In this post, I will give an overview of the process of selection of the Indian team to the International Olympiads in Math, Physics and Chemistry.
The first round is a written theoretical exam, called the National Standard Examination in Physics (NSEP) conducted every year by the Indian Association of Physics Teachers (IAPT). This exam in held in November (or thereabouts) every year. Here are some old question papers.
The students who get selected in NSEP appear in the Indian National Physics Olympiad (INPhO) in January (or thereabouts). Here is some useful information. This is also a theoretical, written exam. This is however conducted by the Homi Bhabha Center for Science Education (HBCSE, Mumbai).
The students, numbering about 30, who get selected in INPhO are summoned to an Orientation cum Selection camp in HBCSE, Mumbai. At the camp theoretical and experimental examinations are held in order to select the top 5 students who will represent India at the international event, which is in a different country every year!
Chemistry and Biology
The process is very similar to Physics except that the team that represents India consists of 4 students.
This is well summarized here.
The math olympiad program funded by the National Board for Higher Mathematics (NBHM). The program is run from the MO Cell at Indian Institute of Science, Bangalore. Logistically, the country is divided into several regions.
The first round is called the Regional Math Olympiad (RMO) and is conducted in every region in December. (Some large regions have a pre-RMO round too.)
People who pass the RMO give the INMO (Indian National Mathematics Olympiad) in February.
Students who clear the INMO are summoned to a month long training cum selection camp (called the IMOTC) in HBCSE, Mumbai. They form what is called the Junior batch in this camp. They are joined by Seniors (students from previous years’ camps who are still eligible). From this pool, six people are selected to go to the International Mathematics Olympiad (IMO).
Personally, IMOTC-2004 (the first time I went there) was the single best academic experience in my high school life!
There are also International Olympiads in Astronomy and Informatics.
I know that India does really well in Physics and Astronomy Olympiads.
Please Note: The syllabus for the Math olympiad is significantly different from what we learn in classes 11 and 12 (Broadly, it includes four topics: Euclidean Geometry in the Plane, Algebra, Number Theory and Combinatorics). This is not the case with Physics and Chemistry where most of the topics are covered by the JEE syllabus.