1974 IMO Problems/Problem 5
Problem 5
Determine all possible values of where
are arbitrary positive numbers.
Solution
Note that We will now prove that
can reach any range in between
and
.
Choose any positive number . For some variables such that
and
, let
,
, and
. Plugging this back into the original fraction, we get
The above equation can be further simplified to
Note that
is a continuous function and that
is a strictly increasing function. We can now decrease
and
to make
tend arbitrarily close to
. We see
, meaning
can be brought arbitrarily close to
.
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~Imajinary