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In one month, I'll be writing a second round of a mathematical Olympiad.

My biggest concern is geometry. While I think I'm doing pretty well in number theory, algebra, combinatorics etc., I still can't say I really understood Olympiad geometry (and bashing is not always possible). Obviously, there are always problems one can and one can't solve, regardless of the training, but my question is:

What is the best books/set of exercises/article, that I could study from and get a greater grasp of Olympiad-level geometry a month before the competition?

I mean, a full month, $30$ days, no school or any other activities.

I must say, I'd prefer ones that don't use overly sophisticated notations and don't go into super-advanced theorems you happen to use once or never. And I'm aware that a month is not really a long time and this question may seem a bit like: "quickly, what should I learn to win an Olympiad". I don't mean it. I'm just asking - what to study Olympiad geometry from to greatest benefit while certainly greatly determined to?

EDIT: I'd like to address some comments regarding taking Olympiad too seriously. While I agree to great extent with this reasoning and I appreciate your concern, I'd like to clarify: it's not that I'm going to be working $\frac{24}7$ on maths only. I'd just like to work with the best resources available and thus not waste my time on exercises that aren't of much benefit to my geometry skills.

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    $\begingroup$ See his answer. artofproblemsolving.com/community/forum-user-search/56597/6 $\endgroup$ – Takahiro Waki Jan 28 '17 at 5:37
  • $\begingroup$ Prasolov $\endgroup$ – zhoraster Jan 30 '17 at 6:10
  • $\begingroup$ Schaum's Outline of Geometry. $\endgroup$ – Hazem Orabi Jan 30 '17 at 10:32
  • $\begingroup$ Another possibility is to create more answers than questions, preferably with a geometry tag, at Mathematics Stack Exchange. $\endgroup$ – Han de Bruijn Jan 30 '17 at 16:15
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    $\begingroup$ As an IMO gold medalist, I'm a bit concerned with this "full month without any other activities". Olympiads are made for fun, not for grind. I think it was one of my major mistakes to spend pretty much time for honing my skills in solving these "problems for fun" instead of doing serious mathematics. Just a caution though, maybe it is what you really like to do. $\endgroup$ – Wolfram Feb 1 '17 at 22:31
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For an adequate update over a one-month period, I would suggest you:

  • "Euclidean Geometry in Mathematical Olympiads" by Evan Chen: this is a problem-solving book focused on Euclidean geometry, suitable and specifically written as a preparation for mathematical olympiads;

  • "Problems in Plane Geometry" by Igor F. Sharygin: it has several "non-standard" problems with increasing levels of difficulty, so that it is useful to understand some issues of plane geometry not often described in standard books, such as what additional constructions can be made, which "alternative" pathways could be used to arrive at the solution, and so on;

  • "Geometry Unbound" by Kiran S. Kedlaya, a very good paper structured in the form of a textbook, which starts from rudiments and arrives to the most modern areas of geometry, including inversion and projective geometry;

  • lastly, a complete book is "Problems in plane and solid geometry" by Viktor Prasolov (already cited in one of the comments), a comprehensive 600-page text with thousands of problems and detailed solutions covering all areas of plane and solid geometry.

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I'm supporting Wolfram's comment, You shouldn't take Math Olympiad too much seriously. Take this as fun/relaxation. I would suggest the following books (Please read the whole post) -

  • Geometry Revisited by Coxeter & S.L. Greitzer. This book is must. You should have completed it at least. This contains the very basics of Olympiad Level Problems.
  • 103 Trigonometry Problems from USA Math Camp by Titu Andrescu. This is also a must. All the books from USA Math Camp is really cool. i.e. 101 Problems in Algebra, 102 Combinatorial Problems, 103 Trigonometry Problems from USA Math Camp, 104 Number Theory Problems, all by Titu Andrescu. You must take a look at these books, as geometry is not all in Math Olympiads. In our country we are assured that if anyone completes these 4 books, he doesn't need to worry about Nationals.
  • Problems in Plane and Solid Geometry by Viktor Parasolov. This book covers all sides of plane and solid geometry. The book is of 495 pages, only with problems. So, I don't think you can cover this in 30 days only. But you should solve at least from chapter 1 to 5. By this your idea about the theorems will become very clear. And the solutions of this book are really cool.
  • Geometry Unbound by Kiran S. Kedlaya. At last this. This books is specially written for the students preparing for IMO. You should touch this book after reading the previous books I suggested. Our country coaches said that if you can solve at least 35% of this book, no one will be able to defeat you to take your place in IMO Team.

Geometry is not all of the Math Olympiad. Almost all coaches say that you may not solve Algebra, you may not solve Combinatorics or a number theory problem, But you should be able to solve the Geometry one.
Here are solving books that you should try, for getting better place in Olympiad-

  • Number Theory Structures, Examples, and Problems also by Titu Andrescu. I think this is the best book on Number Theory I seen. It covers all sides of Number Theory. This is also written mainly for the students preparing for IMO.
  • Principles and Techniques in Combinatorics, this cover many thing from beginner to advanced. But I think 103 Combinatorial problems will be sufficient, If you have previous knowledge.
  • Functional Equations also by Titu Andrescu. All you need about Functional Equations.
  • Elementary Number Theory by David M. Burton. It contains most of the theorems you need. But I prefer Titu Andrescu's book most. If you ave time you can take a look at this after Titu's book. Then you'll need to see only the exercises.

As you don't have so much time, here is my suggestion in which order you should read the books -

  • One fourth of the problems in Olympiad will be from Geometry. It is not a good idea to spend all time in Geometry. I think you should read the 4 books from USA Math Camp. Each of them will need 3 days if you try heart and soul. One book may take near 3 days or 4. You should be able to read all theories of each of these books in 12 hours, then 24 hours to solve the Introductory problems, and then 36 hours to solve the Advanced problems. So, here near 14 days will be gone.
  • After this you need to see in which side you are weak. Then read books on that side on this 4 days. Now each 4 days you need to cover a whole topic. Spend more time where you are weak and less time where you are strong.
  • If you see you are weak in geometry. Then first read the book of Viktor Parasolov. I don't expect that one can cover this in even 6 months. Try heart and soul. For a second round you will not need all. Try to solve 5 chapters.
  • You must look at Titu Andrescu's Number Theory book. He is Director of MOP, Head Coach of the USA IMO Team and Chairman of the USAMO. His book is also great. But you need all of them to be solved. Just see the techniques used in the book, the approach to solve NT Problems. You should take this book after everything covered on other sides. And try this book till the end.
  • As I am a contest programmer I didn't need Combinatorics much. They always seem easy to me. I think this is easy to all. You shouldn't spend more than 2 days on it.
  • I don't expect that after all these you will have any time. But if you have time. See Geometry Unbound. Try to solve at least 10%. If you can't solve a problem you don't need to stuck at that problem. Just continue to next problem.

Note: You should see more and more SOLUTIONS even if you solved them. You need to see the techniques. Seeing solutions has a bad effect but I think not in this area. Weather or not you solve the problem, see the solution / technique.

Good luck! Wish you get chance into IMO Team :)

P.S: Ignore my bad English. I am not responsible if the links I provided violates any copyright Law. As I just searched google and gave the first link that came up.

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