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

**Problem 1 :**

Let be a continuous function. Suppose

for all and all . Then show that there exists a constant such that for all .

**Problem 2 :**

Suppose that is a polynomial with real coefficients such that for some positive real numbers and for all natural numbers we have

Prove that has a real zero.

**Problem 3 :**

Let be a complex number such that are collinear in the complex plane. Show that is a real number.

**Problem 4 :**

Let be integers. Show that there exists integers and such that the sum

is divisible by .

**Problem 5 :**

If a polynomial with integer coefficients has three distinct integer zeroes, then show that for any integer .

**Problem 6 :**

Let denote the binomial coefficient and be the th Fibonacci number given by and for all . Show that

Here, the above sum is over all pairs of integers with .

**Problem 7 :**

Let,

How many squares can be formed in the plane all of whose vertices are in and whose sides are parallel to the -axis and -axis?

**Problem 8 :**

Let,

.

Prove that,

.

**Problem 9 :**

For determine all real solutions of the system of equations:

**Problem 10 :**

If is a prime number and is a natural number, then show that the greatest common divisor of the two numbers and is either or .

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

**Problem 1 :**

Let be a continuous function. Suppose

for all and all . Then show that there exists a constant such that for all .

**Problem 2 :**

Suppose that is a polynomial with real coefficients such that for some positive real numbers and for all natural numbers we have

Prove that has a real zero.

**Problem 3 :**

Let be a complex number such that are collinear in the complex plane. Show that is a real number.

**Problem 4 :**

Let be integers. Show that there exists integers and such that the sum

is divisible by .

**Problem 5 :**

If a polynomial with integer coefficients has three distinct integer zeroes, then show that for any integer .

**Problem 6 :**

Let denote the binomial coefficient and be the th Fibonacci number given by and for all . Show that

Here, the above sum is over all pairs of integers with .

**Problem 7 :**

Let,

How many squares can be formed in the plane all of whose vertices are in and whose sides are parallel to the -axis and -axis?

**Problem 8 :**

Let,

.

Prove that,

.

**Problem 9 :**

For determine all real solutions of the system of equations:

**Problem 10 :**

If is a prime number and is a natural number, then show that the greatest common divisor of the two numbers and is either or .

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