Difference between revisions of "2001 IMO Shortlist Problems/A3"
(New page: == Problem == Let <math>x_1, x_2, \dotsc, x_n</math> be arbitrary real numbers. Prove the inequality <cmath> \frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \dotsb + \frac{x_n}{1+ x_...) |
m (typo) |
||
Line 6: | Line 6: | ||
== Solution == | == Solution == | ||
− | We prove the following general | + | We prove the following general inequality, for arbitrary positive real <math>k</math>: |
<cmath> \sum_{j=1}^n \frac{x_j}{k^2 + \sum_{i=1}^j x_i^2} \le \sqrt{n}/k , </cmath> | <cmath> \sum_{j=1}^n \frac{x_j}{k^2 + \sum_{i=1}^j x_i^2} \le \sqrt{n}/k , </cmath> | ||
with equality only when <math>n=0</math>. | with equality only when <math>n=0</math>. |
Revision as of 11:03, 29 December 2007
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
.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.