IMO Problem Shortlist

Let $n\ge 2$ be an integer and let $\{T_1,\dots ,T_N\}$ be any set of triples of nonnegative integers satisfying the conditions (1) and (2). Since the $a$-coordinates are pairwise distinct we have $$\sum _{i=1}^N{a}_i\ge \sum _{i=1}^N(i-1)=\frac{N(N-1)}{2}$$ Analogously, $$\sum _{i=1}^N{b}_i\ge \frac{N(N-1)}{2}\text{ and }\sum _{i=1}^N{c}_i\ge \frac{N(N-1)}{2}.$$ Summing these three inequalities and applying (1) yields $$3\frac{N(N-1)}{2}\le \sum _{i=1}^N{a}_i+\sum _{i=1}^N{b}_i+\sum _{i=1}^N{c}_i=\sum _{i=1}^N\left(a_i+b_i+c_i\right)=nN,$$ hence $3\frac{N-1}{2}\le n$ and, consequently, $$N\le \lfloor \frac{2n}{3}\rfloor +1$$ By constructing examples, we show that this upper bound can be attained, so $N(n)=\lfloor \frac{2n}{3}\rfloor +1$. We distinguish the cases $n=3k-1,n=3k$ and $n=3k+1$ for $k\ge 1$ and present the extremal examples in form of a table.

$n=3k-1$
$\lfloor \frac{2n}{3}\rfloor +1=2k$
$a_i$	$b_i$	$c_i$
0	$k+1$	$2k-2$
1	$k+2$	$2k-4$
$⋮$	$⋮$	$⋮$
$k-1$	$2k$	0
$k$	0	$2k-1$
$k+1$	1	$2k-3$
$⋮$	$⋮$	$⋮$
$2k-1$	$k-1$	1

$n=3k$
$\lfloor \frac{2n}{3}\rfloor +1=2k+1$
$a_i$	$b_i$	$c_i$
0	$k$	$2k$
1	$k+1$	$2k-2$
$⋮$	$⋮$	$⋮$
$k$	$2k$	0
$k+1$	0	$2k-1$
$k+2$	1	$2k-3$
$⋮$	$⋮$	$⋮$
$2k$	$k-1$	1

$n=3k+1$
$\lfloor \frac{2n}{3}\rfloor +1=2k+1$
$a_i$	$b_i$	$c_i$
0	$k$	$2k+1$
1	$k+1$	$2k-1$
$⋮$	$⋮$	$⋮$
$k$	$2k$	1
$k+1$	0	$2k$
$k+2$	1	$2k-2$
$⋮$	$⋮$	$⋮$
$2k$	$k-1$	2

It can be easily seen that the conditions (1) and (2) are satisfied and that we indeed have $\lfloor \frac{2n}{3}\rfloor +1$ triples in each case.