Solution:
Given a,b,c>0
a+b+c=1
this is possible when a=b=c=1/3
a^3/(a²+b²)+b^3/(b²+c²)+c^3/(c²+a²)>=1/2
then (1/3)²/(1/9+1/9)+ (1/3)²/(1/9+1/9)+(1/3)²/(1/9+1/9)
=1/2
hence proved
Solution:
Given a,b,c>0
a+b+c=1
this is possible when a=b=c=1/3
a^3/(a²+b²)+b^3/(b²+c²)+c^3/(c²+a²)>=1/2
then (1/3)²/(1/9+1/9)+ (1/3)²/(1/9+1/9)+(1/3)²/(1/9+1/9)
=1/2
hence proved
This is not a proof, you only showed it was true for one set of values for a, b and c
Edit: I didn't realise this question was from 2008.