
 Let and be two permutations of the numbers . Show that

 Let a,b,c,d be distinct digits such that the product of the 2digit 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 nonempty, 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 nth person in the queue, where n can be 2, . . . , 100, enters the hall after (n1)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 100th person occupies seat number 100.