# 2004 JBMO Problems/Problem 1

## Contents

## Problem

Prove that the inequality holds for all real numbers and , not both equal to 0.

## Solution

Since the inequality is homogeneous, we can assume WLOG that xy = 1.

Now, substituting , we have:

, thus we have

Now squaring both sides of the inequality, we get:
after cross multiplication and simplification we get:
or, which is always true since .

## Solution 2

Again, since the inequality is homogenous, we can assume WLOG that .

By AM-GM we gave and by QM-AM we have that .

Substituting we have

## Solution 3

By Trivial Inequality,

Then by multiplying by on both sides, we use the Trivial Inequality again to obtain which means which after simplifying, proves the problem.