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 ( $n+1$ is given so that there are $n$ repetitions of $\frac {\pi}{1003}$ ). 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