1973 AMC 12 Problem 11

Question

A circle with a circumscribed and an inscribed square centered at the origin

O

of a rectangular coordinate system with positive

x

- and

y

-axes

OX

and

OY

is shown in each figure

\mathrm{I}

to

\mathrm{IV}

below.

The inequalities

\begin{aligned} |x|+|y| &\le \sqrt{2(x^2+y^2)}\\ &\le 2\operatorname{Max}(|x|,|y|). \end{aligned}

are represented geometrically by the figure numbered

Geometric representation: An inequality

f(x,y)\le g(x,y)

for all

x,

y

is represented by a figure showing, for a typical real number

a,

the containment

\begin{aligned} &\{(x,y):g(x,y)\le a\}\\ &\quad\subset \{(x,y):f(x,y)\le a\}. \end{aligned}

\mathrm{I}

\mathrm{II}

\mathrm{III}

\mathrm{IV}

none of these

Answer: B

Concepts:inequality coordinate geometry circle square (geometry)

Difficulty rating: 2190

Small hint:

Identify the shapes of the sublevel sets for

|x|+|y|,

\sqrt{2(x^2+y^2)},

and

2\operatorname{Max}(|x|,|y|)

Big hint:

The inequality reverses the order of containment of the corresponding sublevel sets

Solution:

For a fixed positive

a,

the set

2\operatorname{Max}(|x|,|y|)\le a

is an axis-aligned square. The set

\sqrt{2(x^2+y^2)}\le a

is its circumscribed circle, and

|x|+|y|\le a

is the diamond circumscribed about that circle. Thus the required nesting is an inner axis-aligned square, then a circle, then an outer diamond, which is figure

\mathrm{II}.

Therefore, the correct answer is B.