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Part of 2010 India IMO Training Camp

Problems(1)

Indian Team Selection Test 2010 ST1 P3

Source:

5/22/2010
For any integer n≥2n\ge 2, let N(n)N(n) be the maximum number of triples (aj,bj,cj),j=1,2,3,⋯ ,N(n),(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n), consisting of non-negative integers aj,bj,cja_j,b_j,c_j (not necessarily distinct) such that the following two conditions are satisfied:
(a) aj+bj+cj=n,a_j+b_j+c_j=n, for all j=1,2,3,⋯N(n)j=1,2,3,\cdots N(n); (b) j≠kj\neq k, then aj≠aka_j\neq a_k, bj≠bkb_j\neq b_k and cj≠ckc_j\neq c_k.
Determine N(n)N(n) for all n≥2n\ge 2.
floor functionmodular arithmeticcombinatorics unsolvedcombinatorics