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