Problem:
The number is between and . How many pairs of integers are there such that and
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Because and , there are either two or three integer powers of strictly between any two consecutive integer powers of . Thus for each there is at most one satisfying the given inequalities, and the question asks for the number of cases in which there are three powers rather than two. Let (respectively, ) be the number of nonnegative integers less than such that there are exactly two (respectively, three) powers of strictly between and . Because , it follows that and . Solving the system yields .
The problems on this page are the property of the MAA's American Mathematics Competitions