Olympiad-difficulty grad problem sets?

[arachnotron]

Hi all,

Does anybody know of any graduate level math texts with extremely difficult problem sets (olympiad level or harder)?

 

[Petek]

Hi all,

Does anybody know of any graduate level math texts with extremely difficult problem sets (olympiad level or harder)?

https://www.amazon.com/dp/3540636404/?tag=pfamazon01-20 and https://www.amazon.com/dp/3540636862/?tag=pfamazon01-20 by Polya and Szego have lots of difficult problems (with solutions). Most of the problems concern real and complex analysis and were designed to lead students into doing their own research. One possible disadvantage is that the problems are from 1925 and earlier.

https://www.amazon.com/dp/0071002766/?tag=pfamazon01-20 by Walter Rudin also is well-known for its difficult exercises (no solutions, though). It's more recent (1987) than the the Polya and Szego volumes.

That's just off the top of my head. Please post again if you're looking for some topics other than analysis.

Petek

 

[snipez90]

Uh, I'm not exactly sure what you mean here. If you are talking about contest mathematics type problems that are specifically meant to test cleverness, then I would guess there are very few, if any graduate texts with that specific aim.

I think most people would agree that you don't need to be able to solve olympiad-level problems to do college math. On the one hand, being able to solve olympiad problems certainly demonstrates mathematical maturity and good problem-solving skills, which are important. On the other hand, higher mathematics demands greater competency with abstract reasoning.

But really, if you are reading a graduate text, and the problems aren't sufficiently challenging for you, then either you're a genius or you chose a rather poor text. Chances are, there will be problems that require as much thought as one would need for an olympiad problem.

 

[arachnotron]

Petek - Thanks! While I'm interested in analysis, I'd be more than happy for a listing of other books in other topics as well! :)

snipez90 - well, I have been spending 6-10 hours per day this summer working on developing my skills on olympiad problems because I wanted to prepare myself for an eventual career as a research mathematician. Now that I have my problem solving skills sharpened, I want to apply them to "real" mathematics. And I was hoping to find a challenge even greater than the olympiad problems.

On a similar note, are research mathematics problems usually harder than the olympiad stuff?

 

[JCVD]

"Research mathematics problems" are by definition unsolved by anyone (or else you wouldn't be researching them), so yes, they are pretty difficult. One of the big differences between a research problem and an olympiad problem is that research problems often require a lot of specific background knowledge to understand the problem, let alone solve it, while olympiad problems typically don't require too much knowledge beneath the surface. Furthermore, olympiad problems are often contrived so that a little bit of cleverness will go a long way, while making progress on research problems usually requires a lot of mucking around.

Also, 6-10 hours a day working on olympiad problems? It's the summer!

 

[snipez90]

Right, as JCVD pointed out, olympiad problems and likely a large majority of the problems in the texts mentioned by Petek have known solutions, and that is a big difference.

As for the 6-10 hours comment, to be fair, it can often take a few hours or longer to solve one.

 

[Petek]

You might want to check out the books in Springer's http://www.springer.com/math/geometry?SGWID=4-10046-580-173621548-DISPLAY_TYPE=DISPLAY_TYPE_TITLES&RESULTS_PER_PAGE=10&resultStart=0 series. Other than that, look for graduate-level texts in whatever subjects interest you.

Petek

 

[bpet]

The puzzles on Project Euler (online) will keep you busy for a while.