Find The Largest X

Find the largest positive integer \(X\) for which there exist \(X\) triples \((a_1, b_1, c_1), (a_2, b_2, c_2), \cdots , (a_X, b_X, c_X) \) consisting of non-negative integers such that:

  • For all \(1 \leq i \neq j \leq X,\) \(a_i \neq a_j, b_i \neq b_j, c_1 \neq c_j.\)

  • For all \(1 \leq i \leq X,\) \(a_i+b_i+c_i= 2014.\)

Details and assumptions

  • This problem is not original.

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