- 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).

- 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.

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

- 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 \(\mathbf{x \in (0,1)}\)

- 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?

- 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).

- 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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- 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?

Problems 1 and 3 requested