# 2006 SMT/Algebra Problems/Problem 6

## Contents

## Problem

Let be real numbers satisfying:

Determine all possible values of .

## Solution

From the first equation, we have . Plugging this into the third equation, we get . Multiplying both sides by , we get .

Now we plug that into the second equation. We have . Getting rid of the fractions, we have . We can factor that as , so or .

If , then and , so .

If , then and , so .

Therefore, the possible values of are .

## Solution 2

We can rearrange the equations as follows:

Then, using Simon's Favorite Factoring Trick we get:

Multiplying the three equations together yields

If , then dividing this equation by the factored equations yields:

and

If , then dividing this equation by the factored equations yields:

and

Thus, .