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1. 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 $\mathbf{\sum_{i=1}^n i(n+1-i) \le \sum_{i=1}^n a_ib_i \le \sum_{i=1}^n i^2 }$

1. 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.

1. 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.

1. 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).

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 $\mathbf{h \circ g}$.

1. 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?

1. 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’.

1. 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: $\mathbf{F(n,r)={n+1\over r+1}}$.

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.

1. 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.