2001 IMO Shortlist Problems/N5
Problem
Let be positive integers and suppose that
![$ac + bd = (b + d + a - c)(b + d - a + c).$](http://latex.artofproblemsolving.com/4/0/9/40987fe427069b0fea5c78a8c84a1ae1f2992a08.png)
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.