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Here, 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 $100$.

Here, 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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