Are high-school contest problems "harder" than college mathematics?

Question

At my own college and on this site as well, in fact, anywhere where there are a lot of people who study mathematics, I have noticed an interesting pattern.

What I've noticed is that, a lot of people at college level or above can solve problems such as this with little to no difficulty if they've learned the material well. In fact, I showed this very same problem to some of my college classmates and they too were able to solve it with no difficulty. There's nothing unique about this problems, it's something you'd learn to solve in college in a mathematics course, just a simple ODE problem.

However, and this where the interesting part is, when they encounter problems such as this, a problem that has been derived from another that appeared in a High-school mathematics contest, they seem to struggle, quite a lot. The same group of classmates that I asked to solve the previous problem, were not able to solve this, only one of them managed to give me the right answer and the solution was, let's just say, not very good and not something that I believe would be awarded a high score on that contest. The same seems to be true (as far as I have seen) on this site, too. A lot of people generally struggle with such contest-type problems, as I have observed by having posted many such questions here.

So, my question is really simple, why is this the case? Why is it that a college-level calculus problem, something you'd never ask a high-schooler to even look at, is little more than a trivial calculation yet problems like these, that appear in contests, seem to be more challenging and difficult? Are they genuinely more difficult? Are there other factors involved that are being overlooked? Or is it simply a lack of exposure to contest-oriented problems that drive this "issue" rather than a lack of mathematical prowess? What do you think?

Answer 1

Computation is only one aspect (albeit, an important one) of the mathematical problem-solving process in general. The "difficulty" of a math problem, as perhaps characterized by the likelihood* of an individual with the prerequisite knowledge to understand the question to be able to solve it, is not merely a function of the complexity of the computation involved; it is also to a great extent determined by what one might call "creativity" or "original thought" in the approach.

To a degree, it is possible to train students to get better at problem solving by exposing them to a wider array of strategies. Learning how to attack problems by teaching previously solved problems is a form of pattern recognition development that is commonly used in contest preparation, much in the same way that we teach computational techniques through practice problems. However, this approach is not guaranteed to be successful, especially when faced with a question that might be solved only through some line of reasoning that is profoundly different than what was prepared before.

And I would argue that this is the essence of mathematics research: the ability to craft novel logical statements that might build on previous knowledge but require a certain kind of what I referred to above as "creativity." The ability to do this is something of great interest to mathematicians (and non-mathematicians alike!), and if history is any indication, success in math contests seems to be a fairly good predictor of future ability in this regard.

That said, I would actually like to turn your claim around and question it, because I suggest that some the problem-solving methods that we regard as routine today (e.g., the differential equation example you provided), would not have been straightforward in Newton's time, when the calculus was still being discovered and formalized. Similarly, a sophisticated high school student would have no problem applying the cubic formula, but at the time the formula was discovered, it was a secret known only to a select few. Or consider the general solution in integers to the Pell equation $$x^2 - N y^2 = \pm 1,$$ which again, is something we can do today with minimal effort, but was at one point highly nontrivial. Even in the realm of contest math, the "Vieta jumping" technique that was used to solve the notoriously difficult IMO 1988 #6, is now part of the arsenal of strategies regularly taught as part of contest preparation today. This is the nature of mathematics discovery and education.

In closing I should note that there exist contests that require a higher level of prerequisite knowledge (e.g., undergraduate mathematics instead of high school or middle school), the most famous of these is the William Lowell Putnam competition.

*"Likelihood" in this case might be established by imagining that we could gather a random sample of students with the required knowledge and see what proportion could answer the question--in other words, do something not unlike entering them into a contest.