IN-TST 2010 Problem 3

For any integer $n\ge 2$ , let $N(n)$ be the maximum number of triples $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$ consisting of non-negative integers $a_j,b_j,c_j$ (not necessarily distinct) such that the following two conditions are satisfied:

(a) $a_j+b_j+c_j=n,$ for all $j=1,2,3,\cdots N(n)$ ; (b) $j\neq k$ , then $a_j\neq a_k$ , $b_j\neq b_k$ and $c_j\neq c_k$ .

Determine $N(n)$ for all $n\ge 2$ .