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It's pretty common for people to learn and apply university-level math yet not be able to solve competition problems.

Isn't this sentence a contradiction? Please unpack this sentence? Is https://redd.it/lpbxos relevant ?

I agree, weak math graduates can fail to solve competition problems (henceforward CP) at their degree level.

But I am wildered if B.Sc. grads can't solve high school (HS) CP, eg International Mathematical Olympiad.

I am wildered if M.Sc. grads can't solve undergrad CP, eg William Lowell Putnam Mathematical Competition.
I am more wildered if MSc's can't solve HS CP!

I am bewildered if Ph.D.'s can't solve undergrad CP!
I am even bewildered if PhD's can't solve HS CP!

If a graduate can't solve CP designed for a LOWER degree level, then she failed to learn and apply university-level math. Correct? What's wrong with my reasoning?

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There's a huge difference between math degree programs and math competitions.

  • Degrees are about content knowledge. The way a student graduates from a degree program is by learning and evidencing a base level of competency in some further fields of math. The problems here are intentionally selected, organized, scaffolded to be fairly solvable if the student paid attention to what was covered in class.

  • Competitions are about problem-solving insight. It doesn't matter if your base level of competency extends into levels of math further than what's tested in the competition. It's about how good you are with the tools at the level of the competition. The whole point of a competition problem is for it to be very difficult to solve even if you know the underlying content. The goal is to "spread out" students' performance on the basis of their ability to think insightfully about these kinds of tricky problems.

And more concretely: have you seen the caliber of some of these competition problems? I mean, just look at Problem 3 on the 2023 IMO. Sure, if you're an average student with an undergrad math degree, then you can probably do basic induction and divisibility proofs on the fly, and you're probably able to (or at least were able to at one point) reproduce proofs of key theorems in real analysis and abstract algebra, but you're probably going to have no chance against these IMO problems.

And the Putnam?!?! Don't even get me started on the Putnam. The top scores are generally somewhere around 100 points out of 120 possible, and the median score is usually... wait for it... usually no higher than 2 points out of 120 possible. Typically the median is 1 point, and sometimes it's literally 0 points. And typically the students who even take the Putnam at all have mathematical ability well above the average math major. If you take a graduating math major who scored a 0 on the Putnam and put them through 2 years of master's level mathematics, is that going to bring them to the level of collecting half or more (60+) of the total 120 points? Heck no! It would be a "win" to put any points on the board at all.

Addendum. I realized that I haven't talked explicitly about "application" of math in my answer, and one might argue that being unable to solve competition problems is a symptom of being unable to apply what you've learned. The thing is, there are many many ways to apply math, and the vast vast majority have no bearing on competition problems.

Here's a concrete example. Going back to that average student with an undergrad math degree (who can do basic induction, etc.), suppose that student also took a machine learning class and learned how to fit a mathematical model to data using gradient descent. After graduation they become a data scientist and complete a bunch of work projects in which they fit models to data and use those models to make predictions that are useful to their employer. Clearly, they are applying what they've learned -- but have they gotten any better at competition math through this process? Nope!

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Terrell Owens is one of the all-time greatest wide receivers in American football. In 2009 he competed in the ABC Superstars sporting game show. Here's the clip where he fails miserably to get through the first obstacle course, and so gets eliminated in the very first episode:

Terrell Owens on Superstars 2009

How could this happen to a world-class athlete like T.O.? They're simply different games, is all.

No practicing mathematician really cares about game-show competitions.

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    $\begingroup$ I am not American, don't know or care much about football. Can you simplify your example pls? $\endgroup$
    – user21013
    Commented Mar 29 at 9:27
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    $\begingroup$ @user1147844 The context of football is not really important here. The point is that a world class athlete, one of the top 1% of the top 1% in his field, who has significant athletic prowess, failed in an athletic challenge which was not related to his area of exceptionalism. You don't need to understand football to understand that a football game and a televised obstacle course are different. $\endgroup$
    – Xander Henderson ♦
    Commented Mar 29 at 14:03
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    $\begingroup$ Similar examples are, say, basketball players throwing out the first pitch at baseball games. Or for something not so American, there's a video of Adam Ondra (an elite climber) training with an MMA fighter. The way he moves, you'd be convinced he couldn't possibly be good at any sort of physical activity! $\endgroup$
    – Thierry
    Commented Mar 29 at 15:59
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    $\begingroup$ "No practicing mathematician really cares about game-show competitions." Added that to the existing 18 item list of reasons why I'm not a mathematician. Now scoring at 19 :lol: $\endgroup$
    – fedja
    Commented Apr 2 at 22:02
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Focusing just on one aspect of the bewilderment....

But I am wildered if B.Sc. math graduates can't solve high school competition problems, eg International Mathematical Olympiad.

Based on this listing, recent IMOs have included one or two geometry problems. Anecdotally, most math undergrads I knew had not engaged with synthetic geometry since high school, and an average-case high school geometry experience (freshman year??) is not on par with these Olympiad problems. The specialized terminology (e.g., incenter, circumcircle, concyclic points) in some of the problem statements and lack of diagrams would make even content knowledge / topic recall an issue in addition to the overall difficulty level and required ingenuity.

Furthermore, there are suggestions online that success at these problems is related not only to "innate problem-solving ability" but also to accumulating key facts/techniques and pattern-recognizing when to apply certain lines of attack -- skills that are highly correlated with time spent working on similar problems. IMO participants spend a significant amount of time practicing past exam problems, likely far more than a math undergrad for the given class of problems. This effort asymmetry is especially acute for the geometry problems.

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Mathematics competitions involve nonroutine problems and most mathematics courses more heavily emphasize the routine (not to be confused with easy!) components of various subjects. Rather than addressing the original question in its fullness, here is just a fragment:

Can students who are doing well in the Calculus sequence (say, they have taken Calc I, Calc II, Calc III, and are currently taking Differential Equations) solve nonroutine Calculus problems?

The answer is mostly no. See, for example:

Selden, Annie, et al. "Do Calculus Students Eventually Learn to Solve Non-Routine Problems? Technical Report. No. 1999-5." Online Submission (1999). Link (no paywall!).

Even students with full/substantial requisite knowledge of the corresponding mathematical topic, who are presently studying in a relevant mathematics course, are unable to provide completely/substantially correct solutions to nonroutine problems – examples of which are included in the linked study above, and none of which is near the difficulty of a typical Olympiad problem.

Readers may find that this answer offers little else by way of insight as pertains to the original question, although I wish to advocate for first answering a weaker version of a question (can students doing well amid the Calculus sequence and who have the requisite knowledge successfully solve nonroutine Calc problems?) before zooming outward to a stronger question (can students who may be far removed from the material – e.g., years away from a geometry course – and/or have never seen the material – e.g., unfamiliar with stars and bars, proof by induction, generating functions – successfully solve Olympiad style problems?).

I hope the Selden et al piece is of interest for those who are curious about the manifestation of the routine/nonroutine phenomenon presented by the Putnam, IMO, etc.

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Reminds me of Arnold's diatribe

A student who takes much more than five minutes to calculate the mean of sin(x)^100 with 10% accuracy has no mastery of mathematics, even if he has studied non-standard analysis, universal algebra, supermanifolds, or embedding theorems.

Arnol'd, V. I. (1991). A mathematical trivium.

where he prioritizes some arbitrary complicated thing - nevertheless amenable to mechanical solution - in detriment to that which is deeper, more abstract and, dare say, more interesting

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    $\begingroup$ Do you have an interpretation of what Arnold wanted to say here? (I'm sincerely asking because I find it hard to follow Arnold's sentiments) And how it's related to the original question? $\endgroup$
    – Michał Miśkiewicz
    Commented Mar 31 at 20:02
  • $\begingroup$ @MichałMiśkiewicz I understand, from both this article and the "On teaching mathematics" one - where he says "From my French friends I heard that the tendency towards super-abstract generalizations is their traditional national trait. I do not entirely disagree that this might be a question of a hereditary disease, [...]" - that he considers the more 'concrete' parts of mathematics a lot more important than the more 'abstract' ones, and feels that the former serves as the 'true gauge' of mastery of the discipline. The original question makes a similar, maybe the exact same, mistake. $\endgroup$
    – ac15
    Commented Apr 1 at 0:38
  • $\begingroup$ @MichałMiśkiewicz I think Arnold means one should be able to apply basic analysis to such a problem, one he estimates would take much more than five minutes to carry out the mechanical solution, to estimate the value in five minutes or less. IIRC, he makes a similar remark in Huygens and Barrow, Newton and Hooke about $$\lim_{x\rightarrow 0} {\arcsin(\tan x) - \arctan(\sin x) \over \tan x - \sin x}$$ although it's quite possible I've misremembered the limit. $\endgroup$
    – user1815
    Commented Apr 1 at 1:22