ARML 2007 Individual Tiebreaker #1: Compute the largest positive real solution to the equation



Solution: The first thing most people thought of when tackling this problem, I imagine, is the straightforward expansion into a quartic and subsequent factoring into two quadratics. The two quadratics do exist, but I doubt most people have the computational speed necessary to perform the factoring in tiebreaker time. This is probably why only five people at the Las Vegas site (if I recall correctly) correctly answered this question.

The first thing I noticed about the equation was that the LHS could be written . After some musing, this led to the substitution .

This is a very important step! Superficially, it doesn't make much sense to change an equation in one variable into a system of equations in two variables. On the other hand, I have encountered several systems of equations in two variables that only became more complicated if I attempted to solve them with substitution, so why shouldn't the converse hold? In other words, splitting an equation into a system can sometimes make it easier to solve.

And how! The above substitution produces the simple binary quadratic form




Case: . We also know , so .

Case: . Then . The positive root here is the largest root of the original equation, and we are done.

---

The above train of thought took me about a minute and a half to run through on paper. Now, if only I'd been in the tiebreaker round...

In any case, this solution is simply a very elegant way to reveal the factoring of the original quartic (after squaring out), which is . Answering this problem quickly was largely a matter of considering the substitution as soon as possible, except for those of us who possess some unnatural talent for factoring quartics.