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The Fibonacci sequence is defined by $f_1 = 0, f_2 = 1$, and $f_{n+2} = f_{n+1}+f_n$ for $n\ge 1$. Prove that there is a strictly increasing arithmetic progression whose no term is in the Fibonacci sequence.

Just stating the problem as it was in the contest for the sake of the Resources section. BTW: The problem is from the 2004 contest.

$\mod 11$ also works.

The $n$-th Pisano period for $n = F_{2k}+F_{2k+2}$ is small, namely precisely $4k+2$ long. So for $k>1$, all these numbers $n$ work ($11,29,76,\ldots$), by missing residues in the Fibonacci sequence. See http://en.wikipedia.org/wiki/Pisano_period.