1. Let g(x) and h(x) be polynomials with real coefficients such that

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1 1. Let g(x) and h(x) be polynomials with real coefficients such that g(x)(x 2 3x + 2) = h(x)(x 2 + 3x + 2) and f(x) = g(x)h(x) + (x 4 5x 2 + 4). Prove that f(x) has at least four real roots. 2. Let M be a point inside an acute-angled triangle ABC such that AMB = BMC = CMA. Show that a2 + b MA + MB + MC = 2 + c where a, b, c are the sides and is the area of the triangle. 3. Find the area of the region {(x, y) y 2 4x 8 and y 2 8x 0}. 4. Consider an ellipse with centre O and major and minor axes 2a and 2b respectively. Let P be a variable point on the ellipse and let N be the foot of the perpendicular drawn from O to the tangent to the ellipse at P. Show that the maximum possible area of the triangle OP N is (a 2 b 2 )/4. 5. On plain paper, sketch the graph of y = 1 x 2 1 x, x 0, showing the coordinate axes and indicating points of discontinuity, minima and maxima, if any, of the curve. 6. Let f be a function from the set of all integers to the set of all real numbers such that f(n + 9) = f(n) and f(n + 16) = f(n) for every integer n. Show that f is a constant function. 7. Let X = {a 1, a 2,..., a 5 } where a 1, a 2,..., a 5 are integers which are perfect squares. Show that there exists a subset Y = {b 1, b 2, b 3 } of X such that b 1 + b 2 + b 3 is divisible by Let N be the set of all positive integers and X = {(m, n) m, n N}. Consider the order on the set X defined by: (m, n) (p, q) if (i) n < q or (ii) n = q and m < p. (a) Give examples to show that for (m, n) (p, q), the set {(x, y) X (m, n) (x, y) and (x, y) (p, q)} may be finite or infinite. 1

2 (b) Suppose f is a function from X to N with the following property: for any m, n, p, q N, if (m, n) (p, q) then f((m, n)) < f((p, q)). Show that f cannot be one-to-one. 9. Let f(x) = (x u 1 )(x u 2 )... (x u n )+1 where u 1, u 2,..., u n are distinct integers and n is odd. Show that f(x) cannot be factorized as f(x) = g(x)h(x), where g(x) and h(x) are polynomials with integer coefficients and the degrees of g(x) and h(x) are strictly less than n. 10. Let m < n < p be three positive integers and let α be a complex root of the polynomial x p + x n + x m + 1. If α = 1, show that (α p n m 1)(α n m + 1) = Let f(x) and g(x) be two quadratic polynomials all of whose coefficients are rational numbers. Suppose f(x) and g(x) have a common irrational root. Show that g(x) = rf(x) for some rational number r. 12. Suppose that all roots of the polynomial equation x 4 4x 3 + ax 2 + bx + 1 = 0 are positive real numbers. Show that all the roots of the polynomial are equal. 13. Show that the number with 3 n digits is divisible by 3 n. 14. Prove that n C 0 + n C 3 + n C n C 3k 1 3 (2n + 2), where n is a positive integer and k is the largest integer for which 3k n. 15. Let E be the ellipse with centre at origin O whose major and minor axes are of length 2a and 2b respectively. Let θ be the acute angle at which E is cut by a circle with centre at the origin (i.e., θ is the acute angle of intersection of their tangents at a point of intersection). Prove that the maximum possible value of θ is tan 1 ( a2 b 2 2ab ). 16. A function f : R R is called periodic if for some constant a > 0, f(x + a) = f(x) for every real number x. Show that the function 3 f(x) = cos x + cos( 2 x) is not periodic. 2

3 17. Show that the equation x 3 + 7x 14(n 2 + 1) = 0 has no integral root for any integer n. 18. Suppose f is a function that f(x) > 0 and f (x) is continuous at every real number x. If f (t) f(t) for all t, then show that for all x 1. 1 f(x) f(1) + (x 1) Use the derivatives and the left and right limits at points of discontinuities, if any, of the function f(x) = x x x + 2 x to make conclusions about the behaviour of the function as x ranges over all possible values. Using this, draw a rough sketch of the graph of the function f(x) on plain paper. 20. For a finite set A, let A denote the number of elements in the set A. (a) Let F be the set of all functions satisfying f : {1, 2,..., n} {1, 2,..., k} (n 3, k 2) f(i) f(i + 1) for every i, 1 i n 1. Show that F = k(k 1) n 1. (b) Let c(n, k) denote the number of functions in F satisfying f(n) f(1). For n 4, show that (c) Using (b) prove that for n 3, c(n, k) = k(k 1) n 1 c(n 1, k). c(n, k) = (k 1) n + ( 1) n (k 1). 21. Find all positive integers x such that [x/5] [x/7] = 1, where, for any real number t, [t] is the greatest integer less than or equal to t. 22. Consider the equation x 3 + Gx + H = 0, where G and H are complex numbers. Suppose that this equation has a pair of complex conjugate roots. Show that both G and H are real. 3

4 23. Consider the parabola y 2 = 4x. Let P = (a, b) be any point inside the parabola, i.e., b 2 < 4a, and let F be the focus of the parabola. Find the point Q on the parabola such that F Q+QP is minimum. Also, show that the normal to the parabola at Q bisects the angle F QP. 24. Let a be a non-zero complex number such that a = 1. Let P be the point a in the complex plane, and let Q be the point 1/a. Let C 1 be the circle {z : z = 1} and let C 2 be any circle passing through P and Q. Show that C 1 and C 2 intersect orthogonally. [Two circles are said to intersect orthogonally if the tangents at a point of intersection are perpendicular to each other.] 25. Let N = {1, 2,..., n} be a set of elements called voters. Let C = {S : S N} be the set of all subsets of N. Members of C are called coalitions. Let f be a function from C to {0, 1}. A coalition S N is said to be winning if f(s) = 1; it is said to be a losing coalition if f(s) = 0. Such a function f is called a voting game if the following conditions hold. (a) N is a winning coalition. (b) The empty set φ is a losing coalition. (c) If S is a winning coalition and S S, then S also is winning. (d) If both S and S are winning coalitions, then S S φ, i.e., S and S have a common voter. Show that the maximum number of winning coalitions of a voting game is 2 n 1. Also, find a voting game for which the number of winning coalitions is 2 n Find the number of rational numbers m/n, where m, n are relatively prime positive integers satisfying m < n and mn = 25!. 27. Suppose p is a prime number such that (p 1)/4 and (p + 1)/2 are also primes. Show that p = Let n 2 be an integer. Let m be the largest integer which is less than or equal to n, and which is a power of 2. Put l n = the least common multiple of 1, 2,..., n. Show that l n /m is odd, and that for every integer k n, k m, l n /k is even. Hence prove that is not an integer n 29. Suppose f is a real-valued differentiable function defined on [1, ) with f(1) = 1. Suppose, moreover, that f satisfies f (x) = 1 x 2 + f 2 (x). Show that f(x) 1 + π/4 for every x 1. 4

5 30. Consider the real-valued function f, defined over (, ) by f(x) = x 4 + bx 3 + cx 2 + dx + e, where b, c, d, e are real numbers and 3b 2 < 8c. Show that the function f has a unique minimum. 31. Let P (z) = az 2 + bz + c, where a, b, c are complex numbers. (a) If P (z) is real for all real numbers z, show that a, b, c are real numbers. (b) In addition to (a) above, assume that P (z) is not real whenever z is not real. Show that a = For real numbers x, y and z, show that x + y + z x + y z + y + z x + z + x y. 33. Show that there is no real constant c > 0 such that cos x + c = cos x for all real numbers x Let f(x, y) = x 2 + y 2. Consider the region, including the boundary, enclosed by y = x 2, y = x 2 and x = y Find the maximum value of f(x, y) in this region. 35. For any positive integer n, let n denote the integer nearest to n. (a) Given a positive integer k, describe all positive integers n such that n = k. (b) Show that n=1 2 n + 2 n 2 n = For a natural number n, let a n = n If d n denotes the greatest common divisor of a n and a n+1, then show that d n divides Let ABCD be a cyclic quadrilateral with lengths of sides AB = p, BC = q, CD = r and DA = s. Show that AC ps + qr = BD qp + rs. 38. Find the vertices of the two right angled triangles each having area 18 and such that the point (2,4) lies on the hypotenuse and the other two sides are formed by the x and y axes. 39. Consider the squares of an 8 8 chessboard filled with the numbers 1 to 64 as in the figure below. If we choose 8 squares with the property that there is exactly one from each row and exactly one from each column, and add up the numbers in the chosen squares show that the sum obtained is always

6 Suppose that one moves along the points (m, n) in the plane where m and n are integers in such a way that each move is a diagonal step, that is, consists of one unit to the right or left followed by one unit either up or down. (a) Which points (p, q) can be reached from the origin? (b) What is the minimum number of moves needed to reach such a point (p, q)? 41. If a circle intersects the hyperbola y = 1/x at four distinct points (x i, y i ), i = 1, 2, 3, 4, then prove that x 1 x 2 = y 3 y Let a 1, a 2,, a n be n numbers such that each a i is either 1 or -1. If then prove that 4 divides n. a 1 a 2 a 3 a 4 + a 2 a 3 a 4 a a n a 1 a 2 a 3 = 0, 43. Let l, b be positive integers. Divide the l b rectangle into lb unit squares in the usual manner. Consider one of the two diagonals of this rectangle. How many of these unit squares contain a segment of positive length of this diagonal? 44. Find all (x, y) such that sin x + sin y = sin(x + y) and x + y = Find the maximum among 1, 2 1/2, 3 1/3, 4 1/4,. 46. Using Calculus, sketch on plain paper the graph of the function f(x) = x 2 + x + 1 x + 1 for x 0. x2 Show that the function f defined as above for positive real numbers attains a unique minimum. What is the minimum value of the function? What is the value of x at which the minimum is attained? 47. Let S = {1, 2,..., n}. Find the number of unordered pairs {A, B} of subsets of S such that A and B are disjoint, where A and B or both may be empty. 6

7 48. Let f(x, y) = xy, where x 0 and y 0. Prove that the function f satisfies the following property: f(λx + (1 λ)x, λy + (1 λ)y ) > min{f(x, y), f(x, y )} for all (x, y) (x, y ) and for all λ (0, 1). 49. Find all possible real numbers a, b, c, d, e which satisfy the following set of equations: 3a = (b + c + d) 3, 3b = (c + d + e) 3, 3c = (d + e + a) 3, 3d = (e + a + b) 3, 3e = (a + b + c) At time 0, a particle is at the point 0 on the real line. At time 1, the particle divides into two and instantaneously after division, one particle moves 1 unit to the left and the other moves one unit to the right. At time 2, each of these particles divides into two, and one of the two new particles moves one unit to the left and the other moves one unit to the right. Whenever two particles meet, they destroy each other leaving nothing behind. How many particles will be there after time ? 51. For any positive integer k, prove that 2( k + 1 k) < 1 k < 2( k k 1). Also, compute the integral part of , Let a 1 = 1 and a n = n (a n 1 + 1) for all n 2. Define Compute lim n P n. P n = (1 + 1 a 1 ) (1 + 1 a n ). 53. Let ABCD be a quadrilateral such that the sum of a pair of opposite sides equals the sum of the other pair of opposite sides (i.e., AB + CD = AD + BC). Prove that the circles inscribed in triangles ABC and ACD are tangent to each other. 54. For a set S we denote its cardinality by S. Let e 1,..., e k be non-negative integers. Let A k (respectively B k ) be the set of all k-tuples (f 1,..., f k ) of integers such that 0 f i e i for all i and odd). Show that A k B k = 0 or 1. k i=1 f i is even (respectively 7

8 55. Find the point in the closed unit disc D = { (x, y) x 2 + y 2 1 } at which the function f (x, y) = x + y attains its maximum. 56. Let a 0 = 0 < a 1 < a 2 < < a n be real numbers. Suppose p(t) is a real valued polynomial of degree n such that a j+1 a j p(t)dt = 0 for all 0 j n 1. Show that, for 0 j n 1, the polynomial p (t) has exactly one root in the interval (a j, a j+1 ). 57. Let M be a point in the triangle ABC such that Area (ABM) = 2.Area (ACM). Show that the locus of all such points is a straight line. 58. In how many ways can one fill an n n matrix with ±1 so that the product of the entries in each row and each column equals 1? 59. Bishops on a chessboard move along diagonals (i.e. on lines parallel to the two main diagonals). Prove that the maximum number of non-attacking bishops on an n n chessboard is 2n 2. (Two bishops are said to be attacking if they are on a common diagonal.) 60. Prove that there is no non-constant polynomial P (x) with integer coefficients such that P (n) is a prime number for all positive integers n. 61. Find all the roots of the equation 1 x 1 + x(x 1) 2! n x(x 1)... (x n + 1) + ( 1) = 0. n! 62. Let f : R R be a function that is differentiable n + 1 times for some positive integer n. The ith derivative of f is denoted by f (i). Suppose f(1) = f(0) = f (1) (0) =... = f (n) (0) = 0. Prove that f (n+1) (x) = 0 for some x (0, 1). 63. A domino is a 2 by 1 rectangle. For what integers m and n can we cover an m by n rectangle with non-overlapping dominoes? 64. You are standing at the edge of a river which is one km wide. You have to go to your camp on the opposite bank. The distance to the camp from the point on the opposite bank directly across you is 1 km. You can swim at 2 km/h and walk at 3 km/h. What is the shortest time you will take to reach your camp? (Ignore the speed of the river and assume that the river banks are straight and parallel.) 8

9 65. In a triangle ABC, D is a point on BC such that AD is the internal bisector of A. Suppose B = 2 C and CD = AB. Prove that A = 72 a. (Hint: You may use the sine rule sin A = b sin B = c sin C, if you wish.) 66. Let S be the set of all integers k, 1 k n, such that gcd(k, n) = 1. What is the arithmetic mean of the integers in S? 9

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