What is the NO-SHORTCUT approach for learning great Mathematics?

Learn MoreHere, you will find all the questions of ISI Entrance Paper 2010 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 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 }$

**Problem 2:**

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

**Problem 3:**

** **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.

**Problem 4: **

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 ($0,1$).

**Problem 5:**

** **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 $\mathbf{h \circ g}$.

**Problem 6: **

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?

**Problem 7:**

** **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 $AB'D'$.

**Problem 8: **

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}}$.

**Problem 9: **

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.

**Problem 10: **

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 $2 \cdot 100$, 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 $100$.

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