- Let \(\mathbf{a_1,a_2,\cdots, a_n }\) and \(\mathbf{ b_1,b_2,\cdots, b_n }\) be two permutations of the numbers \(\mathbf{1,2,\cdots, n }\). Show that \(\mathbf{\sum_{i=1}^n i(n+1-i) \le \sum_{i=1}^n a_ib_i \le \sum_{i=1}^n i^2 }\)

- Let a,b,c,d be distinct digits such that the product of the 2-digit numbers \(\mathbf{\overline{ab}}\) and \(\mathbf{\overline{cb}}\) is of the form \(\mathbf{\overline{ddd}}\). Find all possible values of a+b+c+d.

- Let \(\mathbf{I_1, I_2, I_3}\) be three open intervals of \(\mathbf{\mathbb{R}}\) such that none is contained in another. If \(\mathbf{I_1\cap I_2 \cap I_3}\) 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 \(\mathbf{x_0}\) is said to be a fixed point of f if \(\mathbf{f(x_0)=x_0}\). 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 \(\mathbf{f:\mathbb{R} to \mathbb{R}}\) such that f(xy)=xf(y) for all \(\mathbf{x,y \in \mathbb{R}}\).(a) If \(\mathbf{f \in A}\) then show that f(x+y)=f(x)+f(y) for all x,y \(\mathbf{\in \mathbb{R}}\)(b) For \(\mathbf{g,h \in A}\), define a function \( \mathbf{g \circ h} \) by \(\mathbf{(g \circ h)(x)=g(h(x))}\) for \(\mathbf{x \in \mathbb{R}}\). Prove that \(\mathbf{g \circ h}\) is in A and is equal to \(\mathbf{h \circ g}\).

- Consider the equation \(\mathbf{n^2+(n+1)^4=5(n+2)^3}\)(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 \(\mathbf{1\le r\le n}\), and consider all subsets of r elements of the set \(\mathbf{{1,2,\ldots,n}}\). Each subset has a smallest element. Let F(n,r) be the arithmetic mean of these smallest elements. Prove that: \(\mathbf{F(n,r)={n+1\over r+1}}\).

- Let \(\mathbf{f: \mathbb{R}^2 to \mathbb{R}^2}\) be a function having the following property: For any two points A and B in \(\mathbf{\mathbb{R}^2}\), 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 \(\mathbf{\mathbb{R}^2}\). 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 \(\mathbf{\mathbb{R}^2}\) and suppose that l(E,F) intersects l(C,D) at an angle \(\mathbf{\alpha}\). 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.