Question

if a+b+c=1 ab+bc+ca=1/3 (a+b):c?

Answer 1

If it is mentioned that a,b,c are positive then you can apply am-gm-hm inequality.

Apply am-hm inequality-》

(a+b+c)/(3)>=3/(1/a+1/b+1/c)[ab+bc+ca=abc(1/a+1/b+1/c)-》1/a+1/b+1/c=1/3abc]

So abc <=1/27

Now apply,am-gm inequality

(a+b+c)/3>=(abc)^1/3

=》(a+b+c)/3>=(1/27)^1/3

See that equality holds here,i.e. a=b=c

hence a:b:c=1:1:1