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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!!

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$2+3+100 > 49 + 50$
but
$2 \times 3 \times 100 < 49 \times 50$

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