\
-z&=&e^{\frac{2\pi ik}{1997}}
+z=e^{\frac{2\pi ik}{1997}}<\math>
-\end{eqnarray*}<\math>
 == See also ==
-* [[1997 AIME Problems]]</math>
+* [[1997 AIME Problems]]
+</math>

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Difference between revisions of "1997 AIME Problems/Problem 14"

Revision as of 20:12, 7 March 2007

Problem

Solution