Syllabus for Mathematical Olympiads
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, systems of linear equations, permutations and combinations, factorisation of polynomials, 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 IX to Class XII levels, but the problems under each topic are of exceptionally high level in difficulty and sophistication. The difficulty level increases from RMO to INMO to IMO.
There is no prescribed syllabus for mathematical olympiads. Broadly speaking, students appearing at RMO must be well versed with all the mathematics taught upto Class X level. The major areas from which problems are chosen are Number Theory, Geometry, Algebra and Combinatorics. This includes Arithmetic of Integers, Geometry, Quadratic equations and expressions, Trigonometry, Coordinate Geometry, Systems of linear equations, Permutations and Combinations, Factorization of polynomials, 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 problems under each topic are of exceptionally high level in difficulty and sophistication. The difficulty level increases from RMO to INMO to IMO. The book ''An Excursion In Mathematics'' gives a fairly good idea of the background needed to solve these problems. The students should also try the old exams of RMO and INMO. All the problems can be solved without using Calculus or calculators. However these problems are not routine text book problems. They are considerably harder, calling for ingenuity on the part of the solver. Therefore, it is highly advisable that a student solves as many new problems as possible with no or with minimum help. Indeed, if you are unable to solve at least one problem from each old RMO exam without help, RMO is not for you.
RMO Books reference :
The following book treats the topics which are covered in the olympiads and also is a rich source of problems; (highly recommended)
 V. Krishnamurthy, C. R. Pranesachar, K. N. Ranganathan and B. J. Venkatachala, Challenge and Thrill of PreCollege Mathematics, New Age International Publishers.
 C. R. Pranesachar, S. A. Shirali, B. J. Venkatachala, and C. S. Yogananda, Mathematical Challenges from the Olympiads, Prism Books Pvt. Ltd. (Contains problems and solutions of International Mathematical Olympiad from 19861994).
 C. R. Pranesachar, B. J. Venkatachala, and C. S. Yogananda, Problem Primer for the Olympiad, Prism Books Pvt. Ltd., #1865, 32nd. Cross, BSK II Stage, Bangalore 560 070. or 49, Sardar Sankar Road, Kolkata 700029. Phone: 24633890/24633944.
 M. R. Modak, S. A. Katre, V. V. Acharya, An Excursion in Mathematics, Bhaskaracharya Pratisthan, 56/14 Erandavane, Damle Path, Pune 411 004.
The books listed below form the recommended reading for the various math competitions. Some are elementary, and some are not so elementary. As far as possible there are indicators to the type of the book but, of course, these can only be indicators....

GEOMETRY
 Durrel M. A., Modern Geometry, Macmillan & Co., London.
 H. S. M. Coxeter and S. L. Greitzer, Geometry Revisited, Mathematical Association of America.
 S. L. Loney, Plane Trigonometry, Macmillan & Co., London.

NUMBER THEORY
 I. Niven & H. S. Zuckerman, An Introduction to the Theory of Numbers, Wiley Eastern Ltd. New Delhi.
 David Burton, Elementary Number Theory, Universal Book Stall, New Delhi.
 G. H. Hardy & Wright, An introduction to the theory of numbers, Oxford University Publishers.

PROBLEM BOOKS
 I M O Problem Collections
 S. L. Greitzer, International Mathematical Olympiad 19591977, MAA.
 M. S. Klamkin, International Mathematical Olympiad 19781985, MAA.
 General
 M. S. Klamkin, USA Mathematical Olympiads 19721985, MAA.
 D. O. Shklyarshky, N. N. Chensov and I. M. Yaglom, Selected problems and Theorems in Elementary Mathematics.
 W. Sierpenski, 250 Problems in Elementary Number Theory, American Elsevier.
 I. R. Sharygin, Problems in Plane Geometry, MIR Publishers.

JOURNALS
 Samasya, journal devoted to problem solving, published by Leelavati Trust, Bangalore.
 Bona Mathematica, published by Bhaskaracharya Prathistana, Pune.
 General Reading
 Arthur Engel, ProblemSolving Strategies, Springer.
 S. A. Shirali, A Primer On Number Sequences, University Press.
 S. A. Shirali, First Steps In Number Theory A Primer On Divisibility, University Press.
 B. J. Venkatachala, Functional EquationsA Problem Solving Approach, Prism Books Pvt. Ltd., #1865, 32nd. Cross, BSK II
Stage, Bangalore 560 070. or 49, Sardar Sankar Road, Kolkata 700029. Phone: 24633890/24633944.
 S. Barnard & J.M. Child, Higher Algebra, Macmillan & Co., London, 1939; reprinted Surjeet Publishers, Delhi, 1990
 W. S Burnside & A.W. Panton, The Theory of Equations, Vol. 1 (13th Edition), S. Chand & Co., New Delhi, 1990
 D. M. Burton, Elementary Number Theory, Second Edition, Universal Book Stall, New Delhi, 1991
 RA. Brualdi, Introductory Combinatorics, Elsevier, NorthHolland, New York, 1977
 H.S.M. Coxeter & S.L. Greitzer, Geometry Revisited, New Mathematical Library 19, The Mathematical Association of America, New York, 1967
 C.V. Durell, Modern Geometry, Macmillan & Co., London, 1961
 D. Fomin, S. Genkin & 1. Itenberg, Mathematical Circles, First Reprinted Edition, University Press, New Delhi, 2000
 H.S. Hall & S.R Knight, Higher Algebra, Macmillan & Co., London; Metric Edition, New Delhi, 1983
 R Honsberger, Mathematical Gems, Part I (1973), Part II (1976), Part III (1985), The Mathematical Association of America, New York
 N.D. Kazarinoff, Geometric Inequalities, New Mathematical Library 4, Random House and The L.W. Singer Co., New York, 1961
 P.P. Korovkin, Inequalities, Little Mathematics Library, MIR Publishers, Moscow, 1975
 I. Niven, H.S. Zuckerman & H.L. Montgomery, An Introduction to the Theory of Numbers, Fifth Edition, Wiley Eastern, New Delhi, 2000
 A.W. Tucker, Applied Combinatorics, Second Edition, John Wiley & Sons, New York, 1984
 G.N. Yakovlev, High School Mathematics, Part II, MIR Publishers, Moscow, 1984
LIST OF BOOKS FOR MATH. OLYMPIADS:
Short list:
1. Problem primer for olympiads: C.R. Pranesachar, B J Venkatachala and C S
Yogananda (Prism Books Pvt Ltd, Jayangar, Bangalore)
2. Challenge and thrill of precollege mathematics: V. Krishnamurthy, C R
Pranseachar, K N Ranganathan and B. J. Venkatachala (New age international
publishers, New Delhi)
3. An Excursion in Mathematics, Ed. M R Modak, S A Katre and V V Acharya
(Bhaskaracharya Pratishthana, Pune)
4. Functional Equations, B J Venkatachala (Prism Books Pvt Ltd, Jayanagar,
Bangalore)
5. Mathematical Circles: Russian Experience (University Press, Hyderabad).
Long list:
1. Klamkin, M.S. U.S.A. Maths Olympiad, 1972  1986
2. Yaglom, I.M. The USSR Olympiad Problem Book (Dover)
3. Sierpenski. W 250 Problems in Elementary Number Theory (Elsevier)
4. Niven & Zukerman An Introduction to the theory of Numbers (Wiley)
5. Coxeter, H.S.M. Geometry Revisited (MAA)
6. Larson, L.C. Problem Solving through Problems (Springer)
7. Bottema. O. Geometric Inequalities (MAA)
8. V. Krishnamoorthy etal. Challenges and thrill of Precollege Mathematics (New Age Publ.)
9. Pranesachar C.R. Mathematical challenges from Olympiads. (Interline Publ)
10. Lozansky E., Rousseau, C Winning Solutions (Springer)
11. M.K. Singal, A.R. Singal Olympiad Mathematics (Pitambar Publ.)
12. S.A. Katre An excursion in Mathematics
13. V. Seshan Mastering Olympiad Mathematics (Frank Brothers)
14. Engel A. Problem Solving Strategies (Springer)
15. Shirali S. A First steps in Number Theory (Universities Press)
16. Shirali S.A. Adventures in Problem Solving ,, ,,
17. Steven G. Krantz Techniques of Problem Solving ,, ,,
18. Titu Andreescu & Mathematical Olympiad Challenges Razvan Gelca; (Universities Press)
19. Burton Elementary Number Theory (UBS)
20. Venkatachala B. J. Functional Equations. A problem solving approach
21. Durrell C. V. Geometry
22. Bonnie Averback and Problem solving through recreational Mathematics Oria Chein (Dover)
23. Alfred Posamentiar Challenging Problems in Geometry (Dover) and Charles T. Salkind
24. Beiler A.H. Recreations in the theory of numbers (Dover)
25. A Gardiner The Mathematical Olympiad Hand book OUP (2000)
26. T. Andreesan Mathematical Olympiad Challenges R. Gelca Birkhauser (2000)
27. S. Muralidharan Gems from the Mathematics Teacher, AMTI (1997) G.R. Vijayakumar
28. V.K. Krishnan (Ed.) Nonroutine problems in Mathematics, AMTI (2000)
29. R. Roy Choudhary 501 Difficult problems in Mathematics, BM Pub (2000)
30. T. Andreescu Mathematical Olympiad Treasures, Birkhauser (2004)
31. Bernard and Child Higher Algebra (Mc Millan)
32. Stein haus One Hundred Problems in Elementary Mathematics (Dover)
33. Eves H. College Geometry (Narosa) (1995)
34. Williams K.S.; Hardy, K The red book of mathematical problems (Dover)
35. I. Reiman International Mathematics Olympiad Vol. IIII (Anthem Press)
