If a,b,c are positive real nos. such that a+b+c=1,then prove that

Adding 3 to both sides we get:

(a+b +c) (1/b+c  +  1/c+a  + 1/a+b) >= 9/2

or

2(a+b +c) (1/b+c  +  1/c+a  + 1/a+b) >= 9

or

((b+c) + (c+a) + (a+b)) (1/b+c  +  1/c+a  + 1/a+b) > =9

which is true by AM >=HM

Hence proved , no use of a+b+c =1 and no hardcore inequalities this is the best way to prove the Nesbitt's inequality.

LHS can be written as (a/1-a)+(b/1-b)+c/1-c

value of expression is min when each term is min

take LCM

We find that the expression formed when a=b=c

substitute and hope to get the ans

CHENGLU