Mock AIME 1 2005-2006/Problem 1

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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Mock AIME 1 2005-2006/Problem 1

Problem 1

Solution