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.

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The above train of thought took me about two minutes 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.