Problem:
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line , the line , and the line ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Note that , and . Therefore there are 31 -by-1 squares of the desired type in the strip -by- 1 squares in the strip -by-1 squares in
the strip , and 121 -by- 1 squares in the strip . Furthermore there are 22 -by- 2 squares in the strip -by-2 squares in the strip , and 82 -by-2 squares in the strip . There is 1 3 -by-3 square in the strip , and there are 43 -by-3 squares in the strip . There are no 4 -by- 4 or larger squares. Thus in all there are squares of the desired type within the given region.
The problems on this page are the property of the MAA's American Mathematics Competitions