1970 IMO Problems/Problem 2
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Problem
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:
,
,
.
Prove:
if and only if
.
Solution
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.
Resources
1970 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |