Consecutive Triples Of Numbers

Let \(\{a_1, a_2, \cdots , a_{2013}\}\) be a set of \(2013\) positive integers. Three distinct integers \(1 \leq i, j, k \leq 2013\) are called good if \(a_k - a_j = a_j - a_i = 1.\) Find the last three digits of the maximum possible number of good triples \(\{i, j, k\}\).

Details and assumptions

  • Two good triples \(\{i, j, k\}\) and \(\{m, n, p\}\) are considered distinct if they differ in at least one number.

  • Yes, it's \(2013,\) not \(2014\).

  • This problem is not original.

  • A mistake in the wording has been fixed. Apologies.


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