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 $ {\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 $100$.

Some useful Links:

• ISI BStat and B.Math 2011 Problems and Solutions
• Our ISI and CMI Entrance Program
• ISI Entrance Number Theory || TOMATO Subjective 35- Watch & learn

Here, you will find all the questions of ISI Entrance Paper 2009 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

Problem 1:

Two train lines intersect each other at a junction at an acute angle $ \mathbf{\theta}$. A train is passing along one of the two lines. When the front of the train is at the junction, the train subtends an angle $\mathbf{\alpha}$ at a station on the other line. It subtends an angle $\mathbf{\beta (<\alpha)}$ at the same station, when its rear is at the junction. Show that $\mathbf{ \tan\theta=\frac{2\sin\alpha\sin\beta}{\sin(\alpha-\beta)}}$

Problem 2:

Let $f(x)$ be a continuous function, whose first and second derivatives are continuous on $\mathbf{[0,2\pi]}$ and $\mathbf{f''(x) \geq 0 }$ for all $x$ in $ \mathbf{[0,2\pi]}$. Show that
$\mathbf{\int_{0}^{2\pi} f(x)\cos x dx \geq 0}$

Problem 3:

Let $ABC$ be a right-angled triangle with $BC=AC=1$. Let $P$ be any point on $AB$. Draw perpendiculars $PQ$ and $PR$ on $AC$ and $BC$ respectively from $P$. Define $M$ to be the maximum of the areas of $BPR$, $APQ$ and $PQCR$. Find the minimum possible value of $M$.

Problem 4:

A sequence is called an arithmetic progression of the first order if the differences of the successive terms are constant. It is called an arithmetic progression of the second order if the differences of the successive terms form an arithmetic progression of the first order. In general, for $ \mathbf{k\geq 2}$, a sequence is called an arithmetic progression of the $k$-th order if the differences of the successive terms form an arithmetic progression of the $(k-1)$-th order.
The numbers $4,6,13,27,50,84$ are the first six terms of an arithmetic progression of some order. What is its least possible order? Find a formula for the $n$-th term of this progression.

Problem 5:

A cardboard box in the shape of a rectangular parallelopiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is $60$ centimetres and its base has radius $30$ centimetres, find the volume of the largest box that can be completely enclosed inside the container with the lid on.

Problem 6:

Let $f(x)$ be a function satisfying $xf(x)=\ln x$ for $x>0$
Show that $\mathbf{f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)}$ where $ \mathbf{f^{(n)}(x) }$ denotes the $n$-th derivative evaluated at $x$.

Problem 7:

Show that the vertices of a regular pentagon are concyclic. If the length of each side of the pentagon is $x$, show that the radius of the circumcircle is $ \mathbf{\frac{x}{2}cosec 36^{\circ}}$.

Problem 8:

Find the number of ways in which three numbers can be selected from the set $ \mathbf{{1,2,\cdots ,4n}}$, such that the sum of the three selected numbers is divisible by $4$.

Problem 9:

Consider $6$ points located at $\mathbf{P_0=(0,0), P_1=(0,4), P_2=(4,0), P_3=(-2,-2), P_4=(3,3), P_5=(5,5)}$. Let $R$ be the region consisting of all points in the plane whose distance from $P_0$ is smaller than that from any other $ \mathbf{P_i, i=1,2,3,4,5}$. Find the perimeter of the region $R$.

Problem 10:

Let $\mathbf{x_n}$ be the $n$-th non-square positive integer. Thus $ \mathbf{x_1=2, x_2=3, x_3=5, x_4=6}$ , etc. For a positive real number $x$, denotes the integer closest to it by $\langle x\rangle$ . If $x=m+0.5$, where $m$ is an integer, then define $\langle x\rangle=m$. For example, $\langle 1.2\rangle=1,\langle 2.8\rangle=3,\langle 3.5\rangle=3 . \text { Show that } x_{n}=n+\langle\sqrt{n}\rangle$

Some useful Links :

• ISI BStat and B.Math 2010 Problems and Solutions
• Our ISI and CMI Entrance Program
• ISI 2020 subjective 3 solution-Part2| Beauty of Complex Numbers- watch & learn

Here, you will find all the questions of ISI Entrance Paper 2008 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

Problem 1:

Of all triangles with given perimeter, find the triangle with the maximum area. Justify your answer

Problem 2:

A $40$ feet high screen is put on a vertical wall $10$ feet above your eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye?

Problem 3:

Study the derivatives of the function
$\mathbf{y=\sqrt[3]{x^3-4x}}$
and sketch its graph on the real line.

Problem 4:

Suppose $P$ and $Q$ are the centres of two disjoint circles $\mathbf{C_1}$ and $\mathbf{C_2}$ respectively, such that $P$ lies outside $\mathbf{C_2}$ and $Q$ lies outside $\mathbf{C_1}$. Two tangents are drawn from the point $P$ to the circle $\mathbf{C_2}$, which intersect the circle $\mathbf{C_1}$ at point $A$ and $B$. Similarly, two tangents are drawn from the point $Q$ to the circle $ \mathbf{C_1}$, which intersect the circle $\mathbf{C_2}$ at points $M$ and $N$. Show that $AB=MN$

Problem 5:

Suppose $ABC$ is a triangle with inradius $r$. The incircle touches the sides $BC$, $CA$, and $AB$ at $D,E$ and $F$ respectively. If $BD=x$, $CE=y$ and $AF=z$, then show that $\mathbf{r^2=\frac{xyz}{x+y+z}}$

Problem 6:

Evaluate: $\lim_{n \to\infty} \frac{1}{2n} \ln {{2n} \choose{n}}$

Problem 7:

Consider the equation $\mathbf{x^5+x=10}$. Show that
(a) the equation has only one real root;
(b) this root lies between $1$ and $2$;
(c) this root must be irrational.

Problem 8:

In how many ways can you divide the set of eight numbers $ \mathbf{{2,3,\cdots,9}}$ into $4$ pairs such that no pair of numbers has $ \mathbf{\text{gcd} }$ equal to $2$?

Problem 9:

Suppose $S$ is the set of all positive integers. For $\mathbf{a,b \in S}$, define
$\mathbf{a * b=\frac{\text{lcm}[a,b]}{\text{gcd}(a,b)}}$
For example $8 * 12=6$.
Show that exactly two of the following three properties are satisfied:
(i) If $\mathbf{a,b \in S}$, then $\mathbf{a * b \in S}$.
(ii) $\mathbf{(a*b)*c=a*(b*c)}$ for all $\mathbf{a,b,c \in S}$.
(iii) There exists an element $ \mathbf{i \in S}$ such that $\mathbf{a *i =a}$ for all $\mathbf{a \in S}$

Problem 10:

Two subsets $A$ and $B$ of the ($x,y$)-plane are said to be equivalent if there exists a function $f: A$ to $B$ which is both one-to-one and onto.
(i) Show that any two line segments in the plane are equivalent.
(ii) Show that any two circles in the plane are equivalent.

Some useful links :

• ISI BStat and B.Math 2009 Problems and Solutions
• Our ISI and CMI Entrance Program
• ISI Entrance Solution || TOMATO Subjective 176 || Auxiliary Polynomial - Watch & Learn

Here, you will find all the questions of ISI Entrance Paper 2007 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

Problem 1:

Suppose \(a\) is a complex number such that \( { a^2+a+\frac{1}{a}+\frac{1}{a^2}+1=0 }\) If \(m\) is a positive integer, find the value of \({a^{2m}+a^m+\frac{1}{a^m}+\frac{1}{a^{2m}}}\)
Discussion

Problem 2:

Use calculus to find the behaviour of the function \( { y=e^x\sin{x} -\infty <x< +\infty}\) and sketch the graph of the function for \( {-2\pi \le x \le 2\pi}\). Show clearly the locations of the maxima, minima and points of inflection in your graph.

Problem 3:

Let \(f(u)\) be a continuous function and, for any real number \( u \), let \( [u] \) denote the greatest integer less than or equal to \( u \). Show that for any \( x>1\), \( {\int_{1}^{x} [u]([u]+1)f(u)du = 2\sum_{i=1}^{[x]} i \int_{i}^{x} f(u)du }\)

Problem 4:

Show that it is not possible to have a triangle with sides \( a,b \), and \(c\) whose medians have length \(\frac{2}{3}a, \frac{2}{3}b\) and \(\frac{4}{5}c\).

Problem 5:

Show that \( {-2 \leq \cos \theta\left(\sin \theta + \sqrt{\sin ^2 \theta +3}\right) \leq 2 }\) for all values of \( {\theta}\).

Problem 6:

Let \( {S={1,2,\cdots ,n}}\) where \(n\) is an odd integer. Let \(f\) be a function defined on \((i,j): i\in S, j \in S\) taking values in \(S\) such that

 (a) \(f(s,r)=f(r,s)\) for all \(r,s \in S\)

(b) \(f(r,s): s \in S=S\) for all \(r\in S\) Show that \( {{f(r,r): r\in S}=S}\)

Problem 7:

Consider a prism with triangular base. The total area of the three faces containing a particular vertex \(A\) is \(K\). Show that the maximum possible volume of the prism is \( {\sqrt{\frac{K^3}{54}}}\) and find the height of this largest prism.

Problem 8:

The following figure shows a \( {3^2 \times 3^2 }\) grid divided into \( {3^2}\) subgrids of size \( {3 \times 3}\). This grid has \(81\) cells, \(9\) in each subgrid.

Now consider an \( {n^2 \times n^2}\) grid divided into \( {n^2}\) subgrids of size \( {n \times n}\). Find the number of ways in which you can select \(n^2\) cells from this grid such that there is exactly one cell coming from each subgrid, one from each row and one from each column.

Problem 9:

Let \(X\) \( {\subset \mathbb{R}^2}\) be a set satisfying the following properties:

1. if \( {(x_1,y_1)}\) and \( {(x_2,y_2)}\) are any two distinct elements in \(X\), then
   \( {\text{ either, } x_1 > x_2 \text{ and } y_1 > y_2 \text{ or, } x_1 < x_2 \text{ and } y_1 < y_2}\)

2. there are two elements \( {(a_1,b_1)}\) and \( {(a_2,b_2)}\) in \(X\) such that for any \( {(x,y) in X}\),
\( {a_1\le x \le a_2 \text{ and } b_1\le y \le b_2 }\)

 3. if \( {(x_1,y_1) \text{and} (x_2,y_2)}\) are two elements of \(X\), then for all \( {\lambda \in [0,1], \left(\lambda x_1+(1-\lambda)x_2, \lambda y_1 + (1-\lambda)y_2\right) \in X }\)
Show that if \( {(x,y) \in X}\), then for some \( {\lambda in [0,1], x=\lambda a_1 +(1-\lambda)a_2, y=\lambda b_1 +(1-\lambda)b_2 }\)

Problem 10: 

Let \(A\) be a set of positive integers satisfying the following properties:

1. if \(m\) and \(n\) belong to \(A\), then \(m+n\) belong to \(A\);
2. there is no prime number that divides all elements of \(A\). (a) Suppose \( { n_1 \text{and} n_2 }\) are two integers belonging to \(A\) such that \( {n_2-n_1 > 1}\). Show that you can find two integers \( {m_1 and m_2 }\) in \(A\) such that \( {0 < m_2-m_1 < n_2-n_1}\)
   (b) Hence show that there are two consecutive integers belonging to \(A\).
   (c) Let \(n_0\)and \(n_0+1\) be two consecutive integers belonging to \(A\). Show that if \( {n\geq n_0^2 }\) then \(n\) belongs to \(A\).

Some useful links:

• ISI BStat and B.Math 2006 Problems and Solutions
• Our ISI and CMI Entrance Program
• ISI BStat BMath 2020 | Problem 1 Solution | Polynomial - Watch and Learn

Here, you will find all the questions of ISI Entrance Paper 2006 from Indian Statistical Institute's B. Stat Entrance. You will also get the solutions soon of all the previous year problems.

Problem1 :

If the normal to the curve \(\displaystyle{ x^{\frac{2}{3}}+y^{\frac23}=a^{\frac23} }\) at some point makes an angle \(\displaystyle{\theta}\) with the \(X\)-axis, show that the equation of the normal is

$$\displaystyle{y\cos\theta-xsin\theta =a\cos 2\theta}$$

Problem 2 :

Suppose that \( a \) is an irrational number.
(a) If there is a real number \( b \) such that both \( (a+b) \) and \( ab \) are rational numbers, show that \( a \) is a quadratic surd. (\( a \) is a quadratic surd if it is of the form \(\displaystyle{r+\sqrt{s}}\) or \(\displaystyle{r-\sqrt{s}}\) for some rationals \( r \) and \( s \), where \( s \) is not the square of a rational number).
(b) Show that there are two real numbers \(\displaystyle{b_1}\) and \(\displaystyle{b_2}\) such that
i) \(\displaystyle{a+b_1}\) is rational but \(\displaystyle{ab_1}\) is irrational.
ii) \(\displaystyle{a+b_2}\) is irrational but \(\displaystyle{ab_2}\) is rational.
(Hint: Consider the two cases, where \( a \) is a quadratic surd and \( a \) is not a quadratic surd, separately).

Problem3 :

Prove that \(\displaystyle{n^4 + 4^{n}}\) is composite for all values of \( n \) greater than \( 1 \).
Discussion

Problem4 :

In the figure below, \( E \) is the midpoint of the arc \( ABEC \) and the segment \( ED \) is perpendicular to the chord \( BC \) at \( D \). If the length of the chord \( AB \) is \(\displaystyle{l_1}\), and that of the segment \( BD \) is \(\displaystyle{l_2}\), determine the length of \( DC \) in terms of \(\displaystyle{l_1, l_2}\).

Discussion

Problem 5 :

Let \( A,B \) and \( C \) be three points on a circle of radius \( 1 \).
(a) Show that the area of the triangle \( ABC \) equals \( \frac{1}{2}(\sin (2 \angle A B C)+\sin (2 \angle B C A)\\+\sin (2 \angle C A B)) \)
(b) Suppose that the magnitude of \(\displaystyle{\angle ABC}\) is fixed. Then show that the area of the triangle \( ABC \) is maximized when \(\displaystyle{\angle BCA=\angle CAB}\)
(c) Hence or otherwise, show that the area of the triangle \(ABC\) is maximum when the triangle is equilateral.

Problem 6 :

(a) Let \(\displaystyle{f(x)=x-xe^{-\frac1x}, x>0 }\). Show that \( f(x) \) is an increasing function on \(\displaystyle{(0,\infty)}\), and \(\displaystyle{\lim_{x\to\infty} f(x)=1}\).
(b) Using part (a) or otherwise, draw graphs of \(\displaystyle{y=x-1, y=x, y=x+1}\) , and \( y=xe^{-\frac{1}{|x|}} \) for \(\displaystyle{-\infty < x < \infty}\) using the same \( X \) and \( Y \) axes.

Problem 7 :

For any positive integer \( n \) greater than \( 1 \), show that \(\displaystyle{2^n < {{2n} \choose{n}} <\frac{2^n}{\prod\limits_{i=0}^{n-1} \left(1-\frac{i}{n}\right)}}\)

Problem 8:

Show that there exists a positive real number \(\displaystyle{x\neq 2}\) such that \(\displaystyle{\log_2x=\frac{x}{2}}\). Hence obtain the set of real numbers \( c \) such that \(\displaystyle{\frac{\log_2x}{x}=c}\) has only one real solution.

Problem 9 :

Find a four digit number \( M \) such that the number \(\displaystyle{N=4\times M}\) has the following properties.
(a) \( N \) is also a four digit number
(b) \( N \) has the same digits as in \( M \) but in reverse order.

Problem 10 :

Consider a function \( f \) on nonnegative integers such that \( f(0)=1 \), \( f(1)=0 \) and \( f(n) \)+\( f(n-1) \)=\( nf(n-1) \)+\( (n-1)f(n-2) \) for \(\displaystyle{n \ge 2}\). Show that \(\displaystyle{\frac{f(n)}{n!}= sum_{k=0}^n \frac{(-1)^k}{k!}}\)

Some useful link :

• ISI B.Stat and B. math 2007 problems & solution
• Our ISI & CMI Entrance program
• ISI BStat BMath 2020 | Subjective Problem 3 | PART 1 - watch & learn

Here, you will find all the questions of ISI Entrance Paper 2005 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

Problem 1:

Let \( a,b \) and \( c \) be the sides of a right angled triangle. Let \( \displaystyle{\theta } \) be the smallest angle of this triangle. If \( \displaystyle{ \frac{1}{a}, \frac{1}{b} } \) and \( \displaystyle{ \frac{1}{c} } \) are also the sides of a right angled triangle then show that \( \displaystyle{ \sin\theta=\frac{\sqrt{5}-1}{2}} \)

Discussion

Problem 2:

Let \( \displaystyle{f(x)=\int_0^1 |t-x|t , dt } \) for all real \(x \). Sketch the graph of \( f(x) \). What is the minimum value of \( f(x) \)?

Problem 3:
Let \( f \) be a function defined on \( \displaystyle{ {(i,j): i,j \in \mathbb{N}} } \) satisfying \( \displaystyle{ f(i,i+1)=\frac{1}{3} } \) for all i \( f(i, j) \) = \( f(i, k) \)+ \( f(k, j) \) - \( 2 f(i, k) f(k, j) \) for all \( k \) such that \( i <k \). Find all real solutions of the equation \( \displaystyle{ \sin^{5}x+\cos^{3}x=1 } \).

Discussion

Problem 4:

Consider an acute-angled triangle \( PQR \) such that \( C,I \) and \( O \) are the circumcentre, incentre and orthocentre respectively. Suppose \( \displaystyle{ \angle QCR, \angle QIR } \) and \( \displaystyle{ \angle QOR } \), measured in degrees, are \( \displaystyle{ \alpha, \beta  and  \gamma } \) respectively. Show that \( \displaystyle{ \frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma} } \) > \( \displaystyle{ \frac{1}{45} } \)

Discussion

Problem 5:

Let \( f \) be a function defined on \( \displaystyle{ (0, \infty ) } \) as follows: \( \displaystyle{ f(x)=x+\frac1x } \) . Let \( h \) be a function defined for all \( \displaystyle{ x \in (0,1) } \) as \( \displaystyle{h(x)=\frac{x^4}{(1-x)^6} } \). Suppose that \( g(x)=f(h(x)) \) for all \( \displaystyle{x \in (0,1)} \). Show that \( h \) is a strictly increasing function. Show that there exists a real number \( \displaystyle{x_0 \in (0,1)} \) such that \( g \) is strictly decreasing in the interval \( \displaystyle{ (0,x_0] } \) and strictly increasing in the interval \( \displaystyle{[x_0,1)} \).

Problem 6:

For integers \( \displaystyle{ m,n\geq 1 } \), Let \( \displaystyle{ A_{m,n} , B_{m,n} } \) and \( \displaystyle{ C_{m,n}} \) denote the following sets:

a) \( A_{m, n} \)=\( \left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{m}\right) \) : \( 1 \leq \alpha_{1} \leq \alpha_{2} \leq \ldots \leq \alpha_{m} \leq n \) given that \( \alpha_{i} \in \mathbb{Z} \) for all \( i \)

b) \( B_{m, n} \)=\( \left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{m}\right) \) : \( \alpha_{1}+\alpha_{2}+\ldots+\alpha_{m} \)=\( n \) given that \( \alpha_{i} \geq 0 \) and \( \alpha_{i} \in \mathbb{Z} \) for all \( i \)

c) \( C_{m, n} \)=\( \left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{m}\right) \) : \( 1 \leq \alpha_{1} \alpha_{2} \ldots ; \alpha_{m} \leq n \) given that \( \alpha_{i} \in \mathbb{Z} \) for all \( i \)

d) Define a one-one onto map from \( \displaystyle{A_{m,n}} \) onto \( \displaystyle{B_{m+1,n-1}} \).

e) Find the number of elements of the sets \( \displaystyle{A_{m,n}} \) and \( \displaystyle{B_{m,n}} \)

Problem 7:

A function \( f(n) \) is defined on the set of positive integers is said to be multiplicative if \( f(mn)=f(m)f(n) \) whenever \( m \) and \( n \) have no common factors greater than \( 1 \). Are the following functions multiplicative? Justify your answer.

Problem 8:

\( \displaystyle{ g(n)=5^k } \) where k is the number of distinct primes which divide \( n \). \( \displaystyle{ h(n)= 0} \) if \( n \) is divisible by \( k^2 \) for some integer \( k>1 \) ...., 1 otherwise.

Problem 9:

Suppose that to every point of the plane a colour, either red or blue, is associated. Show that if there is no equilateral triangle with all vertices of the same colour then there must exist three points \( A,B \) and \( C \) of the same colour such that \( B \) is the midpoint of \( AC \). Show that there must be an equilateral triangle with all vertices of the same colour.

Problem 10:

Let \( ABC \) be a triangle. Take n point lying on the side \( AB \) (different from \( A \) and \( B \)) and connect all of them by straight lines to the vertex \( C \). Similarly, take n points on the side \( AC \) and connect them to \( B \). Into how many regions is the triangle \( ABC \) partitioned by these lines? Further, take \( n \) points on the side \( BC \) also and join them with \( A \). Assume that no three straight lines meet at a point other than \( A,B \) and \( C \). Into how many regions is the triangle \( ABC \) partitioned now?

some useful link

• ISI B.stat & B.math 2007 problems & solutions
• Our ISI & CMI Entrance program
• ISI BStat 2010 Solution and Strategy || Subjective Problem 6 || Number Theory - watch & learn

[et_pb_section fb_built="1" admin_label="Courses Hero" _builder_version="4.3.2" use_background_color_gradient="on" background_color_gradient_start="#474ab6" background_color_gradient_end="#9271f6" background_image="https://www.staging18.cheenta.com/wp-content/uploads/2018/03/coding-background-texture.jpg" background_blend="overlay" custom_padding="100px|0px|0|0px|false|false" animation_direction="top" locked="off"][et_pb_row _builder_version="4.3.2" background_size="initial" background_position="top_left" background_repeat="repeat" custom_margin="|||" custom_width_px="1280px"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_text _builder_version="3.27.4" header_font="|on|||" header_font_size="42px" header_line_height="1.3em" background_size="initial" background_position="top_left" background_repeat="repeat" text_orientation="center" background_layout="dark" max_width="530px" module_alignment="center" custom_padding="|||" header_font_size_tablet="" header_font_size_phone="" header_font_size_last_edited="on|desktop" locked="off"]

Problem Garden

[/et_pb_text][et_pb_text _builder_version="4.3.2" text_text_color="#d4ccff" text_line_height="1.9em" background_size="initial" background_position="top_left" background_repeat="repeat" text_orientation="center" max_width="540px" module_alignment="center" locked="off"]Mathematics is not a spectator sport. In this portal, we have gathered (and are adding) problems, discussions and challenges that you may try your hands on.
[/et_pb_text][/et_pb_column][/et_pb_row][/et_pb_section][et_pb_section fb_built="1" fullwidth="on" _builder_version="4.3.2"][et_pb_fullwidth_header title="Math Olympiad" _builder_version="4.3.2" background_color="#00457a"][/et_pb_fullwidth_header][/et_pb_section][et_pb_section fb_built="1" _builder_version="4.3.2"][et_pb_row column_structure="1_4,1_4,1_4,1_4" _builder_version="4.3.2"][et_pb_column type="1_4" _builder_version="4.3.2"][et_pb_accordion _builder_version="4.3.2"][et_pb_accordion_item title="India" open="on" _builder_version="4.3.2"][/et_pb_accordion_item][/et_pb_accordion][et_pb_button button_url="https://www.staging18.cheenta.com/math-olympiad-portal/" button_text="Enter Portal" button_alignment="center" _builder_version="4.3.2" custom_button="on" button_text_size="14px" button_text_color="#ffffff" button_bg_color="#0c71c3" button_border_color="#0c71c3"][/et_pb_button][/et_pb_column][et_pb_column type="1_4" _builder_version="4.3.2"][et_pb_accordion _builder_version="4.3.2"][et_pb_accordion_item title="U.S.A." open="on" _builder_version="4.3.2"][/et_pb_accordion_item][/et_pb_accordion][et_pb_button button_url="https://www.staging18.cheenta.com/amc-aime-portal/" button_text="Enter Portal" button_alignment="center" _builder_version="4.3.2" custom_button="on" button_text_size="14px" button_text_color="#ffffff" button_bg_color="#0c71c3" button_border_color="#0c71c3"][/et_pb_button][/et_pb_column][et_pb_column type="1_4" _builder_version="4.3.2"][et_pb_accordion disabled_on="on|on|on" _builder_version="4.3.2" disabled="on"][et_pb_accordion_item title="Australia" open="on" _builder_version="4.3.2"][/et_pb_accordion_item][/et_pb_accordion][et_pb_button button_url="https://www.staging18.cheenta.com/math-olympiad-portal/" button_text="Enter Portal" button_alignment="center" disabled_on="on|on|on" _builder_version="4.3.2" custom_button="on" button_text_size="14px" button_text_color="#ffffff" button_bg_color="#0c71c3" button_border_color="#0c71c3" disabled="on"][/et_pb_button][/et_pb_column][et_pb_column type="1_4" _builder_version="4.3.2"][et_pb_accordion disabled_on="on|on|on" _builder_version="4.3.2" disabled="on"][et_pb_accordion_item title="Singapore" open="on" _builder_version="4.3.2"][/et_pb_accordion_item][/et_pb_accordion][et_pb_button button_url="https://www.staging18.cheenta.com/math-olympiad-portal/" button_text="Enter Portal" button_alignment="center" disabled_on="on|on|on" _builder_version="4.3.2" custom_button="on" button_text_size="14px" button_text_color="#ffffff" button_bg_color="#0c71c3" button_border_color="#0c71c3" disabled="on"][/et_pb_button][/et_pb_column][/et_pb_row][/et_pb_section][et_pb_section fb_built="1" fullwidth="on" _builder_version="4.3.2"][et_pb_fullwidth_header title="I.S.I. & C.M.I. Entrance" _builder_version="4.3.2" background_color="#00457a"][/et_pb_fullwidth_header][/et_pb_section][et_pb_section fb_built="1" _builder_version="4.3.2"][et_pb_row _builder_version="4.3.2"][et_pb_column type="4_4" _builder_version="4.3.2"][et_pb_button button_url="https://www.staging18.cheenta.com/isi-entrance-solutions-and-problems/" button_text="Enter Portal" button_alignment="center" _builder_version="4.3.2" custom_button="on" button_text_size="14px" button_text_color="#ffffff" button_bg_color="#0c71c3" button_border_color="#0c71c3"][/et_pb_button][/et_pb_column][/et_pb_row][/et_pb_section][et_pb_section fb_built="1" fullwidth="on" _builder_version="4.3.2"][et_pb_fullwidth_header title="College Mathematics" subhead="TIFR, IIT JAM, I.S.I M,Math, Subject GRE" _builder_version="4.3.2" background_color="#00457a"][/et_pb_fullwidth_header][/et_pb_section][et_pb_section fb_built="1" _builder_version="4.3.2"][et_pb_row _builder_version="4.3.2"][et_pb_column type="4_4" _builder_version="4.3.2"][et_pb_button button_url="https://www.staging18.cheenta.com/problem-garden/college-mathematics/" button_text="Enter Portal" button_alignment="center" _builder_version="4.3.2" custom_button="on" button_text_size="14px" button_text_color="#ffffff" button_bg_color="#0c71c3" button_border_color="#0c71c3"][/et_pb_button][/et_pb_column][/et_pb_row][/et_pb_section]