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Where can I find all the algebraic identities and their proofs of following types,

  • $a^3+b^3+c^3–3abc≡(a+b+c)(a^2+b^2+c^2–ab–bc–ca)$

  • $a^3+b^3+c^3–3abc≡ \dfrac12 (a+b+c)((a-b)^2+(b-c)^2+(a-c)^2$

  • $a+b+c=2s$ then $a^2+b^2–c^2+2ab=4s(s–c)$.

  • If $a+b+c=0$, then $\left[ \dfrac{a}{b+c} + \dfrac{b}{a+c} + \dfrac{c}{a+b} \right] \left[\dfrac{b+c}{a} + \dfrac{a+c}{b} + \dfrac{a+b}{c} \right]=9$

There are many more similar difficult identities. I need a list of all of them with their proofs.

Any help will be appreciated, thanks in advance.

user103816
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    Many math competition problem books have chapters containing what you're looking for, such as: Xu Jiagu, Lecture Notes on Mathematical Olympiad Courses [for Junior Section Volume 1], Mathematical Olympiad Series #6, World Scientific Publishing Company, 2010, xii + 170 pages. [See Lecture 6: Some Methods of Factorization on pp. 35-40]. At this time there appears to be a .pdf copy on the internet, but since this might not be legally available, I am not giving the URL here. – Dave L. Renfro Jun 18 '15 at 19:13
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    Also, several old algebra texts from the 1800s have these kinds of identities. For example, see see the books by George Chrystal, Elias Loomis, Charles Smith, William Steadman Aldis, and Isaac Todhunter that I posed links to in this 5 November 2009 sci.math post at Math Forum. All of these books are freely available on the internet. – Dave L. Renfro Jun 18 '15 at 19:14
  • @DaveL.Renfro Thanks for the valuable resources. – user103816 Jun 19 '15 at 05:46
  • Why do you need resources, can you prove them yourself? – Autolatry Jun 23 '15 at 10:44

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I've found one book out there which covers these type of identities and problem exercises -- SSC Elementary and Advanced Mathematics.

user103816
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