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Lately I have been getting into solving problems in some of the math journals I enjoy reading. More and more I find that solvers employ a theorem or identity that makes solving the problem much easier. Sometimes, that identity or theorem is one I am not familiar with.

For instance a few weeks ago I saw the Stolz-Cesaro Theorem used on a problem in the Fibonacci Quarterly. It was used in a very slick way, and was a theorem that up to that point I was unfamiliar with.

My question is: what are some of the best theorems, identities, inequalities, etc.. that the consummate problem solver should have at their disposal?

FofX
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  • I am pretty sure there is a similar question on the site asking for such a list. Have you tried searching? – Mariano Suárez-Álvarez Dec 10 '14 at 20:32
  • Possible duplicate - http://math.stackexchange.com/questions/178940/proofs-that-every-mathematician-should-know/ – Train Heartnet Dec 10 '14 at 20:38
  • Tychonoff's theorem, Cauchy's integral formula (and its related results), Zorn's lemma, Banach-Alaoglu theorem, Riesz representation theorem and contraction mapping principle are some big ones that have been really useful for me. – Cameron Williams Dec 10 '14 at 20:41
  • "(big-list) Please do not use this as the only tag for a question." – Thomas Dec 10 '14 at 20:55
  • I did a very quick search, and @Thomas: I wasn't sure what other tags I should use. I tried "identities" or "thoerems" but neither of those existed. But I will put those in. – FofX Dec 10 '14 at 20:57
  • @CameronWilliams Tychonoff's Theorem and Zorn's Lemma are the same thing! And I think Banach-Alaoglu is just about equivalent to those. To the OP I would also add other common independent axioms of ZF: CH, $\neg$CH, MA, etc. – Forever Mozart Dec 10 '14 at 21:13
  • @TomCruise Most definitely. The particular interpretations are important I suppose since they get used all over. – Cameron Williams Dec 10 '14 at 21:29
  • Must Have Theorems, Identies, etc. - From what domain? – Lucian Dec 10 '14 at 21:42
  • @Lucian well from analysis, number theory, combinatorics, etc... anything from the big areas of study that would be most useful in solving problems. – FofX Dec 10 '14 at 21:56
  • The answer for each of this individual subdomains is in itself a big list, or even an entire manual. – Lucian Dec 10 '14 at 22:01
  • Big list should surely not be the only tag here - this is too broad for me. – Mark Bennet Dec 10 '14 at 22:42

1 Answers1

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Here is newbies list,a staret pack of a kind:

-Mean value theorem

-Chain,addition,multiplication and other such rules for limits and derivatives

-Cauchy mean value theorem

-Rolles theorem

-Recursion theorem

-Variations of axiom of choice

-Darboux theorem

-Fundamental theorem of arithmetic

-Division algorithm

-Pigeonhole principle

-Binomial theorem

Vanio Begic
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