# 2001 IMO Shortlist Problems/A3

## Contents

## 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.*