I only have experience with the IMO so I won't speak for physics, but I guess the situation is mostly similar. It's 2 years since I participated so some new instructional material that I'm unaware of may have come out in the meanwhile.
Unfortunately 3-4 months is not a lot of time to prepare for olympiad level math. Unless you are a very exceptional genius or have already studied a fair bit of extracurricular math you're unlikely to make the cut. In my experience a lot of Indians think highly of IMO and put in a lot of time to compete. In the end only 6 can represent each country and there are many smart Indian students who have likely prepared for years now.
That being said I will try to give a little guidance in case you want to give it a try anyway (whether for the experience, to challenge yourself, or just because you believe you can overcome the odds). Paul Zeitz' book The Art and Craft of Problem Solving is a great book and many participants at the IMO seem to have started out with this. Everything in it is relevant and it's a pretty good exposition (except for possibly the last chapter which is not that relevant). However on its own it's not enough for sufficient preparation to compete at IMO-level. If you can already solve some recent IMO problems (even if they are problems 1 and 4), then it may be too basic for your needs. You need to supplement with further instruction after Zeitz' book. Common choices are:
- The Art of Problem Solving series: very basic and moves really slowly, but gives good foundations if you have the patience and time to go through them. Given your short timeframe this is likely not that useful for you.
- Problem-Solving Strategies by Arthur Engel: Great book with many excellent problems and good techniques. Fairly advanced, but this is the type of book you need for actual IMO problems. Not really meant to be read from cover to cover, but you should read about specific topics.
- Geometry Revisited by Coxeter: Pretty universally regarded as "the" geometry book for olympiads. At least most participants at IMO seemed to at some point have been in possession of this book and most liked it.
- Books by Titu Andresscu (and co-authors). In particular I recall 104 number theory problems, 102 combinatorial problems, 103 trigonometry problems, mathematical olympiad challenges, complex numbers for A to ..Z. These all have fairly modest prerequisites but quickly moves onto pretty interesting problems and techniques. They all have older IMO problems.
Apart from these books most teams hand out material to people passing the first couple of national tests as part of the training for international competitions. In particular inequalities are usually taught from notes combined possibly with Engel's book. For the very basic techniques like basic application of the AM-GM or Cauchy-Schwarz inequality the art of problem solving books and Paul Zeitz' book are useful, but approaching the IMO this is assumed as known as it was likely used on the national test. Some of these notes are published online. For instance Kiran Kedlaya's article on inqualities:
http://www.artofproblemsolving.com/Resources/Papers/KedlayaInequalities.pdf
was very useful in my preparation for IMO. The only useful actual book on inequalities I encountered was the excellent book "The Cauchy-Schwarz Master Class" by Steele, but for the purposes of the IMO it is likely mostly overkill (except perhaps if you're looking for those last 1 or 2 points on a problem 6).
Geometry is usually also either taught from notes, Coxeter, or just straight classroom instruction + problem solving. Most geometry at IMO is fairly basic Euclidean stuff, but usually quite tricky so the main form of instruction is usually in the form of problem solving. You need to solve a lot of geometry problems. When you know basic stuff like facts about parallel lines, exterior/interior angles, similar triangles, inscribed circles, medians, etc. you are usually set to go. The only advanced theory used in IMO problems I can think of is the operation of inversion which can be quite useful, but if I recall correctly is not in the curriculum for the IMO so all accepted IMO problems have an official solution that does not use inversion.
Number theory may also be taught from notes, but books such as 104 number theory problems also make for a great introduction of precisely the techniques and theory used at the IMO. I have heard the book "the theory of numbers" by niven be recommended to advanced students, but this is way overkill for IMO-level stuff and more useful as a preparation for more advanced college courses if that is what you want.
If you have any specific questions on the IMO feel free to ask. I participated twice for a fairly weak team (compared to India at least) so national qualifiers were not as grueling for me.
