TITLE: What is the value of $\frac{a^2}{b+c} + \frac{b^2}{a+c} + \frac{c^2}{a+b}$ if $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 1$?
QUESTION [5 upvotes]: If $$\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 1$$ then find the values of $$\frac{a^2}{b+c} + \frac{b^2}{a+c} + \frac{c^2}{a+b}.$$ How can I solve it? Please help me. Thank you in advance.

REPLY [6 votes]: You have $$a+b+c = (a+b+c)\left(\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\right) = \frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b} + a+b+c.$$
Then you get $$\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b} = 0.$$

REPLY [4 votes]: HINT:
$$\dfrac{a^2}{b+c}+a=\dfrac{a(a+b+c)}{b+c}$$
$$\sum_{\text{cyc}}\left(\dfrac{a^2}{b+c}+a\right)=(a+b+c)\sum_{\text{cyc}}\dfrac a{b+c}$$