How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?

# 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.