1975 USAMO Problems/Problem 2
Problem
Let denote four points in space and
the distance between
and
, and so on. Show that

Solution
If we project points
onto the plane parallel to
and
,
and
stay the same but
all decrease, making the inequality sharper. Thus, it suffices to prove the inequality when
are coplanar:
Let . We wish to prove that
. Let us fix
and the length
and let
vary on the circle centered at
with radius
. If we find the minimum value of
, which is the only variable quantity, and prove that it is larger than
, we will be done.
First, we express in terms of
, using the Law of Cosines:

.
is a function of
, so we take the derivative with respect to
and obtain that
takes a minimum when

.
$
Define and
:
.
See also
1975 USAMO (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |