<s>[INST] Inequality....(RMO $1994$...question $8$) If $a$, $b$, $c$ are positive real numbers such that $a+b+c = 1$, prove that
$$(1+a)(1+b)(1+c) ≥ 8(1-a)(1-b)(1-c)\text{.}$$
 [/INST] Note that by A.M - G.M we have $$ (b+c)+(c+a) \geq 2 \cdot \sqrt{(b+c)\cdot(c+a)}$$ This says $2c+b+a =1+c\geq 2 \cdot \sqrt{(b+c)\cdot (c+a)}$.  Similary get inequalities for $(1+b)$ and $(1+c)$ multiply and get the answer.
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