John Forbes Nash, Jr. is an American mathematician who is known for his very revolutionary “Nash Equilibrium” and work in Game Theory. His works in Game Theory, Differential Geometry, and Partial Differential Equations have provided insight into the forces that govern chance and events inside complex systems in daily life. His theories are used in market economics, computing, evolutionary biology, artificial intelligence, accounting, politics and military theory. Serving as a Senior Research Mathematician at Princeton University during the latter part of his life, he shared the 1994 Nobel Memorial Prize in Economic Sciences with game theorists Reinhard Selten and John Harsanyi.

01 September 2013 (Class XI and XII)   Marks: 10 X 10 = 100 Time: 1.30 pm to 4.30 pm Answer the following ten questions 1. Prove that $$4(x_1^4+ x_2^4+dots + x_{14}^4)=7(x_1^3+ x_2^3+dots + x_{14}^3)$$ has no solution in positive integers. (Hint: Suppose on the contrary $$sum_{k=1}^{14}(x_k^4-frac{7}{4}x_k^3)=0.$$ Also use $$sum (x_k-1)^4.$$)   2. Find...

01 September 2013 (Class IX and X)   Marks: 10 X 10 = 100 Time: 1.30 pm to 4.30 pm Answer the following ten questions 1. Show that there does not exist a function $$f:Nrightarrow N$$ which satisfy (a) $$f(2)=3,$$ (b) $$f(mn)=f(m)f(n)$$ for all m,n in N; (c) $$f(m)<f(n)$$ whenever $$m<n.$$ (Hint: Suppose the contrary....

01 September 2013 (Class VII and VIII)   Marks: 5 X 20 = 100 Time: 1.30 pm to 4.30 pm Answer the following questions 1. If the radius of a circle is increased by 100%, determine the increase percent in the area of the circle. এটা বৃত্তৰ ব্যাসাৰ্দ্ধ 100% বৃদ্ধি হ’লে বৃত্তটোৰ...

01 September 2013 (Class V and VI)   Marks: 5 X 20 = 100 Time: 1.30 pm to 4.30 pm Answer the following questions 1. Show that 52563744 is divided by 24 without direct division. হৰণ প্ৰক্ৰিয়া প্ৰয়োগ নকৰাকৈ প্ৰমাণ কৰা যে 52563744 সংখ্যাটো 24 ৰে বিভাজ্য।   2. Find the remainder when $$7^{84}$$...

[caption id="attachment_5131" align="alignleft" width="191"]Dr. Eknath Prabhakar Ghate Dr. Eknath Prabhakar Ghate[/caption] Dr. Eknath Prabhakar Ghate, of the School of Mathematical Sciences, Tata Institute of Fundamental Research (TIFR) Mumbai has been selected for the prestigious Shanti Swarup Bhatnagar Prize for Science and Technology for the year 2013 in Mathematical Science category along with seven other scientists in various other disciplines. This year the other awardees include Dr Sathees Chukkurumbal Raghavan (Biological Sciences) from IISc Bangalore, Dr Yamuna Krishnan (Chemical Science) from Tata Institute of Fundamental Research (TIFR) Mumbai, Dr Bikramjit Basu from IISc and Dr Suman Chakraborty from IIT Kharagpur (both from Engineering Sciences), Dr Amol Dighe from TIFR and Dr Vijay Balakrishna Shenoy from IISc (Physical Science). There is no awardee this year in the Earth, Atmosphere, Ocean & Planetary Sciences category.

Figurate NumbersElena Deza and Michel Marie Deza World Scientific, 2011, xviii+456 pp.   This book is about special types of numbers (integers) that have geometric associations and that have intriguing spatial properties. The ancient Greeks were perhaps the first to study what are called “figurate numbers” — numbers that can be represented by regular geometric patterns of points in the plane or in space, such as triangular, polygonal and polyhedral numbers. The first two chapters contain a lot of formulae for all kinds of figurate numbers that arise from geometric patterns in 2 and 3 dimensions. Properties and relations between such figurate numbers and their connections with Diophantine equations have been studied by classical mathematicians like Euler, Fermat, Lagrange, Legendre, Cauchy, Gauss and Dirichlet. Chapter 3 extends the construction of figurate numbers to dimension 4 and beyond. Examples of such numbers are the pentatope numbers which are 4-dimensional analogues of triangular and tetrahedral numbers, and the biquadratic numbers which are the 4-dimensional analogues of square and cubic numbers. Despite the lack of visual pictures and physical intuition, multitudes of formulae are presented and proved.