Problems for RMO, Indian Statistical Institute entrance

Few selected problems to drill your brain with ideal hints for olympiad and entrance tets of ISI in B.Math or B.Stat. courses in INDIA

Olympiad

http://indianolympiad.blogspot.in/ for preparation in RMO and INMO

National Board for Higher Mathematics

Department of Atomic Energy

Government of India.

Mathematics Olympiad

Mathematics Olympiad activity on a national level has been one of the major initiatives

of NBHM (National Board for Higher Mathematics) since 1986. The activity aims to spot

mathematical talent among High School children. NBHM, with Homi Bhabha Centre for

Sciencec Education (HBCSE), also has taken on the responsibility of selecting and training

the Indian team for the International Mathematical Olympiad every year.

For the purpose of the Olympiad contests, the country has been divided in to about 25 regions.

The selection process for participation in the International Mathematical Olympiad

(IMO) consists of the following stages:

Stage 1: Regional Mathematical Olympiad (RMO): RMO is currently held on the

first Sunday of October each year in each of the regions in the country. The Regional coordinator

each region holds the charge of conducting RMO in the region. All school students

from Class XI are eligible to appear in RMO. Students from Class XII may also appear

in RMO, but the number of students selected from Class XII is at most 6. Exceptionally

brilliant students from lower standards may also appear for RMO subject to the approval

of the Regional Coordinator. RMO is a 3-hour written test containg 6 or 7 problems. On

the basis of the performance in RMO, students are selected for the second stage.

The Regional Coordinators may charge a nominal fee to meet the expenses in organising

the contest.

Stage 2: Indian National Mathematical Olympiad (INMO): INMO is currently

held on the third Sunday of January each year at the regional centres in all regions. Only

those students who are selected in RMO are eligible to appear in INMO. This contest is

a 4-hour written test. The evaluation of these papers is centralised, and is undertaken by

the IMO Cell of NBHM. The top 75 contestants in INMO receive Merit Certificates.

Stage 3: International Mathematical Olympiad Training Camp (IMOTC): The

top 30-35 INMO certificate awardees are invited to a month long training camp inMay/June

each year. The training camp is organised by HBCSE, Mumbai. The number of students

from Class XII who are selected for IMOTC is at most 6. In addition to these 35 students,

a certain number of INMO awardees of previous year(s) who have satisfactorily undergone

postal tuition over the year are also invited to a second round of training. A team of six

students is selected from the combined pool of junior and senior batch participants, based

on a number of selection tests conducted during the camp, to represent India in the International

Mathematical Olympiad.

Stage 4: International Mathematical Olympiad (IMO): The six member team selected

at the end of IMOTC, accompanied by a leader and a deputy leader represent India

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at IMO, that is normally held in July each year in one of the chosen for the years IMO.

IMO consists of two 4-and-a-half hour tests held on two consecutive days. The normal

schedule between departure and return of the team takes about two weeks. The students

of Indian team who win gold, silver and bronze medals at IMO receive from NBHM a cash

prize of RS. 5000/-, Rs. 4000/- and Rs. 3000/- respectively. MHRD (Ministry of Human

Resource Development) finances international travel of the 8-member Indian delegation

to IMO, while NBHM (DAE) finances the entire in-country programme and takes care of

other expenditure connected with international participation. The six students representing

India at IMO automatically qualify for Kishore Vaigyanik Protsahan Yojana (KVPY)

scholarship (Rs 3000/- per month and some contingency) instituted by Department of Science

and Technology, Government of India.

Syllabus for Mathematical Olympiad: The syllabus for Mathematical Olympiad (regional,

national and international) consists of pre-degree college mathematics. The difficulty

level increases from RMO to INMO to IMO. Broadly the syllabus for RMO and

INMO is: Algebra (basic set theory, principle of Mathematical Induction,inequalities (AMGM

and Cauchy-Schwarz), theory of equations (remainder theorem, relation between roots

and coefficients, symmetric expressions in roots, applications of the Fundamental theorem

of algebra and its applications), functional equations); Geometry (similarity, congruence,

concurrence, collinearity, parallelism and orthogonality, tangency, concyclicity, theorems

of Appollonius, Ceva, Menelaus and Ptolemy, special points of a triangle such as circumcentre,

in-centre, ex-centres, ortho-centre and centroid); Combinatorics (Basic counting

numbers such as factorial, number of permutations and combinations, cardinality of a

power set, problems based on induction and bijection techniques, existence problems, pigeonhole

principle); Number theory (divisibility, gcd and lcm, primes, fundamental theorem

of arithmetic (canonical factorisation), congruences, Fermat’s little theorem, Wilson’s theorem,

integer and fractional parts of a real number, Pythagorean triplets, polynomials with

integer coefficients). An idea of what is expected in mathematical olympiad can be had

from the earlier question papers (see http://www.isid.ac.in/˜ rbb/olympiads.html) and the

following books:

1. Problem Primer for Olympiads, by C R Pranesachar, B J Venkatachala and C S

Yogananda (Prism Books Pvt. Ltd., Bangalore).

2. 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).

3. An Excursion in Mathematics, Editors: M R Modak, S A Katre and V V Acharya

(Bhaskaracharya Pratishthana, Pune).

4. Problem Solving Strategis, Arthur Engel (Springer-Verlag, Germany).

5. Functional Equations, B J Venkatachala (Prism Books Pvt. Ltd., Bangalore).

6. Mathematical Circles, Fomin and others (University Press, Hyderabad).

Reference to many other interesting books may be found in An Excursion in Mathematics.

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Nurture Programme: The INMO awardees who choose Mathematics as one of the subjects

in their undergraduate studies are eligible for a scholarship by NBHM (which is at

present Rs 1500/= per month) throughout their undergraduate studies. If they further

pursue their studies to masters, they continue to get scholarship (enhanced). Even the students

who do not pursue Mathematics in their undergraduate studies are eligible for certain

benefits under a novel programme instituted by NBHM, called Nurture Programme. Under

this programme, each batch of students (selected from among the INMO awardees through

their responses to a few sets of postal problems) is assigned to an institution. The coordinator

in that institution gives out some reading material which the students can go through

during their leisure time while pursuing their undergraduate studies. At the end of each

year, during summer, they are invited to that institution for a contact programme with

working Mathematicians. Based their performance, they may be recommended to a scholarship

given by NBHM. This programme continue for four years. Thus, even those who

pursue under-graduate studies in some other discipline can still get training in Mathematics

and use it in their further pursuit of knowledge.

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