Problem:
For some positive integers , quadrilateral with positive integer side lengths has perimeter , right angles at and , and . How many different values of are possible?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
In every such quadrilateral, . Let be the foot of the perpendicular from to ; then and . Let and ; then . By the Pythagorean Theorem, , or . Therefore is even, say , and . The perimeter of the quadrilateral is . Increasing positive integer values of give the required quadrilaterals, with increasing perimeter. For the perimeter is , and for the perimeter is . Therefore there are such quadrilaterals.
.jpg)
The problems on this page are the property of the MAA's American Mathematics Competitions