How likely is it that a mathematics student can't solve IMO problems? Is there a fear of embarrassment in being a math Ph.D. who can't solve problems that high-school students can?

Response by Cornelius Goh on Quora:

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) ［Note *］and his PhD student 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, not 100％ were 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).

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