# What is Madhava Mathematics Competition

This blog is especially for the Madhava Mathematics Competition and IITJEE Main and Advanced mathematics and also BSc Pure Mathematics of India.

Official information about the Madhava Mathematics Competition can be found at its official website:

It is conducted by Department of Mathematics, S.P. College, Pune and TIFR (Tata Institute of  Fundamental Research)/Homibhabha Centre for Science Education. IG is like “baby Putnam” , to use the analogy of William Lowell Putnam competition of the USA.

(A Mathematics Competition for Undergraduate Students)

The competition is named after Madhava, who introduced in the fourteenth centurry, profound mathematical ideas that are now part of Calculus. His most famous achivements include the Madhava – Leibnitz series for π, the Madhava – Newton power series for sine and Cosine functions and a numerical value of π that is accurate to eleven decimal places. The “Madhava School”, consisting of a long chain of teachers-students continuity, flourished for well over two centuries, making significant contributions to mathematics and astronomy.

(A Mathematics Competition for Undergraduate Students)

The competition is named after Madhava, who introduced in the fourteenth centurry, profound mathematical ideas that are now part of Calculus. His most famous achivements include the Madhava – Leibnitz series for π, the Madhava – Newton power series for sine and Cosine functions and a numerical value of π that is accurate to eleven decimal places. The “Madhava School”, consisting of a long chain of teachers-students continuity, flourished for well over two centuries, making significant contributions to mathematics and astronomy.

ELIGIBILITY

The competition is basically meant for the S. Y. B. Sc. / S. Y. B. Sc. (Computer Science) students with Mathematics as one of the subjects and thus the question paper will be set with the background of these students in mind. However, interested students of F. Y. B. Sc. and T. Y. B. Sc. (Mathematics) may also appear.The merit list of T. Y. B. Sc. students will be made separately.

A Mathematics Competition for S.Y.B.Sc. Students
Interested students of F.Y.B.Sc. and T.Y.B.Sc. are also eligible.
(A seperate merit list of T.Y.B.Sc. will be prepared.)

TOPICS

Calculus of one variable (sequences, series, continuity, differentiation etc.), elementary algebra (polynomials, complex numbers and integers), elementary combinatorics, general puzzle type/ mixed type problems.

PRIZES

After the analysis of the results, first few students above a certain cut-off, with a reasonable gap after that will get cash prizes. Some more students will get the consolation prizes. All the students appearing for the competition will receive a participation certificate.

A registration fee of Rs. 50/- will be charged to the students for ensuring his/her seriousness in the enrollment and assurance of participation. The colleges will send a list of candidates from their college in the prescribed format along with the registration fees to the center coordinator. A receipt in the name of Principal of that college will be given.

The last date for the registration is November 15, 2014.

There will be no provision for revaluation of the answer sheets.

# Permutations and combinations — an application to Statistical Mechanics

In Statistical Mechanics, one  encounters the situation of putting k particles into r distinct energy levels. The particles can thus be considered as discrete objects and the different energy levels as distinct boxes or cells. Three different situations are obtained by making  three different assumptions. These are:

a) Maxwell-Boltzmann: Here the particles are all distinct and any number of particles can be put into any of the r boxes. The number of possibilities are $r^{k}$ as given by the following theorem:

Let M be a multi set consisting of r distinct objects, each with infinite multiplicity. Then, the total number of d-permutations of M is $r^{d}$.

b) Bose-Einstein: Here the particles are all identical and any number of particles can be put in any one of the r boxes. The number of possibilities is $k+r-1 \choose k$ as given by the following theorem:

The following sets are in bijective correspondence:

i) The set of all increasing sequences of  length  k on an ordered set with r elements.

ii) The set of all the ways of putting k identical objects into  r distinct boxes.

iii) The set of all the k-combinations of a multi-set with r distinct elements.

c) Fermi-Dirac: Here the particles are all identical  but no  box can hold more than one particle. The number of possibilities is $r \choose k$.

More later…

Nalin Pithwa

# a cute problem on infinite series

Evaluate the infinite series $S = \sum_{n=1}^{\infty} \arctan (2/n^{2})$

Solution:

For $n \geq 1$ we have

$\arctan (1/n)-\arctan (1/(n+2))=\arctan \frac{(1/n)-(1/(n+2)}{1+(1/(n(n+2)))}=\arctan (2/(n+1)^{2})$

so that for $N \geq 2$ we have

$\sum_{n=2}^{N} \arctan (2/n^{2}) = \sum_{n=1}^{N-1}\arctan(2/(n+1)^{2})$ which equals the following

$\sum_{n=1}^{N-1} (\arctan (1/n) - \arctan(1/(n+2)) = \arctan (1) + \arctan (1/2) - \arctan (1/N) - \arctan (1/(N+1))$

Letting $N \rightarrow \infty$ we get

$\sum_{n=2}^{\infty} \arctan (2/n^{2}) = \arctan (1) + \arctan (1/2) = (\pi/4) + \arctan (1/2)$ and so

$S= (\pi/4) + \arctan (2) + \arctan (1/2) = ((3\pi)/4)$

More later…

Nalin Pithwa