RMO Part-5!

We try by taking cases!1)n can't be negative(its trivial to prove it)2)if n is 7 or greater than 7 then each of the three term becomes less than 1\3rd and hence can't add up to one 3)for n=6 or 5 leaving the first term the(absolute) magnitude of 2nd and third term is too small with respect to first term(by trial or simple logic they can't add up to one !4)for one or two or three the sum isn't equaling 1 5)try for 4 and u get the required answer!!

We can know that $n≡4(mod 6)$ and $n>0$ easily.

$n$ can't be too large.So we try $n=4$ or $10$ .Then find $4$ is satisfied and $10$ is too large.

So,the answer is $\boxed{4}$