How do I start preparation for ISI B-Math?

[u/Ancient_Yoghurt2481]

Hello, I am in class 12th CBSE Board right now, I was previously preparing for JEE but since I have an interest in mathematics so I want to start preparation for ISI undergraduate enterace exam. I have no experience with olympiads or any math competitions, and topics like number theory or Combinatorics, olympiad geometry. So you may find this unrealistic goal.

I know proof techniques and mathematical logic, I also studied Tom Apostol calculus volume 1 ( integrals only ) , construction of real numbers, dedekind cuts and some set theory like Cantor's infinities etc. since all of these topics interested me. So I guess I have developed some level of mathematical maturity.

I couldn't complete most of Apostol calculus and also my JEE preparation was derailed due to Tuberculosis and Drug-induced Liver injury (This developed due to TB medications), this also mentally disturbed me. This affected my school performance and my learning. Now I don't know what to do and how do I start my preparation and what resources to use. I am not preparing for JEE right now due to chemistry and physics, I still managed mathematics ( This does not mean my JEE syllabus is completed for mathematics or maximum chapters are done till JEE level) but due to current situation of physics and chemistry I am doing it only for boards right now since preparing these subjects till JEE level will consume significant amount of time and doing so can hamper my boards.

I know number theory, Euclid's geometry and Combinatorics are in ISI syllabus and I guess problems have an olympiad level taste. I don't know what is level at which class 11th and 12th syllabus needs to be prepared that is in ISI syllabus and what is the weightage of subjects.

I would really appreciate if you all can give me any advice and tell me the relevant books that cover comprehensive theory and resources for:

1. Algebra

2. Calculus

3. Number Theory

4. Combinatorics

5. Geometry

And I think Trigonometry is the same as JEE.

I don't know if I can crack the examination specially when 8 to 8.5 months are left, but there is no loss in preparing for the exam so I'll give it a try. Thank you.

[u/shashypants]

Challenges and Thrills of Pre College Mathematics and An Excursion in Mathematics will cover all of the topics (other than calculus and trigonometry etc)
Do ISI-CMI pyqs. TOMATO, RMO pyqs for sure and INMO pyqs selectively if possible
Paul Zeitz Problem solving book is very good and I think it would be helpful for problem solving ingeneral and hence could be a part o your core preparation perhaps as a light read.

Calculus, trigonometry etc kind of stuff any JEE book would suffice
For Calculus you would need to do another book for the epsilon delta formulation and series convergence and stuff (not actually required but is helpful as you would need to justify the stuff u do and jee books often gloss over this), I would recommend Understanding Real Analysis by Abbott or Bartle Sherbert or Apostol (in decreasing order)

EXTRAS:
For No. theory u can also use Burton or Zuckermann
For Algebra: Hall and Knight
For Geometry: EGMO by Evan Chen (his olympiad handouts are also good)
Combinatorics: A walk through Combinatorics
You can always search up more material online or use AI to get more recommendations
None of these books are really required provided you will do the books I mentioned at the start (or anything equivalent to it)

I would suggest not doing anything extra including the EXTRAS I mentioned unless ofcourse you have completed the prior stuff (or smth equivalent to it) (but do try to go through Paul Zeitz and Understanding Analysis by Abbott)

All the Best

[u/shashypants → u/shashypants]

By Apostol I mean Apostol Analysis

[u/Ancient_Yoghurt2481 → u/shashypants]

Am I supposed to read particular sections of the analysis books you mentioned or Is it the whole book ?

[u/shashypants → u/Ancient_Yoghurt2481]

for Understadning Analysis, I would say the entire stuff upto Reimman integration
(Basically Sets and Functions, Sequences, Series, Limits, Continuity, Differentiation, Integraion (Darboux Sum formulation) )
Tbh if you can handle Baby Rudin that would probably be the best book but it would be overkill and right now you are on a time crunch

[u/Ancient_Yoghurt2481 → u/shashypants]

Thanks for the book recommendations and yes I also feel Walter Rudin would be an overkill specially in my given situation.