Mock AIME 5 2005-2006 Problems/Problem 15


$2006$ colored beads are placed on a necklace (circular ring) such that each bead is adjacent to two others. The beads are labeled $a_0$, $a_1$, $\ldots$, $a_{2005}$ around the circle in order. Two beads $a_i$ and $a_j$, where $i$ and $j$ are non-negative integers, satisfy $a_i = a_j$ if and only if the color of $a_i$ is the same as the color of $a_j$. Given that there exists no non-negative integer $m < 2006$ and positive integer $n < 1003$ such that $a_m = a_{m-n} = a_{m+n}$, where all subscripts are taken $\pmod{2006}$, find the minimum number of different colors of beads on the necklace.



See also

Mock AIME 5 2005-2006 (Problems, Source)
Preceded by
Problem 14
Followed by
Problem 15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Invalid username
Login to AoPS