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**Maths Olympiad**

**Address **: Homi Bhabha Centre for Science Education, Tata Institute of Fundamental Research V. N. Purav Marg, Mankhurd Mumbai – 400 088** **

**Website** : www.hbcse.tifr.res.in

## Eligibility

All school students of Class XI and Class XII are eligible to appear for Regional Maths Olympiad. The Regional Coordinators may on their discretion admit students of lower classes also.

## Entrance exam

The Maths Olympiad consists of the following stages:

**Stage 1 :** **Regional Mathematical** **Olympiad (RMO)** which is held normally between September and the first Sunday of December each year in more than 20 different regions in the country. The Regional Co-ordinator ensures that at least one centre is provided in each district of his/her region. On the basis of RMO, a certain number from each region is selected to appear for the Stage 2 examination. (The selected students from each region may include a maximum of 6 students of Class XII.)

**Stage 2** : Indian National Mathematical Olympiad (INMO) INMO is held on the first Sunday of February each year at the Centres for different regions. On the basis of INMO, the top 30-35 students from all over the country become INMO awardees (Junior Batch) and receive a Certificate of Merit and two books on Mathematics (A maximum of 6 students of Class XII may be INMO awardees.)

**Stage 3 : International Mathematical Olympiad Training Camp (IMOTC) **The INMO awardees are invited to a month long Training Camp held in May-June each year at the Homi Bhabha Centre for Science Education (HBCSE), Mumbai. INMO awardees of the previous year who have satisfactorily gone through postal coaching throughout the year are invited again to a second round of training (Senior Batch). The senior batch participants who successfully complete the Camp receive a prize of 5,000/- in the form of books and cash. On the basis of a number of selection tests through the Camp, a team of the best six students is selected from the combined pool of junior and senior batch participants.

**Stage 4: International Mathematical Olympiad (IMO) **The six-member team selected at the end of the Camp accompanied by a leader and a deputy leader represent the country at the IMO, held in July each year in a different member country of IMO. The International Maths Olympiad for 2006 was held in July at Slovenia and the Olympiad for 2007 will be held in Vietnam.

## Pattern of exam

RMO is a 3-hour written test containing about 6 to 7 problems.

INMO is a 4-hour written test. The question paper is set centrally and is common throughout the country.

IMO consists of two papers of 4 and 1/2-hours duration held on two days.

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## Syllabus for Mathematics Olympia** **

The syllabus for the Mathematics Olympiad (regional, national and international) is pre-degree college mathematics. The areas covered are arithmetic of integers, geometry, quadratic equations and expressions, trigonometry, co-ordinate 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 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.

## About the Centre

Homi Bhabha Centre for Science Education at the Tata Institute of Fundamental Research is the nodal centre for Olympiad programmes in Physics, Chemistry, Biology, Mathematics and Astronomy. The programmes aim at promoting excellence in science and mathematics among pre-university students. The Mathematics Olympiad is conducted under the aegis of the National Board of Higher Mathematics (NBHM). The programme is financially supported by BRNS (DAE), Department of Science and Technology, Department of Space and Ministry of Human Resource Development.

India has been participating in IMO since 1989. Students of the Indian team who receive gold, silver and bronze medals at the IMO receive a cash prize of 5000/-, 4000/- and 3000/- respectively during the following year at a formal ceremony at the end of the Training Camp. Ministry of Human Resource Developement (MHRD) finances international travel of the 8-member Indian delegation, while NBHM (DAE) finances the entire in-country programme and other expenditure connected with international participation. All the six students of the Indian team automatically qualify for the Kishore Vaigyanik Protsahan Yojana (KVPY) Fellowship ( 3,000/- per month + 6,000/- per annum contingency plus a nurture programme), provided they pursue careers in mathematics or basic sciences. Students aiming to go through the Mathematical Olympiad Programme leading to international participation (for the best 6) should note that RMO is the first essential step for the programme. To appear for RMO, students should get in touch with the RMO Co-ordinator of their region well in advance, for enrolment and payment of any fees, which is nominal.

## How to prepare for the exam

Some of the books you can prepare from are :Challenge and Thrill of Pre-College Mathematics by V. Krishnamurthy, C.R. Pranesachar, K.N. Ranganathan and B.J. Venkatachala (New Age International Publishers, New Delhi); Problem Primer for the Olympiad by C.R. Pranesachar, B.J. Venkatachala and C.S. Yogananda (Prism Books Pvt. Ltd., Bangalore); An Excursion in Mathematics by Regional Mathematical Olympiads Committee (Maharashtra and Goa); Functional Equations: A Problem Solving Approach by B. J. Venkatchala* *(Problems from Mathematical Olympiads and other contexts) (Prism Books Pvt. Ltd., Bangalore).

## Sample test paper for the Indian National Maths Olympiad

1. If O,A,C,D are collinear and OA ^{2} = OC .OD both in magnitude and sign, then prove that the symmetric of A with respect to O is the harmonic conjugate of A with respect to C,D.

2. Find the values of *a* for which one of the roots of x ^{2} + (2a + 1) x + (a ^{2} + 2) = 0 is twice the other root. Find also the roots of this equation for these values of *a*.

3. Find the number of permutations of the set {1,2,…,k} in which the patterns 12,23,…(k-1)k do not appear.

4. Find the integer *n* which has the following property : If the numbers from 1 to *n* are all written down in decimal notation, the total number of digits written down is 1998. What is the next higher number (instead of 1998) for which this problem has an answer ?

5. Sixty four squares of a chess board are filled with positive integers one on each in such a way that each integer is the average of the integers on the neighboring squares. (Two squares are neighbours if they share a common edge or vertex. Thus a square can have 8,5 or 3 neighbours depending on its position.) Show that all the sixty four entries are in fact equal.

6. A person left home between 4 p.m. and 5 p.m., returned between 5 p.m. and 6 p.m., and found that the hands of his watch had exchanged places. When did he go out ?

7. Which one of the following statements is not correct ?

a) if *a* is a rational number and *b* is irrational, the a + b is irrational

b) the product of non zero rational numbers with an irrational number is always irrational

c) addition of any two rational numbers can be an integer

d) division of any two integers is an integer