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

**Problem 1:**

Let $n$ be a positive integer . If $n$ has odd number of divisors ( other than $1$ and $n$ ) , then show that $n$ is a perfect square .

**Problem 2: **

Let $a$ and $b$ be two non-zero rational numbers such that the equation $\mathbf{ax^2+by^2=0}$ has a non-zero solution in rational numbers . Prove that for any rational number $t$ , there is a solution of the equation $\mathbf{ax^2+by^2=t}$.

**Problem 3:**

** **For a natural number n>1 , consider the n-1 points on the unit circle $\mathbf{e^{\frac{2\pi ik}{n}} (k=1,2,...,n-1)}$ . Show that the product of the distances of these points from $1$ is $n$.

**Problem 4: **

Let $ABC$ be an isosceles triangle with $AB=AC=20$ . Let P be a point inside the triangle $ABC$ such that the sum of the distances of $P$ to $AB$ and $AC$ is $1$ . Describe the locus of all such points inside triangle $ABC$.

**Problem 5: **

Let $P(X)$ be a polynomial with integer coefficients of degree $d>0$.(a) If $\mathbf{\alpha}$ and $ \mathbf{\beta}$ are two integers such that $ \mathbf{P(\alpha)=1}$ and $\mathbf{P(\beta)=-1}$ , then prove that $ \mathbf{|\beta - \alpha|}$ divides 2.(b) Prove that the number of distinct integer roots of $\mathbf{P^2(x)-1}$ is at most $d+2$.

**Problem 6: **

In ISI club each member is on two committees and any two committees have exactly one member in common . There are 5 committees . How many members does ISI club have?

**Problem 7: **

Let $\mathbf{0\leq \theta\leq \frac{\pi}{2}}$ . Prove that $\mathbf{sin \theta \geq \frac{2\theta}{\pi}}$.

Solution

**Problem 8: **

Let $\mathbf{P:\mathbb{R} \to \mathbb{R}}$ be a continuous function such that $P(x)=x$ has no real solution. Prove that $P(P(x))=x$ has no real solution.

Solution

**Problem 9:**

** **In a group of five people any two are either friends or enemies , no three of them are friends of each other and no three of them are enemies of each other . Prove that every person in this group has exactly two friends .

**Problem 10: **

The eleven members of a cricket team are numbered 1,2,...,11. In how many ways can the entire cricket team sit on the eleven chairs arranged around a circular table so that the numbers of any two adjacent players differ by one or two ?

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

**Problem 1:**

Let $n$ be a positive integer . If $n$ has odd number of divisors ( other than $1$ and $n$ ) , then show that $n$ is a perfect square .

**Problem 2: **

Let $a$ and $b$ be two non-zero rational numbers such that the equation $\mathbf{ax^2+by^2=0}$ has a non-zero solution in rational numbers . Prove that for any rational number $t$ , there is a solution of the equation $\mathbf{ax^2+by^2=t}$.

**Problem 3:**

** **For a natural number n>1 , consider the n-1 points on the unit circle $\mathbf{e^{\frac{2\pi ik}{n}} (k=1,2,...,n-1)}$ . Show that the product of the distances of these points from $1$ is $n$.

**Problem 4: **

Let $ABC$ be an isosceles triangle with $AB=AC=20$ . Let P be a point inside the triangle $ABC$ such that the sum of the distances of $P$ to $AB$ and $AC$ is $1$ . Describe the locus of all such points inside triangle $ABC$.

**Problem 5: **

Let $P(X)$ be a polynomial with integer coefficients of degree $d>0$.(a) If $\mathbf{\alpha}$ and $ \mathbf{\beta}$ are two integers such that $ \mathbf{P(\alpha)=1}$ and $\mathbf{P(\beta)=-1}$ , then prove that $ \mathbf{|\beta - \alpha|}$ divides 2.(b) Prove that the number of distinct integer roots of $\mathbf{P^2(x)-1}$ is at most $d+2$.

**Problem 6: **

In ISI club each member is on two committees and any two committees have exactly one member in common . There are 5 committees . How many members does ISI club have?

**Problem 7: **

Let $\mathbf{0\leq \theta\leq \frac{\pi}{2}}$ . Prove that $\mathbf{sin \theta \geq \frac{2\theta}{\pi}}$.

Solution

**Problem 8: **

Let $\mathbf{P:\mathbb{R} \to \mathbb{R}}$ be a continuous function such that $P(x)=x$ has no real solution. Prove that $P(P(x))=x$ has no real solution.

Solution

**Problem 9:**

** **In a group of five people any two are either friends or enemies , no three of them are friends of each other and no three of them are enemies of each other . Prove that every person in this group has exactly two friends .

**Problem 10: **

The eleven members of a cricket team are numbered 1,2,...,11. In how many ways can the entire cricket team sit on the eleven chairs arranged around a circular table so that the numbers of any two adjacent players differ by one or two ?

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