1978 AMC 12 Problem 25

Question

Let

a

be a positive number. Consider the set

S

of all points whose rectangular coordinates

(x,y)

satisfy all of the following conditions:

\begin{aligned} &\text{(i) }\frac a2\le x\le2a\\ &\text{(ii) }\frac a2\le y\le2a\\ &\text{(iii) }x+y\ge a\\ &\text{(iv) }x+a\ge y\\ &\text{(v) }y+a\ge x. \end{aligned}

The boundary of set

S

is a polygon with

3

sides

4

sides

5

sides

6

sides

7

sides

Begin with the square described by conditions

\text{(i)}

and

\text{(ii)}

Determine which condition is redundant and which two cut off opposite corners

The first two conditions form the square

[\frac{a}{2},2a]\times[\frac{a}{2},2a].

Within this square,

x+y\ge a

is automatic. The last two conditions are equivalent to

|x-y|\le a;

their boundary lines cut off the corners

(2a,\frac{a}{2})

and

(\frac{a}{2},2a).

Cutting two opposite corners from a square produces a hexagon, so the boundary has six sides.

Therefore, the correct answer is D.

Problem 25 in other years