In a small village, there are 87 families, of which 52 families have atmost 2 children. In a rural development programme 20 families are to be chosen for assistance, of which atleast 18 families must have at most 2 children. In how many ways can the choice be made?

A student has to answer 10 questions, choosing atleast 4 from each of Parts A and B. If there are 6 questions in Part A and 7 in Part B, in how many ways can the student choose 10 questions?

All the letters of the word ‘EAMCOT’ are arranged in different possible ways. The number of such arrangements in which no two vowels are adjacent to each other is …….

Ten different letters of alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have atleast one letter repeated is

The number of signals that can be sent by 6 flags of different colours taking one or more at a time is

The straight

lines l1, l2 and l3 are parallel and lie in

the same plane. A total numbers of 4 points are taken on l1 ; 3

points on l2 , 5 points on l3. The maximum number of

triangles formed with vertices at these points are …

Find the general formula for the nth term of the infinite sequence Un (n = 1, 2, ... ) defined by the conditions U1 = 1, U2 = 3, Un+2 = 4Un+l - 3Un for n = 1, 2, ... .

If we choose any
two coprime numbers, a and b. Then the GCD of 

Let f be a function such that 

Let n be a positive integer. Then the number of lines which go through the origin and precisely one other point with integer coordinates (x, y), 0 ≤ x, y ≤ n, is at least….

Let T denote the 15-element set {10a + b : a, b ∈ Z, 1 ≤ a < b ≤ 6}. Let S be a subset of T in which all six digits 1, 2, . . . , 6 appear and in which no three elements together use all these six digits. Determine the largest possible size of S.

Given a rectangular grid, split into m × n squares, a colouring of the squares in two colours (black and white) is called valid if it satisfies the following conditions:

I. All squares touching the border of the grid are coloured black.

II. No four squares forming a 2 × 2-square are coloured in the same colour.

III. No four squares forming a 2 × 2-square are coloured in such a way that only diagonally touching squares have the same colour.

Which grid sizes m × n (with m, n ≥ 3) have a valid colouring?

Two persons play the following game with integers. The initial number is 2011. The players move in turns. Each move consists of subtraction of an integer between 1 and 2010 inclusive, or division by 2011, rounding down to the closest integer when necessary. The player who first obtains a non-positive integer wins. Which player has a winning strategy?

Let AB and CD be

two diameters of the circle C. For an arbitrary point P on C, let R and S be

the feet of the perpendiculars from P to AB and CD, respectively. Is the

length of RS is independent of the choice of P.

Let P be a point inside a square ABCD such that P A : P B : P C is 1 : 2 : 3. Determine the angle ∠BPA.

The incircle of

a triangle ABC touches the sides BC, CA, AB at D, E, F, respectively. Let G

be a point on the incircle such that F G is a diameter. The lines EG and F D

intersect at H. Is it true that CH intersect AB, if extended.

Let a be any integer. Define the sequence x₀, x₁, . . . by x₀ = a, x₁ = 3 and x_n = 2x_n−1 − 4x_n−2 + 3 for all n > 1. Determine the largest integer k for which there exists a prime p such that p^k divides x₂₀₁₁ −1.

Determine the largest unit positive integer d such that whenever d divides a positive integer n, d will also divide any integer obtained by rearranging the digits of n.

Determine all pairs (p, q) of primes for which both p² + q³ and q² + p³ are perfect squares.

If squared of any

integer is divided by 4 then it remainder will be?

From any three integers, one can always choose two so that a³b−ab³ is divisible by … .

Find smallest positive integer n such that when divided by 4 leaves remainder 2, when divided by 5 leaves remainder 1, and when divided by 7 leaves remainder 1.

How many zeroes are at the end of 300!?

The numbers in the sequence 101,104,109,116,... are of the form a_n = 100+n ² ,n = 1,2,.... For each n let d_n = (a_n,a_n+1). Find max d_n for n≥1.