Difference between revisions of "2001 IMO Shortlist Problems/A3"
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<math>\sum_{n=1}^{k}a_n^2\leq 1</math> | <math>\sum_{n=1}^{k}a_n^2\leq 1</math> | ||
− | Hence Proved by Maths1234RC.<math>\blacksquare</math> | + | Hence Proved by @Maths1234RC.<math>\blacksquare</math> |
Revision as of 06:33, 23 October 2023
Contents
Problem
Let be arbitrary real numbers. Prove the inequality
Solution
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.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.