Mock AIME 5 2005-2006 Problems/Problem 15

Problem

$2006$ 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.

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

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Mock AIME 5 2005-2006 Problems/Problem 15

Contents

Problem

Solution

Solution

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