Could you kindly point out to me, some good contest-preparation book's to develop theory and problem solving skills in Algebra?
It would be good if the book is less of theory and more of problems. I have "Principles and techniques in Combinatorics", but I would like some more difficult books, explaining theory well(this books theory is meh...).
The same for Algebra(polynomials, functional Equations, inequalities, etc.)
Thanks for your help!!!
Let me answer this for enumerative combinatorics and inequalities; others can deal with the rest.
Integer sequences and recursion:
Quoting from elsewhere:
Enumerative combinatorics:
A huge list of undergraduate-level introductions to enumerative combinatorics is being kept on https://math.stackexchange.com/a/1454420/ . Note that high-school olympiads are somewhere between undergraduate and graduate level in combinatorics, so a lot of the sources in this list should work. However, most are not problem books. The ones by Bogart, by Andreescu and Feng, and by Chuan-Chong and Khee-Meng are definitely problem books, and the ones by Knuth and by Loehr have a lot of exercises too.
Once you get past this level, you can start reading Stanley's EC1, which has one of the largest collections of combinatorics problems I have ever seen in a book. Beware, though, that they are essentially unbounded in difficulty, and mostly without solutions (references are given at best).
Inequalities:
Much of what follows is quoted from my FAQ; note however that I haven't updated this part of it for a decade:
Thomas Mildorf has written nice notes on inequalities (2006).
Hojoo Lee is rather known for his "Topics in Inequalities".
Kiran Kedlaya has another text similar to the two above.
Vasile Cîrtoaje, Algebraic Inequalities - Old and New Methods, Gil: Zalau 2006. This one is 480 pages long and features many interesting tactics and examples on solving inequalities. Unfortunately, you are not likely to enjoy all these 480 pages, because many of the modern methods for solving inequalities include applications of calculus and involved computations. However, a lot was done to keep these ugly parts at a minimum while keeping the whole power of the new methods. The RCF ("right convex function"), LCF (guess what this means) and EV (equal variables) theorems as well as the AC (arithmetic compensation) and GC (geometric compensation) methods provide a means to solve >95% of olympiad inequalities using rather straightforward - not nice, but doable - computations. All of these methods are extensively presented with numerous examples. A short chapter underlines applications of the (underrated) generalized Popoviciu inequality. Finally, and - in my opinion - most importantly, a lot of exercises with solutions are given which don't require any strong new methods, but just creative ideas and clever manipulations.
Pham Kim Hung, Secrets in Inequalities (volume 1), Gil: Zalau 2007. This one has 256 pages, and is remarkable for mostly avoiding computations. Numerous creative ideas can be found here - I was particularly surprised about some of the applications of the Chebyshev and rearrangement inequalities. Besides, a good introduction into the applications of convexity is given. I would recommend this book to olympiad participants who look for challenging problems and intelligent techniques without the aim to be able to kill every inequality.
Titu Andreescu, Marius Stanean, 116 Algebraic Inequalities, XYZ Press 2018 (order from AMS). I'm adding this one because it's out on display here in Oberwolfach and because it looks nice. It is a lot more systematic than its title suggests (the problems are grouped by method, and the methods are explained with examples).
G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, 1934 is a classic. The notation is somewhat dated, but there is much to be learned from here.
Polynomials:
Titu Andreescu, Gabriel Dospinescu, Problems from the Book, 2nd edition 2010 (order from AMS) is one of the best advanced problem books I know. It is not fully devoted to polynomials, but chapters 10, 11, 21, 23 are devoted to them and they make several cameos in other parts.
Titu Andreescu, Navid Safaei, Alessandro Ventullo, 117 Polynomial Problems from the AwesomeMath Summer Program, XYZ Press 2019 (order from AMS). Once again, I've got alerted of this one by its presence in the Oberwolfach library. This one appears to be more of a grab-bag than the one on inequalities, and I find the problems a lot less appealing. But this is largely a problem with the topic. Polynomials become much more interesting once you have seen some abstract algebra.
'Problem Solving Strategies' by Arthur Engel is a classic for math olympiad training at a fairly advanced level. Contains hundreds of problems that take one line to state, two lines to solve with basic means and are still difficult enough to humble olympiad veterans.
