# 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.*