I am curious to know whether the following holds or not.
If $n_1,n_2,n_3,m_1,m_2$ are positive integers strictly greater than 1 such that $$n_1+n_2+n_3 > m_1 +m_2$$ then $$n_1n_2n_3 \geq m_1m_2.$$
Please do not give me the full answer (I only want a hint). I have tried using the AM-GM-HM inequality but can't seem to prove the result. I have tried looking for counterexamples but haven't found one yet.
Thanks in advance for any help!!