Question 108
Version 1. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,
$$
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x .
$$
Version 2. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$,
$$
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x
$$
(Ireland)
$$
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x .
$$
Version 2. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$,
$$
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x
$$
(Ireland)