3.2 THE EXTREME PRINCIPLE 75less than 1. Show that all of these points lie within the interior or on the boundary of a
triangle with area less than 4.
Solution: Let triangle ABC have the largest area among all triangles whose ver
tices are taken from the given set of points. Let [ABC] denote the area of triangle ABC.
Then [ABC] < 1. Let triangle LMN be the triangle whose medial triangle is ABC. (In
other words, A, B, C are the midpoints of the sides of triangle LMN below.)
LMThen [LMN] = 4[ABC] < 4. We claim that the set of points must lie on the boundary
or in the interior of LMN. Suppose a point P lies outside LMN. Then we can connect
P with two of the vertices of ABC forming a triangle with larger area than ABC, con
tradicting the maximality of [ABC]. •
Always be aware of order and maximum/minimum in a problem, and alway s as
sume, if possible, that the elements are arranged in order (we call this monotonizing).
Think of this as "free information." The next example illustrates the principle once
again that a close look at the maximum (and minimum) elements often pays off. You
first encountered this as Problem 1.1. 4 on page 2. We break down the solution into
two parts: the investigation, followed by a formal write-up.
Example 3.2.3 I invite 10 couples to a party at my house. I ask everyone present,
including my wife, how many people they shook hands with. It turns out that everyone
questioned-I didn't question myself, of course-shook hands with a different number
of people. If we assume that no one shook hands with his or her partner, how many
people did my wife shake hands with? (I did not ask myself any questions.)
Investigation: This problem seems intractable. There doesn't appear to be enough
information. Nevertheless, we can make it easier by looking at a simpler case, one
where there are are, say, two couples in addition to the host and hostess.
The host discovers that of the five people he interrogated, there are five different
"handshake numbers." Since these numbers range from 0 to 4 inclusive (no one shakes
with their partner), the five handshake numbers discovered are 0,1,2,3 and 4. Let's
call these people PO,p}, ... ,P 4 , respectively, and let's draw a picture, including the