2006 SMT/Algebra Problems/Problem 6
Contents
[hide]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, .