Problem: (1991 India National Olympiad – #7) Solve the following system for real
Solution: Well, we simply go about solving this system the regular way – look for something to eliminate. We try using the first and second equations and find
which we subtract from the second equation to get
Using the third equation, we have so we substitute, multiply through by (it can’t be zero), and divide by to get
So . But from the second equation, we see that gives but they are real so this is impossible. Hence . It remains to solve for and using
Substitute into the first one and again multiply through by to find
We then have with corresponding . Checking, we see that both solutions work, so our final solution set is
Comment: Not bad for an olympiad question – just requires basic algebraic manipulation and solving quadratics. Looking for the right things to square and substitute made this problem considerably easier.
Practice Problem: (1994 India National Olympiad – #2) If , prove that .