The Twenty-Eighth IMO Havana, Cuba, July 5–16, 1987

3.28 The Twenty-Eighth IMO Havana, Cuba, July 5–16, 1987

3.28.1 Contest Problems

First Day (July 10)

1. Let S be a set of n elements. We denote the number of all permutations of S that have exactly k fixed points by p n (k). Prove that

∑ kp n (k) = n!.

k =0

2. The prolongation of the bisector AL (L ∈ BC) in the acute-angled triangle ABC intersects the circumscribed circle at point N. From point L to the sides AB and

AC are drawn the perpendiculars LK and LM respectively. Prove that the area of the triangle ABC is equal to the area of the quadrilateral AKNM.

3. Suppose x 1 ,x 2 ,...,x n are real numbers with x 2 2 1 2 +x 2 + ···+ x n = 1. Prove that for

any integer k > 1 there are integers e i not all 0 and with |e i | < k such that

Second Day (July 11)

4. Does there exist a function f : N → N, such that f ( f (n)) = n + 1987 for every natural number n?

5. Prove that for every natural number n ≥ 3 it is possible to put n points in the Euclidean plane such that the distance between each pair of points is irrational and each three points determine a nondegenerate triangle with rational area.

6. Let f p 2 (x) = x + x + p, for p ∈ N. Prove that if the numbers f (0), f (1), ...,

f ([ p /3 ]) are primes, then all the numbers f (0), f (1), . . ., f (p − 2) are primes.

3.28.2 Longlisted Problems

be n integers. Let n = p + q, where p and q are positive integers. For i = 1, 2, . . . , n, put

1. (AUS 1) Let x 1 ,x 2 ,...,x n

S i =x i +x i +1 + ··· + x i +p−1 and T i =x i +p +x i +p+1 + ··· + x i +n−1 (it is assumed that x i +n =x i for all i). Next, let m (a, b) be the number of indices i

for which S i leaves the remainder a and T i leaves the remainder b on division by

3, where a , b ∈ {0,1,2}. Show that m(1,2) and m(2,1) leave the same remainder when divided by 3.

3.28 IMO 1987 193

2. (AUS 2) Suppose we have a pack of 2n cards, in the order 1 , 2, . . . , 2n. A per- fect shuffle of these cards changes the order to n + 1, 1, n + 2, 2, . . .,n − 1,2n,n; i.e., the cards originally in the first n positions have been moved to the places

2 , 4, . . . , 2n, while the remaining n cards, in their original order, fill the odd posi- tions 1 , 3, . . . , 2n − 1. Suppose we start with the cards in the above order 1 , 2, . . . , 2n and then succes- sively apply perfect shuffles. What conditions on the number n are necessary for the cards eventually to return to their original order? Justify your answer.

Remark. This problem is trivial. Alternatively, it may be required to find the least number of shuffles after which the cards will return to the original order.

3. (AUS 3) A town has a road network that consists entirely of one-way streets that are used for bus routes. Along these routes, bus stops have been set up. If the one-way signs permit travel from bus stop X to bus stop Y 6= X, then we shall say Y can be reached from X . We shall use the phrase Y comes after X when we wish to express that every bus stop from which the bus stop X can be reached is a bus stop from which the bus stop Y can be reached, and every bus stop that can be reached from Y can also

be reached from X . A visitor to this town discovers that if X and Y are any two different bus stops, then the two sentences “Y can be reached from X ” and “Y comes after X ” have exactly the same meaning in this town. Let A and B be two bus stops. Show that of the following two statements, exactly one is true: (i) B can be reached from A; (ii) A can be reached from B.

4. (AUS 4) Let a 1 ,a 2 ,a 3 ,b 1 ,b 2 ,b 3 be positive real numbers. Prove that

(a 1 b 2 +a 2 b 1 +a 1 b 3 +a 3 b 1 +a 2 b 3 +a 3 b 2 ) 2

≥ 4(a 1 a 2 +a 2 a 3 +a 3 a 1 )(b 1 b 2 +b 2 b 3 +b 3 b 1 ) and show that the two sides of the inequality are equal if and only if a 1 /b 1 =

a 2 /b 2 =a 3 /b 3 .

5. (AUS 5) Let there be given three circles K 1 ,K 2 ,K 3 with centers O 1 ,O 2 ,O 3 respectively, which meet at a common point P. Also, let K 1 ∩K 2 = {P,A}, K 2 ∩K 3 = {P,B}, K 3 ∩K 1 = {P,C}. Given an arbitrary point X on K 1 , join X to A to meet K 2 again in Y , and join X to C to meet K 3 again in Z. (a) Show that the points Z , B,Y are collinear. (b) Show that the area of triangle XY Z is less than or equal to 4 times the area

of triangle O 1 O 2 O 3 .

6. (AUS 6) (SL87-1).

7. (BEL 1) Let f : (0, +∞) → R be a function having the property that f (x) =

f (1/x) for all x > 0. Prove that there exists a function u : [1, +∞) → R satisfying u x +1/x 2 = f (x) for all x > 0.

8. (BEL 2) Determine the least possible value of the natural number n such that n! ends in exactly 1987 zeros.

194 3 Problems

9. (BEL 3) In the set of 20 elements {1, 2, 3, 4, 5, 6, 7, 8, 9, 0, A, B, C, D, J, K , L, U , X , Y , Z } we have made a random sequence of 28 throws. What is the probability that the sequence CU BA JU LY 1987 appears in this order in the sequence already thrown?

10. (FIN 1) In a Cartesian coordinate system, the circle C 1 has center O 1 (−2,0) and radius 3. Denote the point (1, 0) by A and the origin by O. Prove that there is a constant c > 0 such that for every X that is exterior to C 1 ,

OX

− 1 ≥ cmin{AX,AX 2 }.

Find the largest possible c.

11. (FIN 2) Let S ⊂ [0,1] be a set of 5 points with {0,1} ⊂ S. The graph of a real function f : [0, 1] → [0,1] is continuous and increasing, and it is linear on every subinterval I in [0, 1] such that the endpoints but no interior points of I are in S. We want to compute, using a computer, the extreme values of g (x,t) =

f (x)− f (x−t) for x − t,x + t ∈ [0,1]. At how many points (x,t) is it necessary to compute g (x,t) with the computer?

f (x+t)− f (x)

12. (FIN 3) (SL87-3).

13. (FIN 4) Let A be an infinite set of positive integers such that every n ∈ A is the product of at most 1987 prime numbers. Prove that there are an infinite set

B ⊂ A and a number p such that the greatest common divisor of any two distinct numbers in B is p.

14. (FRA 1) Given n real numbers 0 <t 1 ≤t 2 ≤ ··· ≤ t n < 1, prove that

15. (FRA 2) Let a 1 ,a 2 ,a 3 ,b 1 ,b 2 ,b 3 ,c 1 ,c 2 ,c 3 be nine strictly positive real numbers. We set S 1 =a 1 b 2 c 3 , S 2 =a 2 b 3 c 1 , S 3 =a 3 b 1 c 2 ; T 1 =a 1 b 3 c 2 , T 2 =a 2 b 1 c 3 , T 3 =a 3 b 2 c 1 .

Suppose that the set {S 1 ,S 2 ,S 3 ,T 1 ,T 2 ,T 3 } has at most two elements. Prove that

S 1 +S 2 +S 3 =T 1 +T 2 +T 3 .

16. (FRA 3) Let ABC be a triangle. For every point M belonging to segment BC we denote by B ′ and c ′ the orthogonal projections of M on the straight lines AC and

BC . Find points M for which the length of segment B ′ C ′ is a minimum.

17. (FRA 4) Consider the number α obtained by writing one after another the dec- imal representations of 1 , 1987, 1987 2 , . . . to the right the decimal point. Show that α is irrational.

18. (FRA 5) (SL87-4).

3.28 IMO 1987 195

19. (FRG 1) (SL87-14).

20. (FRG 2) (SL87-15).

21. (FRG 3) (SL87-16).

22. (UNK 1) (SL87-5).

23. (UNK 2) A lampshade is part of the surface of a right circular cone whose axis is vertical. Its upper and lower edges are two horizontal circles. Two points are selected on the upper smaller circle and four points on the lower larger circle. Each of these six points has three of the others that are its nearest neighbors at a distance d from it. By distance is meant the shortest distance measured over the curved survace of the lampshade.

Prove that the area of the lampshade if d 2 (2 θ + 3 ), where cot θ 2 = 3 θ .

24. (UNK 3) Prove that if the equation x 4 + ax 3 + bx + c = 0 has all its roots real, then ab ≤ 0.

25. (UNK 4) Numbers d (n, m), with m, n integers, 0 ≤ m ≤ n, are defined by

d (n, 0) = d(n, n) = 0 for all n ≥ 0 and

md (n, m) = md(n − 1,m) + (2n − m)d(n − 1,m − 1) for all 0 < m < n. Prove that all the d (n, m) are integers.

26. (UNK 5) Prove that if x , y, z are real numbers such that x 2 +y 2 +z 2 = 2, then

x + y + z ≤ xyz + 2.

27. (UNK 6) Find, with proof, the smallest real number C with the following property: For every infinite sequence {x i } of positive real numbers such that

x 1 +x 2 + ··· + x n ≤x n +1 for n = 1, 2, 3, . . . , we have √

x 1 + x 2 + ··· + x n ≤C x 1 +x 2 + ··· + x n for n = 1, 2, 3, . . ..

28. (GDR 1) In a chess tournament there are n h n 2 i

≥ 5 players, and they have already played 4 + 2 games (each pair have played each other at most once). (a) Prove that there are five players a , b, c, d, e for which the pairs ab, ac, bc,

ad , ae, de have already played. h

(b) Is the statement also valid for the n 4 + 1 games played? Make the proof by induction over n.

29. (GDR 2) (SL87-13).

30. (HEL 1) Consider the regular 1987-gon A 1 A 2 ...A 1987 with center O. Show that the sum of vectors belonging to any proper subset of M = {OA j |j=

1 , 2, . . . , 1987} is nonzero.

31. (HEL 2) Construct a triangle ABC given its side a = BC, its circumradius R (2R ≥ a), and the difference 1/k = 1/c − 1/b, where c = AB and b = AC.

196 3 Problems

32. (HEL 3) Solve the equation 28 x = 19 y + 87 z , where x , y, z are integers.

33. (HEL 4) (SL87-6).

34. (HUN 1) (SL87-8).

35. (HUN 2) (SL87-9).

36. (ISL 1) A game consists in pushing a flat stone along a sequence of squares S 0 ,S 1 ,S 2 , . . . that are arranged in linear order. The stone is initially placed on square S 0 . When the stone stops on a square S k it is pushed again in the same direction and so on until it reaches S 1987 or goes beyond it; then the game stops. Each time the stone is pushed, the probability that it will advance exactly n squares is 1 /2 n . Determine the probability that the stone will stop exactly on square S 1987 .

37. (ISL 2) Five distinct numbers are drawn successively and at random from the set {1,...,n}. Show that the probability of a draw in which the first three numbers

as well as all five numbers can be arranged to form an arithmetic progression is greater than 6 (n−2) 3 .

38. (ISL 3) (SL87-10).

39. (LUX 1) Let A be a set of polynomials with real coefficients and let them satisfy the following conditions:

(i) if f ∈ A and deg f ≤ 1, then f (x) = x − 1; (ii) if f ∈ A and deg f ≥ 2, then either there exists g ∈ A such that f (x) = x 2 +deg g + xg(x) − 1 or there exist g,h ∈ A such that f (x) = x 1 +degg g (x) +

h (x); (iii) for every f , g ∈ A, both x 2 +deg f + x f (x) − 1 and x 1 +deg f f (x) + g(x) belong to A. Let R n ( f ) be the remainder of the Euclidean division of the polynomial f (x) by x n . Prove that for all f ∈ A and for all natural numbers n ≥ 1 we have

R n ( f )(1) ≤ 0

and

R n ( f )(1) = 0 ⇒ R n ( f ) ∈ A.

40. (MNG 1) The perpendicular line issued from the center of the circumcircle to the bisector of angle C in a triangle ABC divides the segment of the bisector inside ABC into two segments with ratio of lengths λ . Given b = AC and a = BC, find the length of side c.

41. (MNG 2) Let n points be given arbitrarily in the plane, no three of them collinear. Let us draw segments between pairs of these points. What is the mini- mum number of segments that can be colored red in such a way that among any four points, three of them are connected by segments that form a red triangle?

42. (MNG 3) Find the integer solutions of the equation

h√ i h √

i 2m = (2 + 2 )n .

3.28 IMO 1987 197

43. (MNG 4) Let 2n + 3 points be given in the plane in such a way that no three lie on a line and no four lie on a circle. Prove that the number of circles that pass through three of these points and contain exactly n interior points is not less than

1 2n +3

be real numbers such that sin θ 1 + ··· + sin θ n = 0. Prove that

44. (MAR 1) Let θ 1 , θ 2 ,..., θ n

|sin θ 1 + 2 sin θ 2 + ··· + nsin θ n |≤

45. (MAR 2) Let us consider a variable polygon with 2n sides (n ∈ N) in a fixed circle such that 2n − 1 of its sides pass through 2n − 1 fixed points lying on a straight line ∆ . Prove that the last side also passes through a fixed point lying on ∆ .

46. (NLD 1) (SL87-7).

47. (NLD 2) Through a point P within a triangle ABC the lines l, m, and n perpen- dicular respectively to AP , BP,CP are drawn. Prove that if l intersects the line BC in Q, m intersects AC in R, and n intersects AB in S, then the points Q, R, and S are collinear.

48. (POL 1) (SL87-11).

49. (POL 2) In the coordinate system in the plane we consider a convex polygon W and lines given by equations x = k, y = m, where k and m are integers. The

lines determine a tiling of the plane with unit squares. We say that the boundary of W intersects a square if the boundary contains an interior point of the square. Prove that the boundary of W intersects at most 4 ⌈d⌉ unit squares, where d is the maximal distance of points belonging to W (i.e., the diameter of W ) and ⌈d⌉ is the least integer not less than d.

50. (POL 3) Let P , Q, R be polynomials with real coefficients, satisfying P 4 +Q 4 = R 2 . Prove that there exist real numbers p , q, r and a polynomial S such that P = pS ,Q = qS and R = rS 2 .

Variants: (1) P 4 +Q 4 =R 4 ; (2) gcd

(P, Q) = 1; (3) ±P 4 +Q 4 =R 2 or R 4 .

51. (POL 4) The function F is a one-to-one transformation of the plane into it- self that maps rectangles into rectangles (rectangles are closed; continuity is not assumed). Prove that F maps squares into squares.

52. (POL 5) (SL87-12).

53. (ROU 1) (SL87-17).

54. (ROU 2) Let n be a natural number. Solve in integers the equation

x n +y n = (x − y) n +1 .

55. (ROU 3) Two moving bodies M 1 ,M 2 are displaced uniformly on two coplanar straight lines. Describe the union of all straight lines M 1 M 2 .

56. (ROU 4) (SL87-18).

198 3 Problems

57. (ROU 5) The bisectors of the angles B ,C of a triangle ABC intersect the op- posite sides in B ′ ,C ′ respectively. Prove that the straight line B ′ C ′ intersects the inscribed circle in two different points.

58. (ESP 1) Find, with argument, the integer solutions of the equation 3z 2 = 2x 3 + 385x 2 + 256x − 58195.

59. (ESP 2) It is given that a 11 ,a 22 are real numbers, that x 1 ,x 2 ,a 12 ,b 1 ,b 2 are com- plex numbers, and that a 11 a 22 =a 12 a 12 (where a 12 is the conjugate of a 12 ). We

consider the following system in x 1 ,x 2 : x 1 (a 11 x 1 +a 12 x 2 )=b 1 ,

x 2 (a 12 x 1 +a 22 x 2 )=b 2 .

(a) Give one condition to make the system consistent.

(b) Give one condition to make arg x 1 − argx 2 = 98 ◦ .

60. (TUR 1) It is given that x = −2272, y = 10 3 + 10 2 c + 10b + a, and z = 1 satisfy the equation ax + by + cz = 1, where a, b, c are positive integers with a < b < c.

Find y.

61. (TUR 2) Let PQ be a line segment of constant length λ taken on the side BC of

a triangle ABC with the order B , P, Q,C, and let the lines through P and Q parallel to the lateral sides meet AC at P 1 and Q 1 and AB at P 2 and Q 2 respectively. Prove that the sum of the areas of the trapezoids PQQ 1 P 1 and PQQ 2 P 2 is independent of the position of PQ on BC.

62. (TUR 3) Let l ,l ′

be two lines in 3-space and let A , B,C be three points taken on l with B as midpoint of the segment AC. If a , b, c are the distances of A, B,C q

from l ′ , respectively, show that b ≤ a 2 +c 2 2 , equality holding if l ,l ′ are parallel.

63. (TUR 4) Compute ∑ 2n k =0 (−1) k a 2 k , where a k are the coefficients in the expansion √

2n

(1 − 2x +x 2 ) n = ∑ a k x k .

k =0

64. (USA 1) Let r > 1 be a real number, and let n be the largest integer smaller than r . Consider an arbitrary real number x with 0 ≤x≤ n r −1 . By a base-r expansion of x we mean a representation of x in the form

r 3 + ··· ,

where the a i are integers with 0 ≤a i < r. You may assume without proof that every number x with 0 ≤x≤ n r −1 has at least one base-r expansion. Prove that if r is not an integer, then there exists a number p, 0 ≤p≤ n r −1 , which has infinitely many distinct base-r expansions.

3.28 IMO 1987 199

65. (USA 2) The runs of a decimal number are its increasing or decreasing blocks of digits. Thus 024379 has three runs: 024, 43, and 379. Determine the average

number of runs for a decimal number in the set {d 1 d 2 ...d n |d k 6= d k +1 ,k=

1 , 2, . . . , n − 1}, where n ≥ 2.

66. (USA 3) (SL87-2).

67. (USS 1) If a , b, c, d are real numbers such that a 2 +b 2 +c 2 +d 2 ≤ 1, find the maximum of the expression

(a + b) 4 + (a + c) 4 + (a + d) 4 + (b + c) 4 + (b + d) 4 + (c + d) 4 .

68. (USS 2) (SL87-19). Original formulation. Let there be given positive real numbers α , β , γ such that α + β + γ < π , α + β > γ , β + γ > α , γ + α > β . Prove that it is possible to draw a triangle with the lengths of the sides sin α , sin β , sin γ . Moreover, prove that its area is less than

1 (sin 2 α + sin 2 β + sin 2 γ ).

69. (USS 3) (SL87-20).

70. (USS 4) (SL87-21).

71. (USS 5) To every natural number k, k ≥ 2, there corresponds a sequence a n (k) according to the following rule:

a n = τ (a n −1 ) for n ≥ 1, in which τ (a) is the number of different divisors of a. Find all k for which the

a 0 = k,

sequence a n (k) does not contain the square of an integer.

72. (VNM 1) Is it possible to cover a rectangle of dimensions m × n with bricks that have the tromino angular shape (an arrangement of three unit squares forming the letter L) if:

(a) m × n = 1985 × 1987; (b) m × n = 1987 × 1989?

73. (VNM 2) Let f (x) be a periodic function of period T > 0 defined over R. Its first derivative is continuous on R. Prove that there exist x, y ∈ [0,T ) such that x 6= y and

f (x) f ′ (y) = f (y) f ′ (x).

74. (VNM 3) (SL87-22).

75. (VNM 4) Let a k

be positive numbers such that a 1 ≥ 1 and a k +1 −a k ≥ 1 (k =

1 , 2, . . . ). Prove that for every n ∈ N,

k =1 a k +1

200 3 Problems

76. (VNM 5) Given two sequences of positive numbers {a k } and {b k } (k ∈ N) such that

(i) a k <b k ,

(ii) cos a k x + cosb k x ≥− 1 k for all k ∈ N and x ∈ R,

prove the existence of lim

k →∞ b k and find this limit.

77. (YUG 1) Find the least natural number k such that for any a ∈ [0,1] and any natural number n,

(1 − a) < (n + 1) 3 .

78. (YUG 2) (SL87-23).

3.28.3 Shortlisted Problems

1. (AUS 6) Let f be a function that satisfies the following conditions: (i) If x > y and f (y) − y ≥ v ≥ f (x) − x, then f (z) = v + z, for some number z between x and y. (ii) The equation f (x) = 0 has at least one solution, and among the solutions of this equation, there is one that is not smaller than all the other solutions; (iii) f (0) = 1. (iv) f (1987) ≤ 1988.

(v) f (x) f (y) = f (x f (y) + y f (x) − xy). Find f (1987).

2. (USA 3) At a party attended by n married couples, each person talks to every- one else at the party except his or her spouse. The conversations involve sets

of persons or cliques C 1 ,C 2 , . . . ,C k with the following property: no couple are members of the same clique, but for every other pair of persons there is exactly one clique to which both members belong. Prove that if n ≥ 4, then k ≥ 2n.

3. (FIN 3) Does there exist a second-degree polynomial p (x, y) in two variables such that every nonnegative integer n equals p (k, m) for one and only one ordered pair (k, m) of nonnegative integers?

4. (FRA 5) Let ABCDEFGH be a parallelepiped with AE kBFkCGkDH. Prove the inequality

AF + AH + AC ≤ AB + AD + AE + AG. In what cases does equality hold?

5. (UNK 1) Find, with proof, the point P in the interior of an acute-angled triangle ABC for which BL 2 + CM 2 + AN 2 is a minimum, where L , M, N are the feet of the perpendiculars from P to BC ,CA, AB respectively.

6. (HEL 4) Show that if a , b, c are the lengths of the sides of a triangle and if 2S = a + b + c, then

2 n −2

S n −1 , n

b c +a a +c ≥ +b 3 ≥ 1.

3.28 IMO 1987 201

7. (NLD 1) Given five real numbers u 0 ,u 1 ,u 2 ,u 3 ,u 4 , prove that it is always possi- ble to find five real numbers v 0 ,v 1 ,v 2 ,v 3 ,v 4 that satisfy the following conditions: (i) u i −v i ∈ N. (ii) ∑ 0 ≤i< j≤4 (v i −v j ) 2 < 4.

8. (HUN 1) (a) Let (m, k) = 1. Prove that there exist integers a 1 ,a 2 ,...,a m and

b 1 ,b 2 ,...,b k such that each product a i b j (i = 1, 2, . . . , m; j = 1, 2, . . . , k) gives a different residue when divided by mk. (b) Let (m, k) > 1. Prove that for any integers a 1 ,a 2 ,...,a m and b 1 ,b 2 ,...,b k there must be two products a i b j and a s b t ((i, j) 6= (s,t)) that give the same residue when divided by mk.

9. (HUN 2) Does there exist a set M in usual Euclidean space such that for every plane λ the intersection M ∩ λ is finite and nonempty?

10. (ISL 3) Let S 1 and S 2 be two spheres with distinct radii that touch externally. The spheres lie inside a cone C, and each sphere touches the cone in a full circle. Inside the cone there are n additional solid spheres arranged in a ring in such

a way that each solid sphere touches the cone C, both of the spheres S 1 and S 2 externally, as well as the two neighboring solid spheres. What are the possible values of n?

11. (POL 1) Find the number of partitions of the set {1,2,...,n} into three subsets

A 1 ,A 2 ,A 3 , some of which may be empty, such that the following conditions are satisfied: (i) After the elements of every subset have been put in ascending order, every two consecutive elements of any subset have different parity. (ii) If A 1 ,A 2 ,A 3 are all nonempty, then in exactly one of them the minimal number is even.

12. (POL 5) Given a nonequilateral triangle ABC, the vertices listed counterclock- wise, find the locus of the centroids of the equilateral triangles A ′ B ′ C ′ (the ver- tices listed counterclockwise) for which the triples of points A ,B ′ ,C ′ ;A ′ , B,C ′ ; and A ′ ,B ′ ,C are collinear.

13. (GDR 2) IMO 5 Is it possible to put 1987 points in the Euclidean plane such that the distance between each pair of points is irrational and each three points determine

a nondegenerate triangle with rational area?

14. (FRG 1) How many words with n digits can be formed from the alphabet {0,1,2,3,4}, if neighboring digits must differ by exactly one?

15. (FRG 2) IMO 3 Suppose x 1 ,x 2 ,...,x n are real numbers with x 2 1 +x 2 2 2 + ··· + x n = 1. Prove that for any integer k > 1 there are integers e i not all 0 and with |e i |<k such that

16. (FRG 3) IMO 1 Let S be a set of n elements. We denote the number of all permu- tations of S that have exactly k fixed points by p n (k). Prove:

202 3 Problems (a) ∑ n k =0 kp n (k) = n!;

(b) ∑ n (k − 1) 2 k =0 p n (k) = n!.

17. (ROU 1) Prove that there exists a four-coloring of the set M = {1,2,...,1987} such that any arithmetic progression with 10 terms in the set M is not monochro- matic.

Alternative formulation. Let M = {1,2,...,1987}. Prove that there is a function

f :M → {1,2,3,4} that is not constant on every set of 10 terms from M that form an arithmetic progression.

18. (ROU 4) For any integer r ≥ 1, determine the smallest integer h(r) ≥ 1 such that for any partition of the set {1,2,...,h(r)} into r classes, there are integers

a ≥ 0,1 ≤ x ≤ y, such that a + x,a + y,a + x + y belong to the same class.

be positive real numbers such that α + β + γ < π , α + β > γ , β + γ > α , γ + α > β . Prove that with the segments of lengths sin α , sin β , sin γ we can construct a triangle and that its area is not greater than

19. (USS 2) Let α , β , γ

1 (sin 2 α + sin 2 β + sin 2 γ ).

20. (USS 3) IMO 6 p Let f (x) = x 2 + x + p, p ∈ N. Prove that if the numbers f (0), f (1), . . ., f ([ p /3 ]) are primes, then all the numbers f (0), f (1), . . ., f (p − 2) are primes.

21. (USS 4) IMO 2 The prolongation of the bisector AL (L ∈ BC) in the acute-angled triangle ABC intersects the circumscribed circle at point N. From point L to the sides AB and AC are drawn the perpendiculars LK and LM respectively. Prove that the area of the triangle ABC is equal to the area of the quadrilateral AKNM.

22. (VNM 3) IMO 4 Does there exist a function f : N → N, such that f ( f (n)) = n + 1987 for every natural number n?

23. (YUG 2) Prove that for every natural number k (k ≥ 2) there exists an irrational number r such that for every natural number m,

[r m ] ≡ −1

(mod k).

Remark. An easier variant: Find r as a root of a polynomial of second degree with integer coefficients.

3.29 IMO 1988 203