For any integer
![n\geq 2](/media/m/e/d/b/edbb3c15913fef4235c90cca2333a608.png)
, let
![N(n)](/media/m/c/7/d/c7d13362b232fbb29a559ae19841ed73.png)
be the maxima number of triples
![(a_i, b_i, c_i)](/media/m/e/0/2/e0240c397468605c90495d82abd57829.png)
,
![i=1, \ldots, N(n)](/media/m/5/e/b/5ebdde34e667e5e66857d2e448519d45.png)
, consisting of nonnegative integers
![a_i](/media/m/2/a/2/2a22407e8a19d2df9d425caa379f34a8.png)
,
![b_i](/media/m/e/8/8/e8844e25c79b8d97c4934d290b410f10.png)
and
![c_i](/media/m/4/f/a/4fac54a7a396893b1035341a7f5e1ba3.png)
such that the following two conditions are satisfied:
![a_i+b_i+c_i=n](/media/m/b/a/b/babd6b80dd7eb99f55a1721955fd5b29.png)
for all
![i=1, \ldots, N(n)](/media/m/5/e/b/5ebdde34e667e5e66857d2e448519d45.png)
, If
![i\neq j](/media/m/d/0/f/d0f78061e1617137185624c9f5992813.png)
then
![a_i\neq a_j](/media/m/a/5/3/a532021964d8031c9bfd7e9e7aa1492e.png)
,
![b_i\neq b_j](/media/m/8/9/d/89d05d6e3ed00db29e9f2487463bf1be.png)
and
![c_i\neq c_j](/media/m/b/b/c/bbc74656abf0ee9afb566b45b18f5044.png)
Determine
![N(n)](/media/m/c/7/d/c7d13362b232fbb29a559ae19841ed73.png)
for all
![n\geq 2](/media/m/e/d/b/edbb3c15913fef4235c90cca2333a608.png)
.
Proposed by Dan Schwarz, Romania
%V0
For any integer $n\geq 2$, let $N(n)$ be the maxima number of triples $(a_i, b_i, c_i)$, $i=1, \ldots, N(n)$, consisting of nonnegative integers $a_i$, $b_i$ and $c_i$ such that the following two conditions are satisfied:
$a_i+b_i+c_i=n$ for all $i=1, \ldots, N(n)$, If $i\neq j$ then $a_i\neq a_j$, $b_i\neq b_j$ and $c_i\neq c_j$Determine $N(n)$ for all $n\geq 2$.
Proposed by Dan Schwarz, Romania