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Explore the Back-StoryHere, you will find all the questions of ISI Entrance Paper 2011 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 constant such that for all What can you say about ? Justify your answer.

**Problem 2 :**

Let for . Define a function on the nonnegative real numbers as follows: for each integer , the graph of the function on the interval is the straight line segment connecting the points and . Find the total area of the region which lies between the curves of and .

**Problem 3 :**

For any positive integer , show that .

**Problem 4 :**

If are not necessarily distinct real numbers such that for all , then show that we can choose three of them such that they are the lengths of the sides of a triangle.

**Problem 5 :**

For any real number let denote the largest integer which is less than or equal to . Let be the sequence of non-square positive integers. If the th non-square positive integer satisfies then show that

**Problem 6 :**

Let and be two cubes with sides of lengths and respectively, where and are positive integers. Show that the difference of their volumes equals the difference of their surface areas, if and only if .

**Problem 7 :**

Let be any triangle and let be a point on the line segment Show that there exists a line parallel to which divides the triangle into two equal parts of equal area.

**Problem 8 :**

Let be real numbers, and consider the function be given by Show that

Here, you will find all the questions of ISI Entrance Paper 2011 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 constant such that for all What can you say about ? Justify your answer.

**Problem 2 :**

Let for . Define a function on the nonnegative real numbers as follows: for each integer , the graph of the function on the interval is the straight line segment connecting the points and . Find the total area of the region which lies between the curves of and .

**Problem 3 :**

For any positive integer , show that .

**Problem 4 :**

If are not necessarily distinct real numbers such that for all , then show that we can choose three of them such that they are the lengths of the sides of a triangle.

**Problem 5 :**

For any real number let denote the largest integer which is less than or equal to . Let be the sequence of non-square positive integers. If the th non-square positive integer satisfies then show that

**Problem 6 :**

Let and be two cubes with sides of lengths and respectively, where and are positive integers. Show that the difference of their volumes equals the difference of their surface areas, if and only if .

**Problem 7 :**

Let be any triangle and let be a point on the line segment Show that there exists a line parallel to which divides the triangle into two equal parts of equal area.

**Problem 8 :**

Let be real numbers, and consider the function be given by Show that

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