3. Which of the numbers 1, 2,..., 1983 have the largest number of positive divisors?

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1 1983 IMO Long List 1. A total of 1983 cities are served by ten airlines. There is direct service (without stopovers) between any two cities and all airline schedules run both ways. Prove that at least one of the airlines can offer a round trip with an odd number of landings. 2. The altitude from a vertex of a given tetrahedron intersects the opposite face in its orthocentre. Prove that all four altitudes of the tetrahedron are concurrent. 3. Which of the numbers 1, 2,..., 1983 have the largest number of positive divisors? 4. Find all possible finite sequences {n 0, n 1,..., n k } of integers such that, for each i = 0, 1,..., k, i appears in the sequence n i times. 5. Let a 0 = 0 and a n+1 = k(a n + 1) + (k + 1)a n + 2 k(k + 1)a n (a n + 1), n = 0, 1, 2,..., where k is a positive integer. Prove that a n is a positive integer for n = 1, 2, 3, Show that there exists infinitely many sets of 1983 consecutive positive integers each of which is divisible by some number of the form a 1983, where a 1 is a positive integer. 7. Let r and s be integers, with s > 0. Show that there exists an interval I of length 1/s and a polynomial P (x) with integral coefficients such that, for all x I, P (x) r < 1 s s Let F : [0, 1] R be a continuous function satisfying { F (2x) = bf (x), 0 x 1/2, F (x) = b + (1 b)f (2x 1), 1/2 x 1, where b = (1 + c)/(2 + c) and c > 0. Prove that 0 < F (x) x < c for all x (0, 1). 9. Let P 1, P 2,..., P n be distinct n points in a plane. Prove that 3 max P ip j > (n 1) min 1 i<j n 2 P ip j. 1 i<j n 10. Prove that if the sides a, b, c of a triangle satisfy 2(bc 2 + ca 2 + ab 2 ) = b 2 c + c 2 a + a 2 b + 3abc, then the triangle is equilateral. Prove also that the equation can be satisfied by positive real numbers that are not the sides of a triangle. 11. Prove that there is a unique infinite sequence {u 0, u 1, u 2,...} of positive integers such that, for all n 0, ( ) n + r u 2 n = u n r. r r=0 12. For a given set X of 1983 members there exists a family of subsets {S 1, S 2,..., S k } such that (i) the union of any three of these subsets is the entire set X, and (ii) the union of any two of these subsets contains at most 1979 members. Determine the largest possible value of k. 1

2 13. There are 1983 points on a given circle, and each is given one of the affixes ±1. Prove that, if the number of points with the affix +1 is greater than 1789, than at least 1207 of the points have the property that the partial sums that can be formed by summing their own affix and those of their consecutive neighbour on the circle up to any other point, in either direction on the circle, are all strictly positive. 14. Show that there exist distinct natural numbers n 1, n 2,..., n k such that ( 1 π 1984 < ) < π n 1 n 2 n k 15. The set {1, 2,..., 49} is partitioned into three subsets. Show that at least one of the subsets contains three different numbers a, b, c, such that a + b = c. 16. Prove that in any parallelepiped the sum of the lengths of the edges does not exceed twice the sum of the lengths of the four principal diagonals. 17. Given non-negative real numbers x 1, x 2,..., x k and positive integers k, m, n such that km n, prove that ( k ) ( k ) n (x m i 1) m (x n i 1). 18. A polynomial P (x) of degree 990 satisfies P (k) = F k, k = 992, 993,..., 1982, where {F k } is the Fibonacci sequence, defined by Prove that P (1983) = F F 1 = F 2 = 1, F n+1 = F n + F n 1, n = 2, 3, 4, Let a and b be integers. Is it possible to find integers p and q such that the integers p + na and q + nb are relatively prime for any integer n? 20. AB is the diameter of a circle γ with centre O. A segment BD is bisected by the point C on γ, and AC and DO intersect at P. Prove that there is a point E on AB such that P lies on the circle with diameter AE. 21. The sum of all the face angles at all but one of the vertices of a given simple polyhedron is Find the sum of all the face angles of the polyhedron. 22. Determine all pairs (a, b) of positive real numbers with a 1 such that log a b < log a+1 (b + 1). 23. A tetrahedron is inscribed in a unit sphere. The tetrahedron is such that the centre of the sphere lies in its interior. Show that the sum of the edge lengths of the tetrahedron exceeds The proper divisors of the natural number n are arranged in increasing order, x 1 < x 2 < < x k. Find all numbers n such that x x = n. 25. A triangle T 1 is constructed with the medians of a right triangle T. Prove that if R 1 and R are the circumradii of T 1 and T respectively, then R 1 > 5R/ Let x 1, x 2,..., x n denote n real numbers lying in the interval [0,1]. Show that there is a number x [0, 1] such that 1 x x i = 1 n 2. 2

3 27. Let ABCD be a convex quadrilateral, and let A 1, B 1, C 1, D 1 be the circumcentres of triangles BCD, CDA, DAB, and ABC respectively. (i) Prove that either all of A 1, B 1, C 1, D 1 coincide in one point, or else they are all distinct. Assuming the latter case, show that A 1 and C 1 are on opposite sides of line B 1 D 1, and that B 1 and D 1 are on opposite sides of line A 1 C 1. (This establishes the convexity of quadrilateral A 1 B 1 C 1 D 1.) (ii) Let A 2, B 2, C 2, D 2 be the circumcentres of triangles B 1 C 1 D 1, C 1 D 1 A 1, D 1 A 1 B 1, A 1 B 1 C 1 respectively. Show that quadrilateral A 2 B 2 C 2 D 2 is similar to quadrilateral ABCD. 28. Determine all integers x such that is a perfect square. x 4 + x 3 + x 2 + x Determine all integer solutions (x, y) to the Diophantine equation x 3 y 3 = 2xy A box contains p white balls and q black balls, and beside the box lies a large pile of black balls. Two balls chosen at random (with equal likelihood) are taken out of the box. If they are of the same colour, a black ball from the pile is put into the box; otherwise, the white ball is put back into the box. The procedure is repeated until the last two balls are removed from the box and one last ball is put in. What is the probability that this last ball is white? 31. Prove that, for every natural number n, the binomial coefficient ( 2n n ) divides the least common multiple of the numbers 1, 2, 3,..., 2n. 32. A regular n-gonal truncated pyramid with base areas S 1 and S 2 and lateral surface area S is circumscribed about a sphere. Let A be the area of the polygon whose vertices are the points of tangency of the lateral faces of the truncated pyramid with the sphere. Prove that AS = 4S 1 S 2 cos 2 (π/n). 33. Given are a circle Γ and line l tangent to it at B. From a point A on Γ, a line AP l is constructed, with P l. If the point M is symmetric to P with respect to AB, determine the locus of M as A ranges on Γ. 34. Determine the permutation α = (a 1, a 2,..., a n ) of (1, 2,..., n) which maximizes and also the permutation α which minimizes Q. Q = a 1 a 2 + a 2 a a n a 1, 35. You are given an algebraic system with an addition and a multiplication for which all the laws of ordinary arithmetic are valid except commutativity of multiplication. Show that 36. Let (a + ab 1 a) 1 + (a + b) 1 = a 1, where x 1 is that element for which x 1 x = xx 1 = e, the multiplicative identity. S = { m + n : m, n positive integers}. m2 + n2 Show that for each (x, y) S S, with x < y, there exists a z S such that x < z < y. 37. Four circles C, C 1, C 2, C 3 and a line l are given, all in the same plane. The circles C 1, C 2, C 3 are all distinct, each touches the other two and touches also C and l. If the radius of C is 1, determine the distance between its centre and l. 3

4 38. Let the numbers u 1, u 2,..., u n be all positive and let Prove that v k = k u 1 u 2 u k, k = 1, 2,..., n. v k e u k. k=1 39. A country has n cities, any two of which are connected by a railroad. A railroad worker has to travel on each line exactly once. If at any stop there is a city he must reach but cannot (having already travelled on the line to or from that city), then he can fly. What is the smallest number of plane tickets he must buy? 40. Let Γ be a unit circle with centre O, and let P 1, P 2,..., P n be points of Γ such that k=1 OP 1 + OP OP n = 0. Prove that P 1 Q + P 2 Q + + P n Q n for all points Q. 41. A convex figure F lies inside a circle with centre O. The angle subtended by F from every point of the circle is 90. Prove that O is a centre of symmetry of F. 42. If (1 + x + x 2 + x 3 + x 4 ) 496 = a 0 + a 1 x + + a 1984 x 1984, (i) determine the greatest common divisor of the coefficients a 3, a 8, a 13,..., a 1983, (ii) show that > a 992 > Solve the equation tan 2 2x + 2 tan 2x tan 3x 1 = Let n be a positive integer having at least two distinct prime factors. Show that there is a permutation (a 1, a 2,..., a n ) of (1, 2,..., n) such that ( ) 2πak k cos = 0. n 45. Three roots of the equation k=1 x 4 px 3 + qx 2 rx + s = 0 are tan A, tan B, tan C, where A, B, C are the angles of a triangle. Determine the fourth root as a function of (only) p, q, r, and s. 46. Let (p ij ) be a given m n matrix with real entries, and let A i = m p ij and B j = p ij. (1) j=1 We way that a real number is rounded off if it is an integer or, if not an integer, when it is replaced by one of its two nearest neighbouring integers. Show that the p ij, A i, and B j can be rounded off so that (1) still remains valid. 47. In the Martian language any finite ordered set of Latin letters is a word. The Martian Word editorial office issues a many-volume dictionary of the Martian language, in which the entries are numbered consecutively in alphabetical order. The first volume contains all the one-letter words, the second volume all the two-letter words, etc., and the numbering of the words in each successive volumes continues the numbering in the preceding one. Determine the word whose number is the sum of the numbers of the words Prague, Olympiad, Mathematics. 48. Let O be the centre of the axis of a right circular cylinder. Let A and B be diametrically opposite points in the boundary of its upper base, and let C be a boundary point of its lower base which does not lie in the plane OAB. Show that \BOC + \COA + \AOB = 2π. 4

5 49. Let x 1, x 2,..., x n be numbers such that 1 x 1 x 2 x n > 0. Prove that if 0 a 1, then (1 + x 1 + x x n ) a 1 + x a a 1 x a n a 1 x a n. 50. Given are the function F (x) = ax 2 + bx + c and G(x) = cx 2 + bx + a, where F (0) 1, F (1) 1, and F ( 1) 1. Prove that, for x 1, (i) F (x) 5/4, (ii) G(x) Let P be a regular convex 2n-gon. Show that there is a 2n-gon Q with the same vertices as P (but in a different order) such that Q has exactly one pair of parallel sides. 52. Show that if n > 2 is an integer and x denotes the greatest integer x, then n(n + 1) n + 1 =. 4n If P, Q, and R are non-proportional polynomials with complex coefficients, prove that the identity P n + Q n + R n 0, where n is a natural number, implies that n < The incircle of triangle A 1 A 2 A 3 touches the sides A 2 A 3, A 3 A 1, A 1 A 2 at the points T 1, T 2, T 3 respectively. If M 1, M 2, M 3 are the mid-points of the sides A 2 A 3, A 3 A 1, A 1 A 2 respectively, prove that the perpendiculars through the points M 1, M 2, M 3 to the lines T 2 T 3, T 3 T 1, T 1 T 2 respectively, are concurrent. 55. Prove that the volume of a tetrahedron inscribed in a closed right circular cylinder (capped by two disks) of volume 1 does not exceed 2/3π. 56. Given that a 1, a 2,..., a 2n are distinct integers such that the equation (x a 1 )(x a 2 ) (x a 2n ) + ( 1) n 1 (n!) 2 = 0 has an integer solution r, show that r = (a 1 + a a 2n )/(2n). 57. Let m and n be non-zero integers. Prove that 4mn m n can be a square infinitely often, but that this is never a square if either m or n is positive. 58. Determine all sequences {a 1, a 2,...} such that a 1 = 1 and for all positive integers m and n. a n a m 2mn m 2 + n A strictly increasing function f defined on [0,1] satisfies f(0) = 0, f(1) = 1, and 1 f(x + y) f(x) 2 f(x) f(x y) 2 for all x, y such that 0 x ± y 1. Show that f(1/3) 76/ ABC is an isosceles right triangle with right angle at A. Determine the minimum value of BP + CP 3AP, where P is any point in the plane of the triangle. 61. You start with a white balls and b black balls in a container and proceed as follows: Step 1. You draw one ball at random from the container (each ball being equally likely). If the ball is white, then stop. 5

6 Step 2. If the drawn ball is black, then add two black balls to the balls remaining in the container and repeat Step 1. Let s denote the number of draws until stop. For the cases a = b = 1 and a = b = 2 only, determine and the expectation E(s) = n 1 na n. a n = Pr(s = n), b n = Pr(s n), lim b n, n 62. A box is to be filled with twenty-four bricks. In how many different ways can this be done if the bricks are indistinguishable? 63. Prove that the product of five consecutive integers cannot be a perfect square. 64. The function f(n) is defined for non-negative integers n by f(0) = 0, f(1) = 1, and ( ) ( ) m(m 1) m(m + 1) f(n) = f n f n 2 2 for m(m 1) 2 < n < Determine the smallest integer n for which f(n) = 5. m(m + 1), m Determine all continuous functions f such that, for all real x and y, f(x + y)f(x y) = [f(x)f(y)] Determine positive integers p, q, r such that the diagonal of a block consisting of p q r unit cubes passes through exactly 1984 of the unit cubes, while its length is a minimum. (The diagonal is said to pass through a unit cube it is has more than one point in common with the unit cube.) 67. Let a, b, c be positive numbers such that Prove that the system a + b + c = 3 2. y a + z a = 1, z b + x b = 1, x c + y c = 1, has exactly one real solution (x, y, z). 68. Let X be an arbitrary non-empty point set in a plane and let A 1, A 2,..., A m and B 1, B 2,..., B n be its images under translations (in the plane). Prove that if the sets A 1, A 2,..., A m are pairwise disjoint and A 1 A 2 A m B 1 B 2 B n, then m n. 69. Let {a n } and {b n }, n = 1, 2, 3,..., be two sequences of natural numbers such that, for all n 1, a n+1 = na n + 1 and b n+1 = nb n 1. Prove that the two sequences can have only a finite number of terms in common. 70. Let S k = x k 1 + x k x k n, where the x i are real numbers. Prove that if then x i = 0 or x i = 1 for every i = 1, 2,..., n. S 1 = S 2 = = S n+1, 6

7 71. Construct a non-isosceles triangle ABC such that a(tan B tan C) = b(tan A tan C), where a and b are the side lengths opposite angles A and B, respectively. 72. Let P be a convex n-gon with equal interior angles, and let l 1, l 2,..., l n be the lengths of its consecutive sides. Prove that a necessary and sufficient condition for P to be regular is that l 1 l 2 + l 2 l l n l 1 = n. 73. Let {a 1, a 2, a 3,...} be an infinite real sequence such that, for all positive integers n and m, Prove that a n /n has a limit as n. a n a n+m a n + a m. 74. A fair coin is tossed repeatedly until there is a run of an odd number of heads followed by a tail. Determine the expected number of tosses. 75. Inside triangle ABC, a circle of radius 1 is externally tangent to the incircle and tangent to sides AB and AC. A circle of radius 4 is externally tangent to the incircle and tangent to sides BA and BC. A circle of radius 9 is externally tangent to the incircle and tangent to sides CA and CB. Determine the inradius of the triangle. 7

8 1984 IMO Long List 1. An integer sequence is defined by a n = 2a n 1 + a n 2 (n > 1), a 0 = 0, a 1 = 1. Prove that 2 k divides a n if and only if 2 k divides n. 2. Let a n = (n + 1) 2 + n 2, n = 1, 2,..., where x denotes the greatest integer part of x. Prove that (i) there are infinitely many positive integers m such that a m+1 a m > 1, (ii) there are infinitely many positive integers m such that a m+1 a m = Let n be a positive integer. Find the number of odd coefficients of the polynomial u n (x) = (x 2 + x + 1) n. 4. The triangle ABC is inscribed in a circle. The interior bisectors of the angles A, B, and C meet the circle again at A, B, and C respectively. Prove that the area of triangle A B C is greater than or equal to the area of triangle ABC. 5. Let k be a positive integer and M k the set of all the integers that are between 2k 2 + k and 2k 3 + 3k, both included. Is it possible to partition M k into 2 subsets A and B such that x 2? x A x 2 = x B 6. An n n chessboard (n 2) is numbered by the numbers 1, 2,..., n 2 (and every number occurs). Prove that there exist two neighbouring (with common edge) squares such that their numbers differ by at least n. 7. Let n be an even positive integer. Let A 1, A 2,..., A n+1 be sets having n elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which n can one assign to every element of the union one of the numbers 0 and 1 in such a manner that each of the sets has exactly n/2 zeros? 8. In a given tetrahedron ABCD let K and L be the centres of the edges AB and CD respectively. Prove that every plane that contains the line KL divides the tetrahedron into two parts of equal volume. 9. If a 0 is a positive real number, consider the sequence {a n } defined by: a n+1 = a2 n 1 n + 1 for n 0. Show that there exists a real number a > 0 such that: (i) for all real a 0 a, the sequence {a n } +, (ii) for all real a 0 < a, the sequence {a n } Let a be the greatest positive root of the equation x 3 3x = 0. Show that a 1788 and a 1988 are both divisible by 17. ( x denotes the integer part of x.) 11. Let u 1, u 2,..., u m be m vectors in the plane, each of length 1, with zero sum. Show that one can re-arrange u 1, u 2,..., u m as a sequence v 1, v 2,..., v m such that each partial sum v 1, v 1 + v 2, v 1 + v 2 + v 3,..., v 1 + v v m has length less than or equal to Show that there do not exist more than 27 half-lines (or rays) emanating from the origin in 3- dimensional space such that the angle between each pair of rays is π/4. 8

9 13. Let T be a triangle with inscribed circle C. A square with sides of length a is circumscribed about the same circle C. Show that the total length of the parts of the edges of the square interior to the triangle T is at least 2a. 14. Let a and b be two positive integers such that ab + 1 divides a 2 + b 2. Show that is a perfect square. a 2 + b 2 ab Let 1 k < n. Consider all finite sequences of positive integers with sum n. Find T (n, k), the total number of terms of size k in all these sequences. 16. If n runs through all positive integers, f(n) = n + n/3 + 1/2 runs through all positive integers skipping the terms of the sequence a n = 3n 2 2n. 17. If n runs through all positive integers, f(n) = n + 3n + 1/2 runs through all positive integers skipping the terms of the sequence a n = (n 2 + 2n)/ Let N = {1, 2,..., n}, n 2. A collection F = {A 1,..., A t } of subsets A i N, i = 1,..., t, is said to be separating, if for every pair {x, y} N, there is a set A i in F so that A i x, y contains just one element. F is said to be covering, if every element of N is contained in at least one set A i F. What is the smallest value f(n) of t, so that there is a set F = {A 1,..., A t } which is simultaneously separating and covering? 19. Let Z m,n be the set of all ordered pairs (i, j) with i {1,..., m} and j {1,..., n}. Also, let a m,n be the number of all those subsets of Z m,n that contain no two ordered pairs (i 1, j 1 ), (i 2, j 2 ) with i 1 i 2 + j 1 j 2 = 1. Then show, for all positive integers m and k, that a 2 m,2k a m,2k 1 a m,2k The lock on a safe consists of 3 wheels, each of which may be set in 8 different positions. Due to a defect in the safe mechanism the door will open if any two of the three wheels are in the correct position. What is the smallest number of combinations which must be tried if one is to guarantee being able to open the safe (assuming the right combination is now known)? 21. Let AB and CD be two perpendicular chords of a circle with centre O and radius r and let X, Y, Z, W denote in cyclical order the four parts into which the disc is thus divided. Find the maximum and minimum of the quantity (E(X) + E(Z))/(E(Y ) + E(W )), where E(U) denotes the area of U. 22. In a triangle ABC, choose any points K BC, L AC, M AB, N LM, R MK, and F KL. Show that if E 1, E 2, E 3, E 4, E 5, E 6, and E denote the areas of the triangles AMR, CKR, BKF, ALF, BNM, CLN, and ABC respectively, then E 8(E 1 E 2 E 3 E 4 E 5 E 6 ) 1/ In a right-angled triangle ABC let AD be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles ABD, ACD intersect the sides AB, AC at the points K, L respectively. Show that if E and E 1 denote the areas of the triangles ABC and AKL respectively, then E/E Find positive integers x 1, x 2,..., x 29 at least one of which is greater than 1988 so that x x x 2 29 = 29x 1 x 2 x Find the total number of different integers the function takes for 0 x 100. f(x) = x + 2x + 5x/3 + 3x + 4x 9

10 26. The circle x 2 + y 2 = r 2 meets the coordinates axes at A = (r, 0), B = ( r, 0), C = (0, r), and D = (0, r). Let P = (u, v) and Q = ( u, v) be two points on the circumference of the circle. Let N be the point of intersection of P Q and the y-axis, and M be the foot of the perpendicular drawn from P to the x-axis. Show that if r 2 is odd, u = p m > q n = v, where p and q are prime numbers, and m and n are natural numbers, then AM = 1, BM = 9, DN = 8, P Q = Assuming that the roots of x 3 + px 2 + qx + r = 0 are all real and positive, find a relation between p, q, and r which gives a necessary and sufficient condition for the roots to be exactly the cosines of the three angles of a triangle. 28. Find a necessary and sufficient condition on the natural number n for the equation x n + (2 + x) n + (2 x) n = 0 to have an integral root. 29. Express the number 1988 as the sum of some positive integers in such a way that the product of these positive integers is maximal. 30. In the triangle ABC, let D, E, and F be the mid-points of the three sides, X, Y, and Z the feet of the three altitudes, H the orthocentre, and P, Q, and R the mid-points of the line segments joining H to the three vertices. Show that the nine points D, E, F, P, Q, R, X, Y, Z lie on a circle. 31. For what values of n does there exist an n n array of entries 1, 0, or 1 such that the 2n sums obtained by summing the elements of the rows and columns are all different? 32. n points are given on the surface of a sphere. Show that the surface can be divided into n connected congruent regions such that each of them contains exactly one of the given points. 33. In a multiple choice test there were 4 questions and 3 possible answers for each question. A group of students was tested and it turned out that for any 3 of them there was a question which the three students answered differently. What is the maximal possible number of students tested? 34. Let ABC be an acute-angled triangle. Three lines L A, L B, and L C are constructed through the vertices A, B, and C respectively according to the following prescription: let H be the foot of the altitude drawn from the vertex A to the side BC; let S A be the circle with diameter AH; let S A meet the sides AB and AC at M and N respectively, where M and N are distinct from A; then L A is the line through A perpendicular to MN. The lines L B and L C are constructed similarly. Prove that L A, L B, and L C are concurrent. 35. A sequence of numbers a n, n = 1, 2,..., is defined as follows: a 1 = 1/2 and for each n 2 Prove that n k=1 a k < 1 for all n 1. a n = 2n 3 2n a n (i) Let ABC be a triangle with AB = 12 and AC = 16. Suppose M is the mid-point of side BC and points E and F are chosen on sides AC and AB respectively, and suppose that the lines EF and AM intersect at G. Find the ratio EG/GF if AE = 2AF. (ii) Let E be a point external to a circle and suppose that two chords EAB and EDC meet at an angle of 40. Find the size of angle ACD if AB = BC = CD. 37. (i) Four balls of radius 1 are mutually tangent, three resting on the floor and the fourth resting on the others. A tetrahedron, each of whose edges has length s, is circumscribed around the balls. Find the value of s. (ii) Suppose that ABCD and EF HG are opposite faces of a rectangular solid, with \DHC = 45 and \F HB = 60. Find the cosine of \BHD. 38. (i) The polynomial x 2k (x + 1) 2k is not divisible by x 2 + x + 1. Find the value of k. (ii) Find the value of p 3 + q 3 + r 3, if p, q, and r are distinct roots of x 3 x 2 + x 2 = 0. 10

11 (iii) Each of the numbers 1059, 1417, and 2312 leave a remainder of r when divided by d, where d is an integer greater than 1. Find the value of d r. (iv) What is the smallest positive odd integer n such that the product of 2 1/7, 2 3/7,..., 2 (2n+1)/7 is greater than 1000? 39. (i) Let g(x) = x 5 + x 4 + x 3 + x 2 + x + 1. What is the remainder when the polynomial g(x 12 ) is divided by the polynomial g(x)? (ii) If k is a positive number and f is a function such that, for every positive x, f(x 2 + 1) x = k. Find the value of ( ) 9 + y 2 12/y f. for every positive number y. y 2 (iii) The function f satisfies the functional equation f(x) + f(y) = f(x + y) xy 1 for every pair x, y of real numbers, and f(1) = 1. Find the number of integers n, for which f(n) = n. 40. (i) Consider a circle K with diameter AB; a circle L inside K tangent to AB and to K and with a circle M inside K, tangent to circle K, circle L, and AB. Calculate the ratio of the area of circle K to the area of circle M. (ii) In triangle ABC, AB = AC and \CAB = 80. Points D, E, and F lie on sides BC, AC, and AB respectively such that CE = CD and BF = BD. Find the size of \EDF. 41. (i) Calculate ( ) (11 6 2) (ii) For each positive number x, let Calculate the minimum value of k. 42. Show that the solution set of the inequality k = (x + 1/x)6 (x 6 + 1/x 6 ) 2 (x + 1/x) 3 + (x 3 + 1/x 3. ) 70 k=1 k x k 5 4 is a union of disjoint intervals, the sum of whose lengths is Find all plane triangles whose sides have integer length and whose incircles have unit radius. 44. Let 1 < x < 1. Show that 6 k=0 1 x 2 1 2x cos(2πk/7) + x 2 = 7(1 + x7 ) 1 x 7. Deduce that csc 2 (π/7) + csc 2 (2π/7) + csc 2 (3π/7) = Let g(n) be defined as follows: g(1) = 0, g(2) = 1, and g(n + 2) = g(n) + g(n + 1) + 1 (n 1). Prove that if n > 5 is a prime, then n divides g(n)(g(n) + 1). 11

12 46. A 1, A 2,..., A 29 are 29 different sequences of positive integers. For 1 i < j 29 and any natural number x, we define N i (x) to be the number of elements of the sequence A i which are less than or equal to x, and N ij (x) to be the number of elements of the intersection A i A j which are less than or equal to x. It is given that for all 1 i 29 and every natural number x, N i (x) x/e, where e = Prove that there exists at least one pair i, j (1 i < j 29) such that N ij (1988) > In the convex pentagon ABCDE, the sides BC, CD, DE are equal. Moreover each diagonal of the pentagon is parallel to a side (AC is parallel to DE, BD is parallel to AE, etc.). Prove that ABCDE is a regular pentagon. 48. Consider 2 concentric circles of radii R and r (R > r) with centre O. Fix P on the small circle and consider the variable chord P A of the small circle. Points B and C lie on the large circle; B, P, C are collinear and BC is perpendicular to AP. (i) For which value(s) of \OP A is the sum BC 2 + CA 2 + AB 2 extremal? (ii) What are the possible positions of the mid-points U of BA and V of AC as \OP A varies? 49. Let f(n) be a function defined on the set of all positive integers and having its values in the same set. Suppose that f(f(m) + f(n)) = m + n for all positive integers n, m. Find all possible values for f(1988). 50. Prove that the numbers A, B, and C are equal, where: A is the number of ways that one can cover a 2 n rectangle with 2 1 rectangles, B is the number of sequences of ones and twos that add up to n, and { ( m ) ( C = 0 + m+1 ) ( 2 + m+2 ) ( m ) ( 2m if n = 2m, m+1 ) ( + m+2 ) ( + m+3 ) ( + + 2m+1 ) if n = 2m The positive integer n has the property that, in any set of n integers, chosen from the integers 1, 2,..., 1988, twenty-nine of them form an arithmetic progression. Prove that n > ABCD is a quadrilateral. A BCD is the reflection of ABCD in BC, A B CD is the reflection of A BCD in CD and A B C D is the reflection of A B CD in D A. Show that if the lines AA and BB are parallel, then ABCD is a cyclic quadrilateral. 53. Given n points A 1, A 2,..., A n, no three are collinear, show that the n-gon A 1 A 2 A n is inscribed in a circle if and only if 2m+1 A 1 A 2 A 3 A n A n 1 A n + A 2 A 3 A 4 A n A n 1 A n A 1 A n + + A n 1 A n 2 A 1 A n A n 3 A n = A 1 A n 1 A 2 A n A n 2 A n, where XY denotes the length of the segment XY. 54. Find the least natural number n such that, if the set {1, 2,..., n} is arbitrarily divided into two non-intersecting subsets, then one of the subsets contains 3 distinct numbers such that the product of two of them equals the third. 55. Suppose α i > 0, β i > 0 for 1 i n (n > 1) and that α i = β i = π. Prove that cos β i sin α i cot α i. 12

13 56. Given a set of 1988 points in the plane. No four points of the set are collinear. The points of a subset with 1788 points are coloured blue, the remaining 200 are coloured red. Prove that there exists a line in the plane such that each of the two parts into which the line divides the plane contains 894 blue points and 100 red points. 57. S is the set of all sequences {a i : 1 i 7, a i = 0 or 1}. The distance between two elements {a i } and {b i } of S is defined as: 7 a i b i. T is a subset of S in which any two elements have a distance apart greater than or equal to 3. Prove that T contains at most 16 elements. Give an example of such a subset with 16 elements. 58. For a convex polygon P in the plane let P denote the convex polygon with vertices at the mid-points of the sides of P. Given an integer n 3, determine sharp bounds for the ratio (area P )/(area P ), over all convex n-gons P. 59. In 3-dimensional space there is given a point O and a finite set A of segments with the sum of the lengths equal to Prove that there exists a plane disjoint from A such that the distance from it to O does not exceed Given integers a 1,..., a 10, prove that there exists a non-zero sequence (x 1,..., x 10 ) such that all x i belong to { 1, 0, 1} and the number 10 x ia i is divisible by Forty-nine students solve a set of 3 problems. The score for each problem is a whole number of points from 0 to 7. Prove that there exist two students A and B such that, for each problem, A will score at least as many points as B. 62. Let x = p, y = q, z = r, w = s be the unique solution of the system of linear equations x + a i y + a 2 i z + a 3 i w = a 4 i, i = 1, 2, 3, 4. Express the solution of the following system in terms of p, q, r, and s: Assume the uniqueness of the solution. x + a 2 i y + a 4 i z + a 6 i w = a 8 i, i = 1, 2, 3, Let p be the product of two consecutive integers greater than 2. Show that there are no integers x 1, x 2,..., x p satisfying the equation p ( p ) 2 x 2 i 4 x i = 1. 4p + 1 or Show there are only two values of p for which there are integers x 1, x 2,..., x p satisfying p ( p ) 2 x 2 i 4 x i = 1. 4p Find all positive integers x such that the product of all digits of x is given by x 2 10x The Fibonacci sequence is defined by a n+1 = a n + a n 1 (n 1), a 0 = 0, a 1 = a 2 = 1. Find the greatest common divisor of the 1960 th and 1988 th terms of the Fibonacci sequence. 66. Let C be a cube with edges of length 2. Construct a solid with fourteen faces by cutting off all eight corners of C, keeping the new faces perpendicular to the diagonals of the cube, and keeping the newly formed faces identical. At the conclusion of this process the fourteen faces so formed have the same area. Find the area of each face of the new solid. 13

14 67. For each positive integer k and n, let S k (n) be the base k digit sum of n. Prove that there are at most two primes p less than 20,000 for which S 31 (p) are composite numbers with at least two distinct prime divisors. 68. In a group of n people, each one knows exactly three others. They are seated around a table. We say that the seating is perfect if everyone knows the two sitting by their sides. Show that, if there is a perfect seating S for the group, then there is always another perfect seating which cannot be obtained from S by rotation or reflection. 69. Let Q be the centre of the inscribed circle of a triangle ABC. Prove that for any point P, a(p A) 2 + b(p B) 2 + c(p C) 2 = a(qa) 2 + b(qb) 2 + c(qc) 2 + (a + b + c)(qp ) 2, where a = BC, b = CA, and c = AB. 70. ABC is a triangle, with inradius r and circumradius R. Show that: sin(a/2) sin(b/2) + sin(b/2) sin(c/2) + sin(c/2) sin(a/2) 5/8 + r/(4r). 71. The quadrilateral A 1 A 2 A 3 A 4 is cyclic, and its sides are a 1 = A 1 A 2, a 2 = A 2 A 3, a 3 = A 3 A 4, and a 4 = A 4 A 1. The respective circles with centres I i and radii ρ i are tangent externally to each side a i and to the sides a i+1 and a i 1 extended (a 0 = a 4 ). Show that 4 a i ρ i = 4(csc A 1 + csc A 2 ) Consider h + 1 chessboards. Number the squares of each board from 1 to 64 in such a way that when the perimeters of any two boards of the collection are brought into coincidence in any possible manner, no two squares in the same position have the same number. What is the maximum value of h? 73. A two-person game is played with nine boxes arranged in a 3 by 3 square and with white and black stones. At each move a player puts three stones, not necessarily of the same colour, in three boxes in either a horizontal or vertical row. No box can contain stones of different colours: if, for instance, a player puts a white stone in a box containing black stones the white stone and one of the black stones is removed from the box. The game is over when the centrebox and the cornerboxes contain one black stone and the other boxes are empty. At one stage of a game x boxes contained one black stone each and the other boxes were empty. Determine all possible values for x. 74. Let {a k } k=1 be a sequence of non-negative real numbers such that: a k 2a k+1 + a k+2 0 and k a j 1 for all k = 1, 2,.... j=1 Prove that 0 a k a k+1 2 k 2 for all k = 1, 2, Let S be an infinite set of integers containing zero, and such that the distances between successive members never exceed a given fixed number. Consider the following procedure: Given a set X of integers we construct a new set consisting of all numbers x ± s, where x belongs to X and s belongs to S. Starting from S 0 = {0} we successively construct sets S 1, S 2, S 3,..., using this procedure. Show that after a finite number of steps we do not obtain any new sets, i.e. S k = S k0 for k k A positive integer is called a double number if its decimal representation consists of a block of digits, not commencing with 0, followed immediately by an identical block. So, for instance, is a double number, but is not. Show that there are infinitely many double numbers which are perfect squares. 14

15 77. A function f defined on the positive integers (and taking positive integer values) is given by: f(1) = 1, f(3) = 3, f(2n) = f(n), f(4n + 1) = 2f(2n + 1) f(n), f(4n + 3) = 3f(2n + 1) 2f(n), for all positive integers n. Determine with proof the number of positive integers 1988 for which f(n) = n. 78. It is proposed to partition the set of positive integers into two disjoint subsets A and B subject to the conditions (i) 1 is in A, (ii) no two distinct members of A have a sum of the form 2 k + 2 (k = 0, 1, 2,...,), and (iii) no two distinct members of B have a sum of that form. Show that this partitioning can be carried out in a unique manner and determine the subsets to which 1987, 1988, and 1989 belong. 79. The triangle ABC is acute-angled. L is any line in the plane of the triangle and u, v, w are the lengths of the perpendiculars from A, B, C respectively to L. Prove that u 2 tan A + v 2 tan B + w 2 tan c 2, where is the area of the triangle, and determine the lines L for which equality holds. 80. The sequence {a n } of integers is defined by a 1 = 2, a 2 = 7, and Prove that a n is odd for all n > < a n+1 a2 n a n for n There are n 3 job openings at a factory, ranked 1 to n in order of increasing pay. There are n job applicants, ranked 1 to n in order of increasing ability. Applicant i is qualified for job j if and only if i j. The applicants arrive one at a time in random order. Each in turn is hired to the highest-ranking job for which he or she is qualified and which is lower in rank than any job already filled. (Under these rules, job 1 is always filled and hiring terminates thereafter.) Show that applicants n and n 1 have the same probability of being hired. 82. The triangle ABC has a right angle at C. The point P is located on segment AC such that triangles P BA and P BC have congruent inscribed circles. Express the length x = P C in terms of a = BC, b = CA, and c = AB. 83. A number of signal lights are equally spaced along a one-way railroad track, labelled in order 1, 2,..., N (N 2). As a safety rule, a train is not allowed to pass a signal if any other train is in motion on the length of track between it and the following signal. However, there is no limit to the number of trains that can be parked motionless at a signal, one behind the other. (Assume the trains have zero length.) A series of K freight trains must be driven from Signal 1 to Signal N. Each train travels at a distinct but constant speed at all times when it is not blocked by the safety rule. Show that, regardless of the order in which the trains are arranged, the same time will elapse between the first train s departure from Signal 1 and the last train s arrival at Signal N. 15

16 84. A point M is chosen on the side AC of the triangle ABC in such a way that the radii of the circles inscribed in the triangle ABM and BMC are equal. Prove that where is the area of triangle ABC. BM 2 = cot(b/2), 85. Around a circular table an even number of persons have a discussion. After a break they sit again around the circular table in a different order. Prove that there are at least two people such that the number of participants sitting between them before and after the break is the same. 86. Let a, b, c be integers different from zero. It is known that the equation ax 2 + by 2 + cz 2 = 0 has a solution (x, y, z) in integer numbers different from the solution x = y = z = 0. Prove that the equation ax 2 + by 2 + cz 2 = 1 has a solution in rational numbers. 87. In a row are written in increasing order all the irreducible positive rational numbers, such that the product of the numerator and denominator is less than Prove that any two adjacent fractions a/b and c/d, a/b < c/d, satisfy the equation bc ad = Seven circles are given as shown in the figure. That is, there are 6 circles inside a fixed circle, each tangent to the fixed circle and tangent to the two adjacent smaller circles. Prove that if the points of contact between the 6 circles and the larger circle are, in order, A 1, A 2, A 3, A 4, A 5, and A 6, then A 1 A 2 A 3 A 4 A 5 A 6 = A 2 A 3 A 4 A 5 A 6 A We match sets M of points in the coordinate plane to sets M according to the rule that (x, y ) belongs to M if and only if xx + yy 1 whenever (x, y) is in M. Find all triangles Y such that Y is the reflection of Y in the origin. 90. Does there exist a number α (0 < α < 1) such that there is an infinite sequence {a n } of positive numbers satisfying 1 + a n+1 a n + α n a n, n = 1, 2,...? 91. A regular 14-gon with side a is inscribed in a circle of radius 1. Prove 2 a 2a > 3 cos(π/7). 92. Let p 2 be a natural number. Prove that there exists an integer n 0 such that n 0 1 i p 1 + i > p. 93. Given a natural number n, find all polynomials P (x) of degree less than n satisfying the following condition: ( ) n P (i)( 1) i = 0. i i=0 94. Let n + 1 (n 1) positive integers be formed by taking the product of n given prime numbers (a prime can appear several times or also not appear at all in a product formed in this way). Prove that among these n + 1 one can find some numbers whose product is a perfect square. 16