Question 146
Let $a$ and $b$ be nonzero real numbers. Prove that at least one of the following inequalities is true:
$$
\begin{aligned}
& \left|\frac{a+\sqrt{a^{2}+2 b^{2}}}{2 b}\right|\lt 1 \\
& \left|\frac{a-\sqrt{a^{2}+2 b^{2}}}{2 b}\right|\lt 1
\end{aligned}
$$
$$
\begin{aligned}
& \left|\frac{a+\sqrt{a^{2}+2 b^{2}}}{2 b}\right|\lt 1 \\
& \left|\frac{a-\sqrt{a^{2}+2 b^{2}}}{2 b}\right|\lt 1
\end{aligned}
$$