2001 IMO Shortlist Problems/N5
Problem
Let be positive integers and suppose that

Prove that is not prime.
Solution
Equality is equivalent to
.
Let be the quadrilateral with
,
,
,
,
, and
. Such a quadrilateral exists by
and the Law of Cosines.
By Strong Form of Ptolemy's Theorem, we find that;
and by rearrangement inequality;
.
Assume is a prime, since
is an integer
must be an integer but this is false since
and
. Thus
can not be a prime.