Olympiad-difficulty grad problem sets?

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In summary, there are several graduate level math texts with extremely difficult problem sets, such as those by Polya and Szego, Walter Rudin, and books in Springer's GTM series. However, these problems may not necessarily be at the level of olympiad contests. Additionally, research mathematics problems are often even more challenging than olympiad problems, requiring specific background knowledge and more time and effort to solve. For those interested in further challenging themselves, they could also check out Project Euler, an online platform for mathematical puzzles.
  • #1
arachnotron
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Hi all,

Does anybody know of any graduate level math texts with extremely difficult problem sets (olympiad level or harder)?
 
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  • #2
arachnotron said:
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
 
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  • #3
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.
 
  • #4
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?
 
  • #5
"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!
 
  • #6
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.
 
  • #7
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
 
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  • #8
The puzzles on Project Euler (online) will keep you busy for a while.
 

1. What is an Olympiad-difficulty grad problem set?

An Olympiad-difficulty grad problem set is a set of challenging mathematical problems that are typically used for training and selecting students to participate in mathematical Olympiads. These problems require advanced problem-solving skills and a deep understanding of mathematical concepts.

2. Who creates Olympiad-difficulty grad problem sets?

Olympiad-difficulty grad problem sets are typically created by experienced mathematicians or educators who have a strong background in problem-solving and Olympiad competitions. They often use previously unsolved or difficult problems, and carefully select and organize them into a set for students to work through.

3. What topics are typically covered in Olympiad-difficulty grad problem sets?

Olympiad-difficulty grad problem sets cover a variety of topics in mathematics, including algebra, geometry, number theory, and combinatorics. These sets often include problems that require students to combine multiple concepts and techniques to find a solution.

4. How can I prepare for Olympiad-difficulty grad problem sets?

To prepare for Olympiad-difficulty grad problem sets, it is important to have a strong foundation in mathematical concepts and problem-solving techniques. It is also helpful to practice with previous Olympiad problems and familiarize yourself with common strategies used to solve them.

5. Can Olympiad-difficulty grad problem sets help me improve my problem-solving skills?

Yes, working through Olympiad-difficulty grad problem sets can greatly improve your problem-solving skills. These problems are designed to challenge and stretch your thinking, helping you to develop new techniques and approaches to solve difficult problems. Regular practice with these sets can also improve your speed and accuracy in problem-solving.

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