2001 IMO Shortlist Problems/A3
Contents
[hide]Problem
Let be arbitrary real numbers. Prove the inequality
Solution 1
We prove the following general inequality, for arbitrary positive real :
with equality only when
.
We proceed by induction on . For
, we have trivial equality. Now, suppose our inequality holds for
. Then by inductive hypothesis,
If we let
, then we have
with equality only if
.
By the Cauchy-Schwarz Inequality,
with equality only when
. Since
, our equality cases never coincide, so we have the desired strict inequality for
. Thus our inequality is true by induction. The problem statement therefore follows from setting
.
Solution 2
By the Cauchy-Schwarz Inequality
For all real numbers.
Hence it is only required to prove
where
for ,
For k=1,
Summing these inequalities, the right-hand side yields
Hence Proved by Maths1234RC
P.S. This is my first solution on AOPS.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.