How likely is it that a mathematics student can't solve IMO problems?

I think the problem was "intermediate level" ...

I'd say it is quite likely. I personally know several mathematicians and mathematics students who would have trouble solving some of the IMO problems. At least by using elementary means. There are some IMO tricks (especially in combinatorics, like double counting) which are used a lot by research mathematicians working in some particular areas. So they wont have much trouble in solving those IMO problems. But I doubt that someone working in algebraic geometry would be able to solve all the inequality problems appearing in IMO. Regarding your second question, I personally don't have any fear of not being able to do something that "a high-school student could". That statement sounds quite absurd to me. There are several things that a high school student could but I can't do and vice versa. In either direction it's just not that big a deal. Moreover, research maths is not a competition. People develop an expertise in a particular area by years and years of preparation and then collaborate with each other to further develop the field of mathematics.

I didn't do many math competitions before college. While I was at Caltech, I took the Putnam twice. I was too embarrassed to actually go get my score, but my best guess that each time I probably got 1 problem (out of 12). In some of the hardest years, the median score is 0. So yes, doing competition problems takes a lot of specialized practice, and it's generally considered a poor indicator of success in math research.

I'd say quite high. It isn't that the problems are difficult, its just that they are broad. Maths is a huge field. For example if you are a specialist in Abstract Algebra, you can be poor at trigonometry say. I teach mathematics at high school, and I am certainly capable of solving all the problems, but probably not straight off. I would have to train for them. Very often I have to revise stuff I haven't used for a few years. I have to do the equivilent of a college maths course/ freshman year at university every year just to keep in touch. One of the hardest things to teach about Mathematics is that being wrong is not a problem. The whole point is I will look at the problems posed and make a reasonable attempt, some can really only be solved by understanding a specific piece of data, which is memory recall, not Mathematics. I'd say if I sat the IMO with my students I wouldn't always get the best score, however if money depended on it and I had a chance to train (only a couple of days), I would win every time. I have absolutely no embarrassment about being wrong, in fact I'd say the opposite was true. If you are never wrong you aren't trying.

It is said that in 1988 (Australia), during problem selection, one of the problems was so hard that several of the top number theorists in Australia failed to solve it. However, they put it on the exam anyway, and multiple IMO contestants received a perfect score on that problem. You can view the problem here: http://www.artofproblemsolving.com/Wiki/index.php/1988_IMO_Problems/Problem_6

Math Research and IMO (International Math Olympiad) Math are different 'things'.A good analogy given by a famous Chinese Math professor, who criticized the "IMO Craze" in China, since 1985 as China wins the World IMO Championship and hundreds of IMO Gold medals for continuously 3 decades :--Math Research = Martial Art (aka Kungfu 武术功夫); IMO Math = Acrobatics (杂技, not real kungfu).Prof S.S. Chern (陈省身 Wolf Prize 1983) and Prof S.T. Yau (邱成桐, Fields Medal 1982) were always surrounded by keen IMO Math students for tough IMO questions, to whom the 2 professors just squarely replied "I don't know how to do it!". It was reported that some Chinese IMO Gold medalists entered PhD class at Harvard and failed, because PhD Math research problem has no quick solution with known techniques, it usually takes many years to see result, unlike IMO Math questions with known solution by astute tricks.Many years ago in a Singapore seminar, I asked Prof. Pierre-Louis Lions (1956 - , Fields Medalist 1994) for his opinion on IMO. He told us when he represented France in the IMO competition, he spent the few days there staring at the ceiling, not knowing how to solve the problems.Looking at the past winner list of Fields Medalists, majority of them were not IMO Medalists -- except Terence Tao (陶轩哲, Australia), Grigori Perelman (Russia), Timothy Gowers (UK), Ngô Báo Châu (吴宝珠, Vietnam), etc.Interesting to note that the "IMO Champion" China has yet to produce a single Fields Medalist todate. In parallel, the "Fields Medal Champion" France (wins 1/3 of the medals so far) has never been the "IMO (team nor individual) Champion". However, a small country Singapore has produced an individual "IMO Champion 2012" (Lim Jeck 林捷) , the only contestant in the world that year with perfect score.In conclusion, IMO Math is like acrobatics doing fantastic Math "stunts" (特技), it is impressive but not the real Math which requires deep thinking, perseverance in finding the Universe's truth, creating new mathematical tools (Category 范畴, Quantum group 量子群, Homology 同调, Homotopy 同伦, Sheaf 束, Motif 动力, Fiber Bundle 光纤丛, ...) to explore the vast scientific frontier...P.S. Fields Medal is the equivalent of Nobel Prize (Math) but tougher to get, only awarded every 4 years and for age below 40. (unlike the yearly Nobel Prize for any living person of any age).[Edit] To illustrate my point, let's use FLT (Fermat's Last Theorem) as example: Let x, y, z be any natural number, xn+yn=znx^n + y^n = z^n is FALSE for any positive integer n > 2 It took 357 years for mathematicians to prove FLT, finally completed by Andrew Wiles in 7 years using almost all the modern math (Galois Group, Elliptic Curve, Modular Form, Taniyama-Shimura-Weil Theorem...) not existing in 1637 AD.However, it takes only a few minutes by IMO-trained kids (with simple Number Theory knowledge ) to solve the similar "FLT problem" below:Prove: (1) 398799+436599=4472993987^{99} + 4365^{99} = 4472^{99} is FALSE ? (Technique) 3987 : 3+9+8+7 = 27 (divisible by 3) 4365: 4+3+6+5 = 18 (divisible by 3) These 2 numbers (3987 & 4365) are divisible by 3, and their power (any, 398799,4365993987^{99}, 4365^{99} ) divisible by 3, so their sums (after power) too divisible by 3.However, 4472: 4+4+7+2 = 17 is not divisible by 3, so the power (any, 4472994472^{99}) too.Hence the equation (1) is FALSE !

Mathematics research is different from mathematics problem solving, though the skills do sometimes intersect and can benefit from synergy if you're good at both. Research often has vague goals because it is not known beforehand where a particular line of study will lead to (otherwise it wouldn't be research..). This is what math Ph.D students work on. They have a problem with no known solution and they are trying to make even incremental progress towards solving it. The problems are nebulous and poorly defined though, because it might have to be reformulated many times after further study shows that certain approaches will not yield satisfactory answers. Problem solving competitions present problems which are completely unambiguous and are already known to be solvable. You won't see a question like: "can we efficiently parallelize algorithm X? How does the resulting algorithm compare to existing ones?" Therefore the skillsets for the two aren't the same. It isn't good enough to be simply "good at math," for either.

If I remember correctly, "The Art of Problem Solving" by Terrence Tao goes through the solution of an IMO number theory problem and also describes how several professional number theorists tried and failed to solve it. This might be the same problem is referring to.

There is a large body of Olympiad-centric problem solving techniques that don't show up all that often in either mathematical research or theory-based study and problem sets, which means that it is very plausible that even a skilled mathematician might struggle with some of these problems. I'm not referring to the basics of proofs like induction, but, say, very particular inequalities which are generally only useful for these sorts of competition problems. I'd say that it will depend very much on the specialty of the mathematician; those working in combinatorics will probably stand a better chance at keeping these sorts of skills fresh. Keep in mind that, especially with the IMO, there are usually one or two Euclidean geometry problems as well, and I'd be willing to wager that many students lose touch with the artistry of that subject soon after beginning their undergraduate studies. As a personal example, I'm fairly bad at IMO problems but I did publish research in mathematics as an undergraduate, so I'd like to think I'm not wholly incompetent. But the research was carried out over the course of several months and involved thinking deeply about a few more theoretical constructions. It was more about probing just a tad more deeply in a particular direction than had been probed before and less about being unusually clever with manipulating inequalities. As another example, the most talented young prodigy I ever met, while doing research another summer (he was years younger than me and still in high school), was not really any good at all at the Putnam. But such is his reputation and talent in research that I'm not going to provide his name, because his name is all over the internet.