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Explore the Back-StoryHere, you will find all the questions of ISI Entrance Paper 2016 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1:**

In a sports tournament of $n$ players, each pair of players plays exactly one match against each other. There are no draws. Prove that the players can be arranged in an order $P_{1}, P_{2}, \ldots, P_{n},$ such that $P_{i}$ defeats $P_{i+1} \forall i=1,2, \ldots, n-1$

**Problem 2:**

Consider the polynomial $a x^{3}+b x^{2}+c x+d,$ where $a d$ is odd and $b c$ is even. Prove that all roots of the polynomial cannot be rational.

**Problem 3: **

$P(x)=x^{n}+a_{1} x^{n-1}+\ldots+a_{n}$ is a polynomial with real coefficients. $a_{1}^{2}<a_{2}$ Prove that all roots of $P(x)$ cannot be real.

**Problem 4: **

Let $A B C D$ be a square. Let $A$ lie on the positive $x$ -axis and $B$ on the positive $y$ -axis. Suppose the vertex $C$ lies in the first quadrant and has co-ordinates $(u, v) .$ Then find the area of the square in terms of $u$ and $v$.

**Problem 5: **

Prove that there exists a right angle triangle with rational sides and area \(d\) if and only if \(x^2,y^2\) and \(z^2\) are squares of rational numbers and are in Arithmetic Progression

Here \(d\) is an integer.

**Problem 6: **

Suppose in a triangle \(\triangle ABC\), \(A\) , \(B\) , \(C\) are the three angles and \(a\) , \(b\) , \(c\) are the lengths of the sides opposite to the angles respectively. Then prove that if \(sin(A-B)= \frac{a}{a+b}\sin A \cos B - \frac{b}{a+b}\sin B \cos A\) then the triangle \(\triangle ABC\) is isoscelos.

**Problem 7: **

\(f\) is a differentiable function such that \(f(f(x))=x\) where \(x \in [0,1]\).Also \(f(0)=1\).Find the value of

$$\int_0^1(x-f(x))^{2016}dx$$

**Problem 8: **

Suppose that \((a_n)_{n\geq 1}\) is a sequence of real numbers satisfying \(a_{n+1} = \frac{3a_n}{2+a_n}\).

- Suppose $0<a_{1}<1$, then prove that the sequence $a_{n}$ is increasing and hence show that $\lim_{n \ to \infty}a_{n} = 1$.
- Suppose $a_{1}>1$, then prove that the sequence $a_{n}$ is decreasing and hence show that $\lim_{n \to \infty}a_{n} =1$.

Here, you will find all the questions of ISI Entrance Paper 2016 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1:**

In a sports tournament of $n$ players, each pair of players plays exactly one match against each other. There are no draws. Prove that the players can be arranged in an order $P_{1}, P_{2}, \ldots, P_{n},$ such that $P_{i}$ defeats $P_{i+1} \forall i=1,2, \ldots, n-1$

**Problem 2:**

Consider the polynomial $a x^{3}+b x^{2}+c x+d,$ where $a d$ is odd and $b c$ is even. Prove that all roots of the polynomial cannot be rational.

**Problem 3: **

$P(x)=x^{n}+a_{1} x^{n-1}+\ldots+a_{n}$ is a polynomial with real coefficients. $a_{1}^{2}<a_{2}$ Prove that all roots of $P(x)$ cannot be real.

**Problem 4: **

Let $A B C D$ be a square. Let $A$ lie on the positive $x$ -axis and $B$ on the positive $y$ -axis. Suppose the vertex $C$ lies in the first quadrant and has co-ordinates $(u, v) .$ Then find the area of the square in terms of $u$ and $v$.

**Problem 5: **

Prove that there exists a right angle triangle with rational sides and area \(d\) if and only if \(x^2,y^2\) and \(z^2\) are squares of rational numbers and are in Arithmetic Progression

Here \(d\) is an integer.

**Problem 6: **

Suppose in a triangle \(\triangle ABC\), \(A\) , \(B\) , \(C\) are the three angles and \(a\) , \(b\) , \(c\) are the lengths of the sides opposite to the angles respectively. Then prove that if \(sin(A-B)= \frac{a}{a+b}\sin A \cos B - \frac{b}{a+b}\sin B \cos A\) then the triangle \(\triangle ABC\) is isoscelos.

**Problem 7: **

\(f\) is a differentiable function such that \(f(f(x))=x\) where \(x \in [0,1]\).Also \(f(0)=1\).Find the value of

$$\int_0^1(x-f(x))^{2016}dx$$

**Problem 8: **

Suppose that \((a_n)_{n\geq 1}\) is a sequence of real numbers satisfying \(a_{n+1} = \frac{3a_n}{2+a_n}\).

- Suppose $0<a_{1}<1$, then prove that the sequence $a_{n}$ is increasing and hence show that $\lim_{n \ to \infty}a_{n} = 1$.
- Suppose $a_{1}>1$, then prove that the sequence $a_{n}$ is decreasing and hence show that $\lim_{n \to \infty}a_{n} =1$.

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