RO-NMO 2016 Problem 121

Prove that there exists an unique sequence $\left( c_n \right)_{n\ge 1}$ of real numbers from the interval $(0,1)$ such that $$\int_0^1 \frac{dx}{1+x^m} =\frac{1}{1+c_m^m } ,$$ for all natural numbers $m,$ and calculate $\lim_{k\to\infty } kc_k^k.$ Radu Pop