INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More

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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