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RMO  The Mathematical Olympiad Programme
in India, which leads to participation of Indian students in the International
Mathematical Olympiad (IMO) is organized by the Homi Bhabha Centre for Science
Education (HBCSE) on behalf of the National Board for Higher Mathematics (NBHM)
of the Department of Atomic Energy (DAE), Government of India. This programme
is one of the major initiatives undertaken by the NBHM. Its main purpose is to
spot mathematical talent among preuniversity students in the country.
Stages
 Stage I: Regional Mathematical Olympiad) (RMO and preRMO,
 Stage II: Indian National Mathematical Olympiad (INMO),
 Stage III: International Mathematical Olympiad Training Camp (IMOTC),
 Stage IV: Predeparture Training Camp (PDT) for IMO,
 Stage V: Participation in International Mathematical Olympiad (IMO).
Eligibility
Students from Class IX, X, XI and XII are eligible to appear for the RMO. Highly motivated and well prepared students from Class VIII may also take the RMO at the discretion of the Regional Coordinator.
Qualification and Eligibility Criteria for the Second Stage
The RMO is a 3hour written test with 6 or 7 problems. On the basis of the performance in RMO, a certain number of students from each region is selected for Stage 2 (INMO).
Mathematical Olympiad
The Maths Olympiad Programme which leads to participation in the IMO International Mathematical Olympiad,constitutes of the following stages:
Stage 1: Regional Mathematical Olympiad (RMO)
Regional Mathematical Olympiad is held in each region normally between September and the first Sunday of December each year. A regional coordinator makes sure that at least one centre is provided in every district of the region. RMO is a 3hour written test which contains about 6 to 7 problems. All highschool students up to class XII are eligible to appear for RMO. To appear for RMO, interested students should get in touch with the RMO coordinator of their region well in advance, for enrolment and payment of a nominal fee.
Each regional coordinator has the freedom to prepare his/her own question paper or to obtain the question paper from NBHM. The regions which opt for the NBHM question paper hold this contest on the 1st Sunday of December. On Based on theirperformance in RMO, certain number of students from each region are selected to appear for the second stage. The regional coordinators charge nominal fees to meet the expenses for organizing the contests.
Stage 2: Indian National Mathematical Olympiad (INMO)
Indian National Mathematical Olympiad is held on the first Sunday of February every year at differentcentres in different regions. Just the students who are selected on the basis of RMO from various regions are eligible to appear for the INMO. It is a 4hour written test. Its question paper is set centrally and the test is common throughout the country. Only the top 3035 performers in INMO receive a merit certificate.
Stage 3: International Mathematical Olympiad Training Camp (IMOTC)
This is a training level for the INMO certificate awardees. They are invited for a month long training camp (for junior batch) conducted in MayJune, each year. Also in addition, INMO awardees of the previous year who have satisfactorily completed the postal tuition throughout the year are again invited for a second round of training (called the senior batch).
Stage 4: International Mathematical Olympiad (IMO)
A leader and deputy leader are chosen by the NBHM from among mathematics teachers/researchers involved in the Mathematics Olympiad activity.So the team selected at the end of the camp, the leader and the deputy leader, represent India at the International Mathematical Olympiad that is normally held in July in a different member country of IMO each year. The IMO consists of two written tests held on two days with a gap of at least one day. Both the tests are of fourandahalfhours.
India has 25 regions along with three independent groups that conduct Regional Math Olympiad. Each region has its own Regional Coordinator, who is responsible for conducting RMO in his/her region.
They are:
Regional Coordinators have the discretion of using central RMO paper set by the HBCSE or set the examination paper themselves. Regions that choose to go with the centrally prepared RMO paper have to conduct a preRMO to screen students for the central RMO examination.
The format of the preRMO paper and criteria for short listing student for RMO is decided solely by the Regional Coordinator.
Eligibility
Only students of class IX, X, XI and XII are eligible to participate in Regional Mathematical Olympiad. However, Regional Coordinators have the discretionary power to allow any Class VIII student with exceptional mathematical talent to sit for the RMO.
Syllabus for Mathematics Olympiads
Syllabus for Mathematics Olympiads (regional, national and international) is class 9^{th} to 12^{th}standard mathematics.
The typical areas of problems are: number theory,algebra, geometry, and combinatorics.
The topics covered under these areas are: number systems, geometry, arithmetic of integers,quadratic equations and expressions, coordinate geometry, trigonometry,systems of linear equations, factorisation of polynomials, permutations and combinations,inequalities, probability theory,elementary combinatorics, number theory, complex numbers, elementary graph theory and , infinite series.
The syllabus does not include statistics and calculus.
Though the syllabus is roughly spread over class IX to class XII levels, still the problems under each topic are of an exceptionally high level in difficulty and sophistication as compared to the text book problems.
The difficulty level keeps increasing from RMO to INMO to IMO.
Books for preparation of Mathematical Olympiads
The following books treats the topic which are covered in the different levels of the olympiad and also are a rich source of problems
S. No.  Book  Author  Publication 
1  Challenge and Thrill of PreCollege Mathematics  V. Krishnamurthy, C. R. Pranesachar, K. N. Ranganathan and B. J. Venkatachala  New Age International Publishers 
2  Mathematical Challenges from the Olympiads*  C. R. Pranesachar, S. A. Shirali, B. J. Venkatachala, and C. S. Yogananda  Prism Books Pvt. Ltd. 
3  Problem Primer for the Olympiad  C. R. Pranesachar, B. J. Venkatachala, and C. S. Yogananda  Prism Books Pvt. Ltd., #1865, 32nd. Cross, BSK II Stage, Bangalore 560 070. or 49, SardarSankar Road, Kolkata 700029. Phone: 24633890/24633944. 
4  An Excursion in Mathematics  M. R. Modak, S. A. Katre, V. V. Acharya  BhaskaracharyaPratisthan, 56/14 Erandavane, Damle Path, Pune 411 004 
5  International Mathematical Olympiad, Vol I, 19591975  IstvanReiman  Anthem Press (Indian Edition available) 
6  International Mathematical Olympiad, Vol II, 19761990  IstvanReiman  Anthem Press (Indian Edition available) 
7  International Mathematical Olympiad, Vol III, 19912004  IstvanReiman  Anthem Press (Indian Edition available) 
8  Mathematical Circles  D. Fomin, S. Genkin& I. Itenberg  First Reprinted Edition, University Press, New Delhi, 2000 
9  ProblemSolving Strategies  Arthur Engel  Springer 
10  A Primer On Number Sequences  S. A. Shirali  University Press 
11  First Steps In Number Theory A Primer On Divisibility  S. A. Shirali  University Press 
12  Functional EquationsA Problem Solving Approach  B. J. Venkatachala  Prism Books Pvt. Ltd 
*(Contains problems and solutions of International Mathematical Olympiad from 19861994)
Apart from the above listed books dedicated for the Olympiad purpose, the following books listed below form the recommended topicwise reading for the various math competitions. From the given reads, some are elementary, and some are not so elementary.
Books on Geometry
S. No.  Book  Author  Publication 
1  Modern Geometry  Durrel M. A.,  Macmillan & Co., London 
2  Geometry Revisited  H. S. M. Coxeter and S. L. Greitzer  Mathematical Association of America 
3  Plane Trigonometry  S. L. Loney  Macmillan & Co., London 
Books on Number Theory
S. No.  Book  Author  Publication 
1  An Introduction to the Theory of Numbers  I. Niven& H. S. Zuckerman  Wiley Eastern Ltd. New Delhi 
2  Elementary Number Theory  David Burton  Universal Book Stall, New Delhi 
3  An introduction to the theory of numbers  G. H. Hardy & Wright  Oxford University Publishers 
Problem Books
 I M O Problem Collections
S. No.  Book  Author  Publication 
1  International Mathematical Olympiad 19591977  S. L. Greitzer  MAA Pubications 
2  International Mathematical Olympiad 19781985  M. S. Klamkin  MAA Pubications 
 General Problems
S. No.  Book  Author  Publication 
1  USA Mathematical Olympiads 19721985  M. S. Klamkin  MAA Pubications 
2  Selected problems and Theorems in Elementary Mathematics  D. O. Shklyarshky, N. N. Chensov and I. M. Yaglom 

3  250 Problems in Elementary Number Theory  W. Sierpenski  American Elsevier 
4  Problems in Plane Geometry  I. R. Sharygin  MIR Publishers 
Books for General Reading
S. No.  Book  Author  Publication 
1  Higher Algebra  S. Barnard & J.M. Child  Macmillan & Co., London, 1939; reprinted Surjeet Publishers, Delhi, 1990 
2  The Theory of Equations, Vol. 1 (13th Edition)  W. S Burnside & A.W. Panton  S. Chand & Co., New Delhi, 1990 
3  Elementary Number Theory, Second Edition  D. M. Burton  Universal Book Stall, New Delhi, 1991 
4  Introductory Combinatorics  RA. Brualdi  Elsevier, NorthHolland, New York, 1977 
5  Geometry Revisited  H.S.M. Coxeter& S.L. Greitzer  New Mathematical Library 19, The Mathematical Association of America, New York, 1967 
6  Modern Geometry  C.V. Durell  Macmillan & Co., London, 1961 
7  Higher Algebra  H.S. Hall & S.R Knight  Macmillan & Co., London; Metric Edition, New Delhi, 1983 
8  Mathematical Gems Part I (1973), Part II (1976), Part III (1985)  R Honsberger  The Mathematical Association of America, New York 
9  Geometric Inequalities  N.D. Kazarinoff  New Mathematical Library 4, Random House and The L.W. Singer Co., New York, 1961 
10  Inequalities  P.P. Korovkin  Little Mathematics Library, MIR Publishers, Moscow, 1975 
11  An Introduction to the Theory of Numbers  Fifth Edition, Wiley Eastern, New Delhi, 2000  
12  Applied Combinatorics  A.W. Tucker  Second Edition, John Wiley & Sons, New York, 1984 
13  High School MathematicsPart II  G.N. Yakovlev  MIR Publishers, Moscow, 1984 
Students who clear INMO but are not selected for International Math Olympiad (IMO) receive postal problems during the period of July to December. Based on their responses, they might be invited to the predeparture training camp for IMO directly or asked to sit for INMO again (without having to sit for the Regional Math Olympiad).
Exam Structure
RMO has six or seven problems that students have to solve in three hours. The syllabus for RMO basically covers predegree college mathematics. The major areas covered in the syllabus are algebra, geometry, number theory and combinatorics. Calculus and statistics are not within the scope of the exam but students are allowed to use approaches based on them to solve problems.
The questions generally have high difficulty level and sophistication which only increase from RMO to INMO to IMO.
One should go through Regional Mathematical Olympiad past year papers as well as Regional Mathematical Olympiad sample papers to fully understand what is to be expected in the exam.
Syllabus
The syllabus for Mathematical Olympiad (regional, national and international) is predegree college mathematics. The areas covered are arithmetic of integers, geometry, quadratic equations and expressions, trigonometry, coordinate geometry, system of linear equations, permutations and combination, factorisation of polynomial, inequalities, elementary combinatorics, probability theory and number theory, finite series and complex numbers and elementary graph theory. The syllabus does not include calculus and statistics. The major areas from which problems are given are number theory, geometry, algebra and combinatorics. The syllabus is in a sense spread over Class XI to Class XII levels, but the problems under each topic involve high level of difficulty and sophistication. The difficulty level increases from RMO to INMO to IMO.
For more information, Visit http://olympiads.hbcse.tifr.res.in/subjects/mathematics