Preparing for the Putnam Exam.

[PFStudent]

Hey,

I'm currently preparing for the putnam exam offered by MAA at my college. However, I am finding an added difficulty from not having a diverse enough knowledge of mathematics to actually atempt some of the questions. As it stands on average I can attempt seven out of the twelve problems, but the rest I am usually lost on because I have not had that particular math.

In paritcular, I am asking if any of you have suggestions for forums/websites that are devoted to preparing for the putnam exam and discussing how to solve the problems.

In addition, I am also asking for suggestions on textbooks likely to cover some more of the proof type problems I am likely to see on this exam. Unconventional integratrion techniques like: integration by series expansion and integration by analysis of a function's graph. In addition, number theory proofs dealing with primes.

Basically, I am trying to learn the mathematics' "tricks" so often referenced in the solutions to these problems.

Lastly is there like a workbook or some comprehensive text devoted to preparing for the Putnam Exam?

Thanks,

-PFStudent

 

[morphism]

1. www.artofproblemsolving.com

2. Larson, Problem Solving Through Problems.

3. You can get past years' Putnams with solutions online or in the books published by the MAA.

 

[PFStudent]

Hey,

Thanks for the reply morphism.

I have all the (available) solutions and problems to the past putnams, but to be honest some of the math I have not had, so even with the solutions it is hard to comprehend some of them, not having had a formal rigorous mathematics course in that particular subject.

In my case, I have only had,

Calculus: I and II, Multivariable Calculus, Differential Equations and Linear Algebra.

So, I am trying to do my best with what I have, but I guess some more guidance on what more I need to learn (in terms of mathematics) to successfully prepare for the putnam would be appreciated.

Thanks,

-PFStudent

 

[ekrim]

Hey,

Thanks for the reply morphism.

I have all the (available) solutions and problems to the past putnams, but to be honest some of the math I have not had, so even with the solutions it is hard to comprehend some of them, not having had a formal rigorous mathematics course in that particular subject.

In my case, I have only had,

Calculus: I and II, Multivariable Calculus, Differential Equations and Linear Algebra.

So, I am trying to do my best with what I have, but I guess some more guidance on what more I need to learn (in terms of mathematics) to successfully prepare for the putnam would be appreciated.

Thanks,

-PFStudent

I thought that the putnam only relied on a knowledge of those exact courses youve taken? I am taking it this year too, but unfortunately have forgotten all the first year math courses and am reviewing with schaums outline of advanced calculus

 

[morphism]

'Technically', all you need to know is first year university-level math. At least, that's what they claim. But it's total BS. Previous exams have had problems that touched on the following topics: single and multivariable calculus, real and complex analysis, linear algebra, differential equations, calculus-based probability, number theory, functional equations, combinatorics/discrete math and classical geometry. And often more sophisticated arguments are really the only way anyone can see a solution to a certain problem, unless of course they've seen the trick that gives the "elementary" solution. For example, I recall a problem whose only solution that didn't seem out of the blue involved some Fourier analysis, while all the others seemed unmotivated and artificial -- but this is the way olympiad-type math is...

 

[PFStudent]

Previous exams have had problems that touched on the following topics: single and multivariable calculus, real and complex analysis, linear algebra, differential equations, calculus-based probability, number theory, functional equations, combinatorics/discrete math and classical geometry. And often more sophisticated arguments are really the only way anyone can see a solution to a certain problem, unless of course they've seen the trick that gives the "elementary" solution. For example, I recall a problem whose only solution that didn't seem out of the blue involved some Fourier analysis, while all the others seemed unmotivated and artificial -- but this is the way olympiad-type math is...

Exactly, that is my point. So basically it is better play this "safe than sorry" so to speak. What I mean is, treat the putnam as if you need to have mastered (or know most of the "tricks") to the subjects listed below.

So far I have used the following texts however, would like suggestions on more formal/rigorous texts ideal for elaborating on the "tricks"/methods of approach to successfully and rigorously solving the putnam problems.

Single and Multivariable Calculus - I used Calculus: 8th Edition (by Larson, Hostetler, Edwards.

I found this text rigorous enough for my calculus courses, however not as rigorous as I would have liked to be, for example there is no section on dealing with limits of two, three, and higher variables and there is no elaboration on the unconventional techniques of integration like: integration by series and integration by function analysis. In addition, I did not like that they did not have proofs of the integrals they listed in their integration table.

Any suggestion on a more rigorous text covering the above?

Real and Complex Analysis - Is this all one course? And is this also commonly referred to as "Advanced Calculus?" In addition, do you have a textbook suggestion for this course(s)?

Linear Algebra - I forgot what text I am using, but it's pretty good. However, if you have a suggestion please let me know.

Differential Equations - My differential equations text is pretty good as well (forgot that textbook name too). Let me know if you have a suggestion.

Calculus-Based Probability - I have not had this course yet. However, I really would like a textbook suggestion for this course.

Elementary Number Theory - I have not had this either, at our college this course is a 300-level (junior course) any suggestions on a textbook for this?

Functional Equations - By functional equations, what exactly do you mean, like f(x,y,z)?

Combinatorics/Discrete Math - Yea, I could use a suggestion for a good text to this subject. Our college's text is not as rigorous as I would like it to be.

Classical Geometry - Beyond geometry covered in high school and the geometry covered in calculus and differential equations, is there any other geometry I should know for the putnam?

In addition, there are several putnam problems that I do not recognize what branch of mathematics they are classified under.

For example summation of an expression summed from some number to infinity and finding a closed form solution, such that you can find the sum for any "nth" term by plugging in the number, "n." What subject goes over techniques and methods about how to find closed form solutions to the example above? I really would like to know that.

Also, I want a more in depth treatment of series functions, beyond just geometric, telescoping, and harmonic. I mean a comprehensive: list, derivation, and explanation of all the fundamental series' functions. Like a text that goes in depth into the various series' functions out there, how they're derived, and used.

A text that elaborates much more on the application of the taylor series and all its various uses. I've seen way too many times how the taylor series is used to solve a lot of problems on the putnam. From solving simultaneous equations by recognizing components of a binomial expansion to integration by taylor expansion.

Counting Problems. This stuff is the subject I have the hardest time with on the puntam (not having had any statistics/probability). Yea, I totally dislike this stuff. I really wish I knew the math to at least begin some of these problems. Any textbook suggestions for this material?

Thanks,

-PFStudent

 

[Hurkyl]

For example, I recall a problem whose only solution that didn't seem out of the blue involved some Fourier analysis, while all the others seemed unmotivated and artificial -- but this is the way olympiad-type math is...

If you could recall the specific problem, I expect that I could demonstrate how someone working on the problem could have uncovered those solutions. It's rather easy for even the most natural solution to appear "unmotivated and artificial" if you've only given superficial thought to the problem!

 

[leon1127]

Exactly, that is my point. So basically it is better play this "safe than sorry" so to speak. What I mean is, treat the putnam as if you need to have mastered (or know most of the "tricks") to the subjects listed below.

So far I have used the following texts however, would like suggestions on more formal/rigorous texts ideal for elaborating on the "tricks"/methods of approach to successfully and rigorously solving the putnam problems.

Single and Multivariable Calculus - I used Calculus: 8th Edition (by Larson, Hostetler, Edwards.

I found this text rigorous enough for my calculus courses, however not as rigorous as I would have liked to be, for example there is no section on dealing with limits of two, three, and higher variables and there is no elaboration on the unconventional techniques of integration like: integration by series and integration by function analysis. In addition, I did not like that they did not have proofs of the integrals they listed in their integration table.

Any suggestion on a more rigorous text covering the above?

Real and Complex Analysis - Is this all one course? And is this also commonly referred to as "Advanced Calculus?" In addition, do you have a textbook suggestion for this course(s)?

Linear Algebra - I forgot what text I am using, but it's pretty good. However, if you have a suggestion please let me know.

Differential Equations - My differential equations text is pretty good as well (forgot that textbook name too). Let me know if you have a suggestion.

Calculus-Based Probability - I have not had this course yet. However, I really would like a textbook suggestion for this course.

Elementary Number Theory - I have not had this either, at our college this course is a 300-level (junior course) any suggestions on a textbook for this?

Functional Equations - By functional equations, what exactly do you mean, like f(x,y,z)?

Combinatorics/Discrete Math - Yea, I could use a suggestion for a good text to this subject. Our college's text is not as rigorous as I would like it to be.

Classical Geometry - Beyond geometry covered in high school and the geometry covered in calculus and differential equations, is there any other geometry I should know for the putnam?

In addition, there are several putnam problems that I do not recognize what branch of mathematics they are classified under.

For example summation of an expression summed from some number to infinity and finding a closed form solution, such that you can find the sum for any "nth" term by plugging in the number, "n." What subject goes over techniques and methods about how to find closed form solutions to the example above? I really would like to know that.

Also, I want a more in depth treatment of series functions, beyond just geometric, telescoping, and harmonic. I mean a comprehensive: list, derivation, and explanation of all the fundamental series' functions. Like a text that goes in depth into the various series' functions out there, how they're derived, and used.

A text that elaborates much more on the application of the taylor series and all its various uses. I've seen way too many times how the taylor series is used to solve a lot of problems on the putnam. From solving simultaneous equations by recognizing components of a binomial expansion to integration by taylor expansion.

Counting Problems. This stuff is the subject I have the hardest time with on the puntam (not having had any statistics/probability). Yea, I totally dislike this stuff. I really wish I knew the math to at least begin some of these problems. Any textbook suggestions for this material?

Thanks,

-PFStudent

From my point of view, your knowledge is not at all enough to archieve good marks on this exam... Rather I suggest you get a book in competition type problem. i.e book by T. Andreescu has a large collection of problem aiming competition type problem solver. Rigourous knowledge of mathematics is a necessary but definitely not sufficient condition on these type of exams.

 

[PFStudent]

Rather I suggest you get a book in competition type problem. i.e book by T. Andreescu has a large collection of problem aiming competition type problem solver. Rigourous knowledge of mathematics is a necessary but definitely not sufficient condition on these type of exams.

Do you have any particular book suggestions by T. Andreescu for specifically preparing for the Putnam. From what I know about the author I believe the majority of his texts are more geared to preparation for the IMO. However, if you have any suggestions on particular texts with "putnam-like" problems please feel free to list them.

Thanks,

-PFStudent

 

[leon1127]

Try to search "Putnam" and "Andreescu" on Amazon. Moreover, his IMO books should cover all the topics you need up to calculus related problem.

 

[PFStudent]

Try to search "Putnam" and "Andreescu" on Amazon. Moreover, his IMO books should cover all the topics you need up to calculus related problem.

Thanks for the suggestion. However, how should I approach the mathematics material that is "post-caclulus" related. Any suggestions on that? Like textbook suggestions?

Thanks,

-PFStudent

 

[leon1127]

Thanks for the suggestion. However, how should I approach the mathematics material that is "post-caclulus" related. Any suggestions on that? Like textbook suggestions?

Thanks,

-PFStudent

I don't know much about patnum exam anyway. For problem solving calculus book, try spivak and apostol. The inequality book by Hardy, Littlewood and Polya is very good in differential inequalities too.

Good luck

 

[mvrusso]

Putnam graders

Does anybody know who grades the Putnam exams and how to contact them?