Problem:
Let be the smallest positive integer whose cube root is of the form , where is a positive integer and is a positive real number less than . Find .
Solution:
We solve the equivalent problem of finding the smallest positive integer for which
This is equivalent to the given problem because
and because if some integer satisfies the double inequality on the right above, then is the smallest such .
Rewriting in the form
we observe that must be near , for the contributions of the other two terms on the right side of are relatively small. Consequently, since , we expect that either or . In the first case, is not satisfied; this can be verified by an easy calculation. It is even easier to show that satisfies , so it is the smallest positive integer with the desired property. The corresponding is the smallest positive integer whose cube root has a positive decimal part which is less than .
The problems on this page are the property of the MAA's American Mathematics Competitions