Mock AIME 1 2005-2006/Problem 1

Revision as of 21:29, 17 April 2009 by Aimesolver (talk | contribs) (Solution)

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}.