Evaluating an intriguing expression.

Question

Let $x_1,x_2...x_{2014} \in R, \not= 1$ such that $$x_1+x_2...+\ x_{2014}=1$$ and $$\frac{x_1}{1-x_1}+\frac{x_2}{1-x_2}...+\frac{x_{2014}}{1-x_{2014}}=1$$

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$$\frac{x_1^2}{1-x_1}+\frac{x_2^2}{1-x_2}...+\frac{x_{2014}^2}{1-x_{2014}}= ?$$

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Any solutions would be appreciated. I've been trying this for a while..but no breakthrough yet...

Answer 1

(From comments:)

Notice that $$ \frac{x}{1-x}-x=\frac{x-x+x^2}{1-x}=\frac{x^2}{1-x} $$

So $$ \sum_{i=1}^{2014} \frac{x_i^2}{1-x_i}=\sum_{i=1}^{2014} \frac{x_i}{1-x_1}-\sum_{i=1}^{2014} x_i = 1-1 = 0 $$