# 2018 USAMO Problems/Problem 1

## Contents

## Problem 1

Let be positive real numbers such that . Prove that

## Solution

WLOG let . Add to both sides of the inequality and factor to get:

The last inequality is true by AM-GM. Since all these steps are reversible, the proof is complete.

## Solution 2

https://wiki-images.artofproblemsolving.com//6/69/IMG_8946.jpg

-srisainandan6

## Solution 3

Similarly to Solution 2, we will prove homogeneity but we will use that to solve the problem differently. Let . Note that , thus proving homogeneity.

WLOG, we can scale down all variables such that the lowest one is . WLOG, let this be . We now have , and we want to prove Adding to both sides and subtracting gives us , or . Let . Now, we have By the trivial inequality, this is always true. Since all these steps are reversible, the proof is complete. ~SigmaPiE