2014-23 Differentiable function

Let $$f:[0,1]\to \mathbb R$$ be a differentiable function with $$f(0)=0$$, $$f(1)=1$$. Prove that for every positive integer $$n$$, there exist $$n$$ distinct numbers $$x_1,x_2,\ldots,x_n\in(0,1)$$ such that $\frac{1}{n}\sum_{i=1}^n \frac{1}{f'(x_i)}=1.$

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