2008 AIME I Problems/Problem 7
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?
The difference between consecutive squares is , which means that all squares above are more than apart.
Then the first sets () each have at least one perfect square because the differences between consecutive squares in them are all less than . Also, since is the largest such that ( is the upper bound which all numbers in must be less than), there are other sets after that have a perfect square.
There are sets without a perfect square.
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