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2014 preRMO p19, sum x_i=1, sum x_i/(1-x_i)=1, sum x^2_i/(1-x_i)=?

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August 9, 2019
Sumalgebra

Problem Statement

Let x1,x2,...,x2014x_1,x_2,... ,x_{2014} be real numbers different from 11, such that x1+x2+...+x2014=1x_1 + x_2 +...+x_{2014} = 1
and x11−x1+x21−x2+...+x20141−x2014=1\frac{x_1}{1-x_1}+\frac{x_2}{1-x_2}+...+\frac{x_{2014}}{1-x_{2014}}=1
What is the value of x121−x1+x221−x2+...+x201421−x2014\frac{x^2_1}{1-x_1}+\frac{x^2_2}{1-x_2}+...+\frac{x^2_{2014}}{1-x_{2014}} ?
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