if a3 + b3 + c3 = 3abc and a,b,c are positive numbers, then prove that a = b = c.
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a^3 +b^3+c^3 =3abc , then , a^3 + b^3+c^3-3abc =0.......1
we know , a^3+v^3+c^3-3abc =(a+b+c)(a^2+b^+c^2-ab-bc-ca)........2
from 1 and 2 , a^2+b^+c^2c-ab-bc-ca= 0,,,,,, since a+b+c is not = 0 a,b,c>0
2 (a^2+b^2+c^2-ab-bc-ca) =0
(a-b)^2+(b-c)^2+(c-a)^2 =0
we have that for LHS to be equal to RHS each term should be equal to zero on LHS
thus ,a=b, b=c,c=a
hence a=b=c
we know , a^3+v^3+c^3-3abc =(a+b+c)(a^2+b^+c^2-ab-bc-ca)........2
from 1 and 2 , a^2+b^+c^2c-ab-bc-ca= 0,,,,,, since a+b+c is not = 0 a,b,c>0
2 (a^2+b^2+c^2-ab-bc-ca) =0
(a-b)^2+(b-c)^2+(c-a)^2 =0
we have that for LHS to be equal to RHS each term should be equal to zero on LHS
thus ,a=b, b=c,c=a
hence a=b=c
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