Books and Compilations
Primarily targeted at the Olympiad
The following books treat, quite comprehensively, the topics that are broadly covered in the Mathematical Olympiads, and provide a rich source of problems  highly recommended.

An Excursion in Mathematics
Editors: M.R. Modak, S.A. Katre and V.V. Acharya (Bhaskaracharya Pratishthana, Pune, 2015) 
Challenge and Thrill of PreCollege Mathematics
V. Krishnamurthy, C.R. Pranesachar, K.N. Ranganathan and B.J. Venkatachala (New Age International Publishers, New Delhi, 2007) 
Problem Primer for the Olympiads
C.R. Pranesachar, B.J. Venkatachala and C.S. Yogananda (Prism Books Pvt. Ltd., Bangalore, 2008) 
Problem Solving Strategies
Arthur Engel (SpringerVerlag, Germany, 1999) 
Mathematical Circles: Russian Experience
Dmitri Fomin, Sergey Genkin and Ilia Itenberg (University Press, Hyderabad, 2008) 
Mathematical Olympiad Challenges
Titu Andreescu and Razvan Gelca (Springer India, 2014) 
Functional Equations
B.J. Venkatachala (Prism Books Pvt. Ltd., Bangalore, 2008)
Most of these books are available at the College Street area of Kolkata, or online, through Flipkart or Amazon. You may also contact Saraswati Pustakalaya, 138/1 Gopal Lal Thakur Road, Kolkata 700 108.
Sample Problems
Previous Olympiad Papers
The students must try to solve the problems from the old exams of RMO and INMO. All these 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.
RMO  Problems (1990 to 1999)
RMO 1990 
RMO 1991 
RMO 1992 
RMO 1993 
RMO 1994 
RMO 1995 
RMO 1996 
RMO 1997 
RMO 1998 
RMO 1999 
RMO  Problems and Solutions (2000 to 2010)
RMO 2000 
RMO 2001 
RMO 2002 
RMO 2003 
RMO 2004 
RMO 2005 
RMO 2006 
RMO 2007 
RMO 2008 
RMO 2009 
RMO 2010 
RMO  Problems and Solutions (2011 onwards)
RMO 2011 :
CRMO 1  renamed to CRMO (4 different papers in general)
RMO 2012 :
CRMO 1 
CRMO 2 
CRMO 3 
CRMO 4 
RMO Mumbai
RMO 2013 :
CRMO 1 
CRMO 2 
CRMO 3 
CRMO 4 
RMO Mumbai
RMO 2014 :
CRMO 1 
CRMO 2 
CRMO 3 
CRMO 4 
RMO Mumbai
RMO 2015 :
CRMO 1 
CRMO 2 
CRMO 3 
CRMO 4 
RMO Mumbai
RMO 2016 :
CRMO 1 
CRMO 2 
CRMO 3 
CRMO 4
INMO  Problems (1989 to 1999)
INMO 1989 
INMO 1990 
INMO 1991 
INMO 1992 
INMO 1993 
INMO 1994 
INMO 1995 
INMO 1996 
INMO 1997 
INMO 1998 
INMO 1999
INMO  Problems and Solutions (2000 onwards)
INMO 2000 
INMO 2001 
INMO 2002 
INMO 2003 
INMO 2004 
INMO 2005 
INMO 2006 
INMO 2007 
INMO 2008 
INMO 2009 
INMO 2010 
INMO 2011 
INMO 2012 
INMO 2013 
INMO 2014 
INMO 2015 
INMO 2016 
INMO 2017
Previous Question Papers and their Solutions on the HBCSE Olympiad Website.
Reference Books
Recommended reading
The following books form the recommended reading for various mathematical competitions at the precollege level. Some of these are elementary, and some quite advanced.
 S. Barnard and J.M. Child, Higher Algebra
Macmillan & Co., London, 1939; reprinted Surjeet Publishers, Delhi, 1990  H.S. Hall and S.R Knight, Higher Algebra
Macmillan & Co., London; Metric Edition, New Delhi, 1983  W. S Burnside and A.W. Panton, The Theory of Equations
S. Chand & Co., New Delhi, 1990  P.P. Korovkin, Inequalities
MIR Publishers, Moscow, 1975  R.A. Brualdi, Introductory Combinatorics
Elsevier, NorthHolland, New York, 1977  A.W. Tucker, Applied Combinatorics
John Wiley & Sons, New York, 1984  D.M. Burton, Elementary Number Theory
Universal Book Stall, New Delhi, 1991  I. Niven, H.S. Zuckerman and H.L. Montgomery, An Introduction to the Theory of Numbers, Wiley Eastern, New Delhi, 2000
 G.H. Hardy and E.M. Wright, An Introduction to the Theory
of Numbers
Oxford University Publishers, UK, 2008  C.V. Durell, Modern Geometry
Macmillan & Co., London, 1961  H.S.M. Coxeter and S.L. Greitzer, Geometry Revisited
The Mathematical Association of America, New York, 1967  N.D. Kazarinoff, Geometric Inequalities
Random House and The L.W. Singer Co., New York, 1961  S.L. Loney, Plane Trigonometry
Macmillan & Co., London.  G.N. Yakovlev, High School Mathematics
MIR Publishers, Moscow, 1984  R. Honsberger, Mathematical Gems
The Mathematical Association of America, New York  D. O. Shklyarshky, N. N. Chensov and I. M. Yaglom, The USSR Olympiad Problem Book
Dover Publications Inc., 1993  W. Sierpenski, 250 Problems in Elementary Number Theory
Elsevier Science Ltd., 1971  I.R. Sharygin, Problems in Plane Geometry
Mir Publishers, 1988
This list of references is always a workinprogress. Please feel free to suggest more such useful resources for the Mathematical Olympiads, in case we have missed out on any. Your input will be highly appreciated.
Lecture Notes, Handouts and Links
Miscellaneous online resources for the Olympiad
Just like the IMO Training Camp organized by HBCSE, various countries organize their own training camps for the IMO. The lecture notes and handouts from these training camps provide a rich source of preparatory material and problems for the Mathematical Olympiad. Several noted mathematicians and previous champions of the Olympiads contribute to these materials regularly. The students are encouraged to consult the following resources.
Lecture Notes from the Canada IMO Training Camps (since 1998)
Olympiad Training Handouts from Yufei Zhao (since 2007)
Olympiad Training Handouts from Alexander Remorov (since 2010)
Lecture Notes from the Indian IMO Training Camps (since 2013)
Lecture Notes on Inequalities by Kiran Kedlaya (1999)
Lecture Notes on Inequalities by Thomas J. Mildorf (2005)
Lecture Notes on Number Theory by Thomas J. Mildorf (2010)
Lecture Notes on Number Theory by Naoki Sato
Art of Problem Solving (AoPS) offers an amazing online community of likeminded students and educators who enjoy discussing interesting mathematical problems. You may find AoPS community and their resources extremely useful while preparing for the Olympiads. Please note that AoPS may hold quick answers to many mathematical problems  but the goal for you should be to arrive at the solution by yourself  even if it is really painstaking. So, have fun discussing problems, but please avoid any shortcut to problemsolving.
This list of references is always a workinprogress. Please feel free to suggest more such useful resources for the Mathematical Olympiads, in case we have missed out on any. Your input will be highly appreciated.