AHSME 1973 Problem 11

A circle with a circumscribed and an inscribed square centered at the origin of a rectangular coordinate system with positive and axes and is shown in each figure to below.

The inequalities $$|x|+|y|\leq\sqrt{2(x^{2}+y^{2})}\leq 2\text{Max}(|x|, |y|)$$ are represented geometrically* by the figure numbered $\textbf{(A)}\ I \textbf{(B)}\ II \textbf{(C)}\ III \textbf{(D)}\ IV \textbf{(E)}\ \text{none of these}$

• An inequality of the form $f(x, y) \leq g(x, y)$, for all $x$ and $y$ is represented geometrically by a figure showing the containment $\{\text{The set of points }(x, y)\text{ such that }g(x, y) \leq a\} \subset\\ \{\text{The set of points }(x, y)\text{ such that }f(x, y) \leq a\}$ for a typical real number $a$.