1998 IMO Shortlist Problems/A2
Revision as of 21:03, 30 December 2007 by Boy Soprano II (talk | contribs) (New page: == Problem == (''Australia'') Let <math>r_1, r_2, \dotsc, r_n</math> be real numbers greater than or equal to 1. Prove that <cmath> \frac{1}{r_1 + 1} + \frac{1}{r_2 + 1} + \dotsb + \frac...)
Problem
(Australia)
Let be real numbers greater than or equal to 1. Prove that
Solution
Let denote the function
.
Lemma 1. For , the function
is decreasing.
Proof. Note that . Since
is increasing for
, the lemma follows.
Lemma 2. For positive ,
is convex.
Proof. Note that the derivative of is
By Lemma 1,
is increasing when
, i.e., when
. Therefore
is convex for nonnegative
.
For all integers ,
, so
. Since
is convex for nonnegative
, it follows from Jensen's Inequality that
as desired.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.