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# ISI Entrance Paper BMath 2006 – Subjective

ISI Entrance Paper BMath 2006 – from Indian Statistical Institute’s B.Stat Entrance

1. Bishops on a chessboard move along the diagonals ( that is, on lines parallel to the two main diagonals). Prove that the maximum number of non-attacking bishops on an n*n chessboard is 2n-2. (Two bishops are said to be attacking if they are on a common diagonal).
2. Prove that there is no non-constant polynomial P(x) with integer coefficients such that P(n) is a prime number for all positive integers n.
3. Find all roots of the equation :-$\mathbf{ 1-\frac{x}{1}+\frac{x(x-1)}{2!} - \cdots +(-1)^n\frac{x(x-1)(x-2)...(x-n+1)}{n!}=0.}$
4. Let $\mathbf{f:\mathbb{R} to \mathbb{R}}$ be a function that is a function that is differentiable n+1 times for some positive integer n . The $\mathbf{i^{th}}$ derivative of f is denoted by $\mathbf{f^{(i)}}$ . Suppose-$\mathbf{f(1)=f(0)=f^{(1)}(0)=...=f^{(n)}(0)=0.}$
Prove that $\mathbf{f^{(n+1)}(x)=0}$ for some $\mathbf{x \in (0,1)}$
5. A domino is a 2 by 1 rectangle. For what integers m and n can we cover an m*n rectangle with non-overlapping dominoes?
6. You are standing at the edge of a river which is 1 km wide. You have to go to your camp on the opposite bank. The distance to the camp from the point on the opposite bank directly across you is 1 km. You can swim at 2 km/hr and walk at 3 km-hr. What is the shortest time you will take to reach your camp? (Ignore the speed of the river and assume that the river banks are straight and parallel).
7. In a triangle ABC , D is a point on BC such that AD is the internal bisector of $\mathbf{\angle A}$ . Now Suppose $\mathbf{\angle B=2\angle C}$ and CD=AB . Prove that $\mathbf{\angle A=72^0}$
8. Let S be the set of all integers k, $\mathbf{1\leq k\leq n}$, such that $\mathbf{gcd(k,n)=1}$. What is the arithmetic mean of the integers in S?

## By Dr. Ashani Dasgupta

Ph.D. in Mathematics, University of Wisconsin, Milwaukee, United States.

Research Interest: Geometric Group Theory, Relatively Hyperbolic Groups.

Founder, Cheenta

## One reply on “ISI Entrance Paper BMath 2006 – Subjective”

Spandansays:

Problems 1 and 3 requested

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