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# ISI Entrance Interview Problems 1. a and b are two numbers having the same no. of digits and same sum of digits (=28). Can one be a multiple of the other? a is not equal to b. (courtesy Abhra Abir Kundu)
2. Is $e^x-sinx$ a polynomial ? (courtesy Tias Kundu)
3. Find the number of onto function from set A containing n elements to set B containing m elements (m<n) (courtesy Tias Kundu)
4. If a+b+c=30,  how many (a,b,c) tuples possible (a,b,c all non-negative). (courtesy Saikat Palit)
5. Can sin(x) be expressed as a polynomial in x? (courtesy Soumik Bhattacharyya)
6. Integers 1-64 are placed in a 8X8 chessboard. How many ways are there to place them such that all numbers in the 1st row and column are in AP? (courtesy Soumik Bhattacharyya)

1. Does there exist a ten-digit number such that all its digits are different and after removing any six digits we get a composite four-digit number?
2. Denote by (a, b) the greatest common divisor of a and b. Let n be a positive integer such that (n, n+ 1) < (n, n+ 2) <
... < (n, n+ 35). Prove that (n, n+ 35) < (n, n+ 36).
3. Suppose n lines are drawn on a plane. Some of them can be concurrent (pass through same point). How many different regions created in this process? (find the least and the greatest number of regions that can be created).
4. Suppose there are 'n' circles no three of which pass through the same point and all of which intersect every other circle at two points. How many regions are created?
5. Suppose there are 100 points on a plane no three of which are on the same straight line. Can you draw a line on the plane such that 50 points are on one side of it?
6. A game is played between two players. There is a round table and unlimited supply of stones (dimension of the stone is unimportant). In each turn of the game a player can put one stone on the table. Whoever fails to find space one the table looses the game. Find a winning strategy for the first player.
7. On a 20 by 20 board a special knight is moving. In each turn the knight moves 1 step in a direction and 5 steps in a direction perpendicular to it. The knight is allowed to take as many turns as required. Can it come back to any of it's four adjacent squares of the square from which it started moving?
8. We often say that $\log (1+x)$ = $x - x^2/2 + x^3/3$ etc. Is the statement true for x = 500 (or may be larger values?) If not, why?
9. Find a point on the plane of a triangle such that the sum of it's distances from three vertices is minimum.

Please post any interview problems that you remember. That will help other prospective applicants.

1. a and b are two numbers having the same no. of digits and same sum of digits (=28). Can one be a multiple of the other? a is not equal to b. (courtesy Abhra Abir Kundu)
2. Is $e^x-sinx$ a polynomial ? (courtesy Tias Kundu)
3. Find the number of onto function from set A containing n elements to set B containing m elements (m<n) (courtesy Tias Kundu)
4. If a+b+c=30,  how many (a,b,c) tuples possible (a,b,c all non-negative). (courtesy Saikat Palit)
5. Can sin(x) be expressed as a polynomial in x? (courtesy Soumik Bhattacharyya)
6. Integers 1-64 are placed in a 8X8 chessboard. How many ways are there to place them such that all numbers in the 1st row and column are in AP? (courtesy Soumik Bhattacharyya)

1. Does there exist a ten-digit number such that all its digits are different and after removing any six digits we get a composite four-digit number?
2. Denote by (a, b) the greatest common divisor of a and b. Let n be a positive integer such that (n, n+ 1) < (n, n+ 2) <
... < (n, n+ 35). Prove that (n, n+ 35) < (n, n+ 36).
3. Suppose n lines are drawn on a plane. Some of them can be concurrent (pass through same point). How many different regions created in this process? (find the least and the greatest number of regions that can be created).
4. Suppose there are 'n' circles no three of which pass through the same point and all of which intersect every other circle at two points. How many regions are created?
5. Suppose there are 100 points on a plane no three of which are on the same straight line. Can you draw a line on the plane such that 50 points are on one side of it?
6. A game is played between two players. There is a round table and unlimited supply of stones (dimension of the stone is unimportant). In each turn of the game a player can put one stone on the table. Whoever fails to find space one the table looses the game. Find a winning strategy for the first player.
7. On a 20 by 20 board a special knight is moving. In each turn the knight moves 1 step in a direction and 5 steps in a direction perpendicular to it. The knight is allowed to take as many turns as required. Can it come back to any of it's four adjacent squares of the square from which it started moving?
8. We often say that $\log (1+x)$ = $x - x^2/2 + x^3/3$ etc. Is the statement true for x = 500 (or may be larger values?) If not, why?
9. Find a point on the plane of a triangle such that the sum of it's distances from three vertices is minimum.

Please post any interview problems that you remember. That will help other prospective applicants.

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### 22 comments on “ISI Entrance Interview Problems”

1. Spandan says:

How to solve the 6th additional problem?

1. Ananya says:

Hi,
Can the answer be that the first player should place his 1st stone at the center of the round table? Then for every other stone kept by the second player there will be a point symmetric about the center diametrically. If second player is able to keep a stone, then the following turn, the first player WILL have a place to keep stone. Hence, player two will run out of places one turn before player one.
My answer might be very stupid. You have my permission to laugh. 😀

1. Ashani Dasgupta says:

Your solution is very nice and I think it is right

1. Sanchal ghosh says:

Any book that can be referred for solving such questions?

1. Ashani Dasgupta says:

Test of Mathematics at 10+2 level is good source.

2. Spandan says:

Thanks a lot Ananya.

3. tanmay says:

no the answer is totally justifying

2. Spandan says:

Hello, I think I just discovered a general case here, Player 1 always wins! Think about it... I will post the solution 2 days l8r...

1. Spandan says:

There are 2n+1 points in the table where the stones can be placed, n diametrically opposite, and 1 at the centre.
(P.S. Consider a stone with a hole in its middle is placed on the centre!)

3. Spandan says:

Hello there! I have been selected for interview this year. Wish me luck (I sure will need it). I will upload the interview problems ASAP.

4. tanmay says:

I wanted to know the answer of the first question( a and b are two nos whose sum of digits is 28............................)

1. Spandan says:

I think its not possible, you know, with the SAME NUMBER OF DIGITS condition.

5. Spandan says:

Hi there! Here are the interview sums that were asked to me-
1. There is a square of side 2 units and sides are parallel to the axes, the sides pass through (1,0), (-1,0), (0,1) and (0,-1). Find its locus.
2. My name is 'Spandan'. Find the total number of permutations of my name such that both the N's are seperate.
3. Sketch the graph of the function x^3 + 4x^2 + ln(x) +3.
4. Show that modulus (x+y) <= modulus(x) + modulus(y)

6. Ashish Sangwan says:

these questions were asked in interviews of ISI b.math or b.stat

1. Ashani Dasgupta says:

From both... this is a mixed set of question submitted by our ex-students.

7. Vishnu Bharadwaja says:

Hi all, the questions which were asked to me during my interview on 10th June 2019 are as follows :
1) consider a function f:R->R , if f(x)=0 for |x|>=5 and integral of f(x+t)dt from 0 to 1 = f(x) , then prove that f(x)=0 for all real x
2) find all n such that n(4^n) is divisible by 5

1. Vishnu Bharadwaja says:

Sorry, the second question is
Find all n such that n(4^n) + 1 is divisible by 5

8. Soumyadeep chakraborty says:

How to solve the problem no 2

1. Shah hetav says:

I have a question which is posted on quora that
If n is a natural no. And a,b,c belongs to the set of integers then prove that there exists n for which
n^3 + an^2+bn+ c is not a perfect square

9. Madhav Garg says:

Spandan did you get selected

10. Arkaprava Roy says:

hello the 1st question i think it should be the recurring part of the inverse of a prime number but how do i equate the sum to 28?

11. Jayant Jha says:

1) f is a continuous function from R to R such that f(0)=0 and f(x)+f(2x)=0 for all real x. Comment about the function f(x).
2) 3^3-3 and 5^3-5 is divisible by 24 is it true for all numbers? If not then find those numbers for which it is true.

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