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, .