2002 IMO Shortlist Problems/A2
Problem
Let be an infinite sequence of real numbers, for which there exists a real number
with
for all
, such that
Prove that .
Solutions
Solution 1
For some fixed value of , let
be the permutation of the first
natural numbers such that
is an increasing sequence. Then we have
Now, by the Cauchy-Schwarz Inequality, we have
Thus for all , we must have
and therefore must be at least 1, Q.E.D.
Solution 2
We proceed to as in Solution 1. We now note that by the AM-HM Inequality,
Thus for any , we have two
that differ by more than
. But this becomes arbitrarily close to 1 as
becomes arbitrarily large. Hence if we had
, then we could obtain a contradiction, so we conclude that
, Q.E.D.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
Notes
The chief difficulty of this problem seems to be obtaining ; once this result has been obtained, there are many ways to conclude.