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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).

1. 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.

1. Find all roots of the equation :-$\mathbf{ 1-\frac{x}{1}+\frac{x(x-1)}{2!} - \cdots +(-1)^n\frac{x(x-1)(x-2)...(x-n+1)}{n!}=0.}$

1. 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 $\mathbf{x \in (0,1)}$

1. 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?

1. 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).

1. 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 $\mathbf{\angle A=72^0}$

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1. 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?