Problem:
Let be the set of all integers such that . For example, is the set . How many of the sets do not contain a perfect square?
Solution:
Note that . All sets contain a perfect square unless is so large that, for some integer with , and with . This implies that , and hence . But , which is a member of , so all sets contain at least one perfect square. Furthermore, for all , each set which contains a perfect square can only contain one such square. The largest value in any of the given set is , and . Disregarding the first squares dealt with above, there are perfect squares, each the sole perfect square member of one of the sets . Thus there are sets without a perfect square, and the answer is .
The problems on this page are the property of the MAA's American Mathematics Competitions