Mock AIME 1 2005-2006/Problem 1

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Problem 1

$2006$ points are evenly spaced on a circle. Given one point, find the maximum number of points that are less than one radius distance away from that point.

Solution

Number the points $p_1$, $p_2$, $\dots$, $p_{2006}$. Assume the center is $O$ and the given point is $p_1$. Then $\angle p_nOp_{n+1}$ = $\frac {\pi}{1003}$, and we need to find the maximum $n$ such that $\angle p_1Op_{n+1} \le 60$ degrees. This can be done in $\frac {\pi}{3}$ divided by $\frac {\pi}{1003}$ = $\frac {1003}{3}$ = $334.333\dots$, so $n$ + $1$ = $335$. We can choose $p_2$, $p_3$, $\dots$, $p_{335}$, so $334$ points. But we need to multiply by $2$ to count the number of points on the other side of $p_1$, so the answer is $\boxed{668}$.

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