- Let and be two permutations of the numbers . Show that

- Let a,b,c,d be distinct digits such that the product of the 2-digit numbers and is of the form . Find all possible values of a+b+c+d.

- Let be three open intervals of such that none is contained in another. If is non-empty, then show that at least one of these intervals is contained in the union of the other two.

- A real valued function f is defined on the interval (-1,2). A point is said to be a fixed point of f if . Suppose that f is a differentiable function such that f(0)>0 and f(1)=1. Show that if f'(1)>1, then f has a fixed point in the interval (0,1).

- Let A be the set of all functions such that f(xy)=xf(y) for all .(a) If then show that f(x+y)=f(x)+f(y) for all x,y (b) For , define a function \( \mathbf{g \circ h} \) by for . Prove that is in A and is equal to .

- Consider the equation (a) Show that any integer of the form 3m+1 or 3m+2 can not be a solution of this equation.(b) Does the equation have a solution in positive integers?

- Consider a rectangular sheet of paper ABCD such that the lengths of AB and AD are respectively 7 and 3 centimetres. Suppose that B’ and D’ are two points on AB and AD respectively such that if the paper is folded along B’D’ then A falls on A’ on the side DC. Determine the maximum possible area of the triangle AB’D’.

- Take r such that , and consider all subsets of r elements of the set . Each subset has a smallest element. Let F(n,r) be the arithmetic mean of these smallest elements. Prove that: .

- Let be a function having the following property: For any two points A and B in , the distance between A and B is the same as the distance between the points f(A) and f(B).Denote the unique straight line passing through A and B by l(A,B)(a) Suppose that C,D are two fixed points in . If X is a point on the line l(C,D), then show that f(X) is a point on the line l(f(C),f(D)).(b) Consider two more point E and F in and suppose that l(E,F) intersects l(C,D) at an angle . Show that l(f(C),f(D)) intersects l(f(E),f(F)) at an angle alpha. What happens if the two lines l(C,D) and l(E,F) do not intersect? Justify your answer.

- There are 100 people in a queue waiting to enter a hall. The hall has exactly 100 seats numbered from 1 to 100. The first person in the queue enters the hall, chooses any seat and sits there. The n-th person in the queue, where n can be 2, . . . , 100, enters the hall after (n-1)-th person is seated. He sits in seat number n if he finds it vacant; otherwise he takes any unoccupied seat. Find the total number of ways in which 100 seats can be filled up, provided the 100-th person occupies seat number 100.