If `a ,b , a n dc` are distinct positive real numbers such that `a+b+c=1,` then prove that `((1+a)(1+b)(1+c))/((1-a)(1-b)(1-c))> 8.`

Explanation

a + b + c = 1
Or a = 1 - b = c
`implies 1 + a = (a - b) + (1 - c) gt 2 sqrt((1 - b)(1 - c))`
Using A.M `ge` G.M
Similarly, `1 + b gt 2 sqrt((1 - c)(1 - a))`
and `1 + c gt 2 sqrt((1 - a)(1 - b))`
Multiply, we get
`(1 + a) (1 + b) (1 + c) gt 8 (1 - a) (1 - b) (1 - c)`
or `((1 + a) (1 + b) (1 + c))/((1- a) (1 - b) (1 - c)) gt 8`