It's a problem that nobody is really talking about right now, but one that definitely appears to exist in the competition sphere today. We're a far cry from the 1970s when problems were almost assuredly original -- and, as a corollary, much easier than today's -- and the competitive math scene was booming (in some sense of the term "booming") following the inception of the USAMO. The influx of new and exciting concepts provided a stark contrast to the cut-and-dry versions the classroom offered, and students became attracted to the competitions for basically identical reasons to today.

Interestingly, the history of math contests is surprisingly difficult to lock down, so it's worth a brief overview here (much of this is adapted from the excellent read Countdown by Olsen). Competitive math in the US started basically in 1950, with the introduction of what is today better known as the AMC 12. Local contests, perhaps most notably NYSML (which continues to run to this day, and provided the groundwork for the creation of ARML), also sprung up during that decade. But it was not until a 1971 editorial sparked interest in the International Math Olympiad -- up until then the battleground for eastern Europe and the Soviet bloc -- that the concept fully began to take off. A year later, in 1972, the USAMO was created, and the US sent their first 8-member team to the IMO in 1974.

That team was special for many reasons: not only was it the first ever US IMO team, it finished second among eighteen teams (just behind the Soviet Union). Perhaps more pertinently though, you might recognize some of the members:

• Eric Lander who, besides co-chairing the PCAST and being an MIT celebrity, is one of the most respected geneticists in the world and an all-around amazing person
• Paul Zeitz, a name any serious math competitor is surely familiar with.

This paved the way for a whole new wave of competitive spirit, and middle schools jumped into the scene in 1983 (1984?) with the introduction of MATHCOUNTS.

Jumping ahead to the next major event in math competition history, AoPS arrived on the scene in late 2002/early 2003. Adding the internet into the competitive sphere drastically changed the scene; no longer were practice materials difficult to come by, and organized training material (AoPS books etc.) was available for essentially the first time. The effect didn't take long to materialize; compare contests from pre-2002 to contests from post-2005 and witness the huge shift in difficulty.

Anyway, history lesson over; back to the main point. This ease of access and the collection of standard strategies/tricks AoPS offered had a downside to it: it became quite difficult to write truly original problems. Some contests, like MATHCOUNTS, became notorious for "borrowing" questions from old editions of other contests, but we're increasingly starting to see this in other contests as well. ARML Local, for example, had a recent event in which an entire contest was basically "borrowed", and even the AMC -- the unquestionable standard to which all competitions are measured (there's a reason we call things AMC/AIME/Olympiad-level) -- has become increasingly unoriginal as well.

The point is not to complain about these contests or anything of the sort (so go put your pitchforks away before continuing); the point is that it's practically impossible to write an original problem these days. And when one is managed, it's often identical to some other problem in spirit. Contest math has always prided itself on utilizing elementary techniques in clever ways, but it increasingly seems like our cleverness has finally caught up with us.

This is largely because of the truly gigantic number of competitions going on today. Besides the major on-site ones (ARML, HMMT, PUMaC, etc.), there is no shortage of smaller, more regular ones (NIMO, USAMTS, many local contests, etc.), and even a steady supply of mock tests to fill the interim -- not to mention nearly every country's olympiad/selection series (AMC/AIME/USAMO and equivalents). In short, there are many more problems than one person could ever hope to do -- but they are relying on a rather finite number of techniques.

Nowhere is this problem more evident than Olympiad number theory -- or, should I say, the total lack of it. NT problems are few and far between, and the few major ones that are classified under NT are... not NT. I believe the reason is rather simple: we've more or less run out of interesting problems to ask in elementary number theory. To get back to interesting, good problems, we have to appeal to higher level number theory, which brings up the major question: why don't we?

From its inception, contest math has been all about applying techniques any high school students knows -- meaning, essentially, the fundamentals of algebra and geometry -- and applying them in clever ways to create and solve problems. It's a romantic ideal, to be sure, but have we simply moved past the point where that is possible? More pertinently, do we seriously believe we're doing this? Yes, it is technically true that all our contest problems utilize only "elementary math", but does anyone seriously believe typical high school students are even vaguely familiar with inversion, Muirhead, or even the basics of graph theory? We accept that all these things are de facto prerequisites, but does this not tacitly acknowledge that the ideal we focused on decades ago has long passed us by?

At some point, we need to revisit our concept of what is "higher math" and what is not. Of course, whatever division is made will be largely arbitrary, so it is a dumb argument to simply say something like "math competitions should have calculus in them" without qualifying that statement. But it is clear that the division we are using is leaving us starved. In an age where resources are available literally at the press of a button, do any of the major arguments against higher math hold much weight anymore? I don't think so...

• Higher math should not be used because students are not generally introduced to it until college.

Ok, but students are not generally introduced to almost anything we treat as basic knowledge for competitions until at least college. It is well-established that schools are generally pretty terrible at teaching math for anyone even slightly above the average student, and it is no longer difficult or unreasonable to learn material outside of school, so why do we cling to this outdated constraint?

• Higher math should not be used because students will not be able to solve problems involving it.

This argument is, quite simply, refuted. PUMaC has, for years, been willing to put high-level math concepts on their power round (albeit at a "baby steps" pace), and the general consensus among top teams is that a) it's hardly anything new to them (note that this hits hard against the first argument as well), and b) it's pretty easy. Consider last year's number theory power round on what is essentially graduate-level material: several teams scored above 200/210. This phenomenon is even better evidenced by HMMT's relatively new initiative, the HMIC, which recognizes that top students are almost invariably already familiar with higher-level math anyway, and there are many many nice problems that can be written using it. Scores on this are quite respectable.

• Higher math should not be used because TRADITION!

The thing is that we're already starting to see a ton of problems that are really based off of higher math anyway. The best example is probably USAMO 2008 Problem 6, in which the solution is basically "do group theory and/or linear algebra and just basically be done". A lot of later-end HMMT problems, in particular, are essentially adaptations of higher math for which elementary solutions can be found. So in reality this is not even that novel a concept.

• Higher math should not be used because there are still plenty of problems in elementary math left to be discovered

I mean, yes, the title of this post is unnecessarily alarmist -- we're hardly out of problems. But we're definitely getting there. But what is the point of constraining ourselves to this unnecessarily restrictive ideal when there's a ton of beautiful problems that we're outright ignoring? It feels more than a bit like the constraint in writing (each word should be the next digit of ): yes, constructing something fulfulling the constraint is very satisfying and often quite beautiful, but there are plenty of other works of literature that are no less appealing simply because they avoid an arbitrary constraint.

• Higher math should not be used because it forms an unnecessary barrier to competition entry

This is the sole argument that makes sense to me, but then again not really. In theory, newbies should be able to approach all the questions without the need for onerous background knowledge; in practice, a newbie wouldn't notice any real difficulty shift if we replaced the last question with the abc conjecture. It is indeed true that contests should not be entirely replaced with higher math -- that would be totally antithetical to the purpose of competitions -- but I think it's high time that we start the process now.

How can this be done? Fairly simply; take it from the top: Make (with proper prior announcement) the IMO involve higher math. Only in the #3/#6 slot, for now, as this is unlikely to affect results in a particularly meaningful way -- in today's world, anyone who's solving #3s/#6s almost certainly has higher math knowledge -- and has the potential to dramatically improve problem quality (there might actually be a real NT question!!). At some point the USAMO will likely follow suit, and then perhaps the AIME will include some easy higher math questions (yes, these exist -- see the early Putnam problems). Other competitions, like HMMT and perhaps PUMaC, will start including these as late problems. This will all further expand the number of people familiar with higher math, and those who don't want to bother with it won't see a huge difference if they don't (if you're not willing to spend much time on higher math, you probably weren't too willing to spend time studying to solve #3s on USAMO either). And, as an added benefit, math competitions will have even more transferrable skills.

Of course, the addition of higher math to the math competition canon is hardly a new one; it comes up every so often and tends to get roundly shouted down. But it seems that now is the time to give it serious consideration once again. The IOI (competitive programming) community is adapting to the changing times; will math follow suit?