<s>[INST] what is the value of $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$? if we have $a+b+c=1$ and $ab+bc+ac=\frac{1}{3}$ then what is the value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$$  
and 
$$\frac{a}{b+1}+\frac{b}{c+1}+\frac{c}{a+1}$$.
from the hypothesis we have also $a^2 +b^2 +c^2 =\frac{1}{3}$ and also we have $a^3 +b^3+c^3=3abc$ but I don't know how should get the result. it will be great if you share your Idea with me,thanks.
 [/INST] Hint:since
$$a+b+c=1,ab+bc+ac=\dfrac{1}{3}$$ so 
$$(a+b+c)^2-3(ab+bc+ac)=0\Longrightarrow (a-b)^2+(b-c)^2+(c-a)^2=0\Longrightarrow  a=b=c$$
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