Mock AIME 1 2005-2006/Problem 1
Problem 1
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 ,
,
,
. Assume the center is
and the given point is
. Then $\anglep_nOp_(n+1)$ (Error compiling LaTeX. Unknown error_msg) =
, and we need to find the maximum
such that $\anglep_1Op_n+1 \le 60$ (Error compiling LaTeX. Unknown error_msg) degrees (
is given so that there are
repetitions of
). This can be done in
divided by
\frac {1003}{3} =
, so
+
=
. We can choose
,
, \dots,
, so
points. But we need to multiply by
to count the number of points on the other side of
, so the answer is \boxed{668}.