1970 IMO Problems/Problem 2
Let , and be integers greater than 1, and let and be the bases of two number systems. and are numbers in the system with base and and are numbers in the system with base ; these are related as follows:
if and only if .
Suppose . Then for all integers , , with equality only when or . (In particular, we have strict inequality for .) In summation, this becomes or which is equivalent to This implies On the other hand, if , then evidently , and if , then by what we have just shown, . Hence if and only if , as desired.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
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