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- IndicaLv 75 years agoFavourite answer
Write p=∑a and q=∑ab

Inequality becomes (p−a)(p−b)(p−c) ≥ abc+q+p+1

→ p³−p²∑a+p∑ab−abc ≥ abc+q+p+1 → pq−q−p+1 ≥ 2(abc+1)

So need to prove (p−1)(q−1) ≥ 4

But AM-GM gives p ≥ 3∛(abc) = 3 and q ≥ 3∛(a²b²c²) = 3 so result is proved.

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