The Thirty-First IMO Beijing, China, July 8–19, 1990

3.31 The Thirty-First IMO Beijing, China, July 8–19, 1990

3.31.1 Contest Problems

First Day (July 12)

1. Given a circle with two chords AB ,CD that meet at E, let M be a point of chord AB other than E. Draw the circle through D, E, and M. The tangent line to the

circle DEM at E meets the lines BC , AC at F, G, respectively. Given AM AB = λ , find GE EF .

2. On a circle, 2n − 1 (n ≥ 3) different points are given. Find the minimal natu- ral number N with the property that whenever N of the given points are colored black, there exist two black points such that the interior of one of the correspond- ing arcs contains exactly n of the given 2n − 1 points.

3. Find all positive integers n having the property that n 2 +1 n 2 is an integer. Second Day (July 13)

be the set of positive rational numbers. Construct a function f : Q + → Q + such that

4. Let Q +

f (x)

f (x f (y)) =

for all x , y in Q + .

5. Two players A and B play a game in which they choose numbers alternately according to the following rule: At the beginning, an initial natural number n 0 >

1 is given. Knowing n 2k , player A may choose any n 2k +1 ∈ N such that

2k ≤n 2k +1 ≤n 2k .

Then player B chooses a number n 2k +2 ∈ N such that

where p is a prime number and r ∈ N. It is stipulated that player A wins the game if he (she) succeeds in choosing the number 1990, and player B wins if he (she) succeeds in choosing 1. For which

natural numbers n 0 can player A manage to win the game, for which n 0 can player B manage to win, and for which n 0 can players A and B each force a tie?

6. Is there a 1990-gon with the following properties (i) and (ii)? (i) All angles are equal; (ii) The lengths of the 1990 sides form a permutation of the numbers 1 2 ,2 2 ,

3.31 IMO 1990 235

3.31.2 Longlisted Problems

1. (AUS 1) In triangle ABC, point O is the circumcenter, H is the orthocenter. De- note by A 1 ,B 1 , and C 1 the circumcenters of the triangles CHB, CHA, and AHB respectively. Prove that the triangles ABC and A 1 B 1 C 1 are congruent and that their nine-point circles coincide.

2. (AUS 2) Prove that

3. (AUS 3) (SL90-1)

4. (CAN 1) (SL90-2)

5. (COL 1) Let b be a positive integer. Assume that there exist exactly 1990 trian- gles ABC with integral side-lengths satisfying the following conditions:

(i) ∠ABC = 1 2 ∠BAC; (ii) AC = b. Find the minimal value for b.

6. (COL 2) Assume that the function f : (Z + ) 3 → N satisfy the following condi- tions: (i) f (0, 0, 0) = 1; (ii) f (x, y, z) = f (x − 1,y,z) + f (x,y − 1,z) + f (x,y,z − 1); (iii) When applying the above relation iteratively, if any of x ′ ,y ′ ,z ′ is negative, then f (x ′ ,y ′ ,z ′ ) = 0. ( f (x,y,z)) Prove that if x, y, z are the side lengths of a triangle, then k

f (mx,my,mz) is not an integers for any integers k , m > 1.

7. (CUB 1) Let A and B be two points in the plane α , and let r be the line passing through A and B. There are n distinct points P 1 ,P 2 , ...,P in one of the half- planes divided by the line r. Prove that there are at least √ n n distinct values among

the distances AP 1 , AP 2 , . . . , AP n , BP 1 , BP 2 , . . . , BP n .

8. (CZS 1) (SL90-3)

9. (CZS 2) (SL90-4)

h p (p−k+1) i

10. (CZS 3) Let p, k, and x be positive integers such that p > k and x <

, where [q] denotes the largest integer not greater than q. Prove that when x balls

2 (k−1)

are put into p boxes arbitrarily, there exist k boxes with the same number of balls.

11. (CZS 4) In a group of mathematicians, every mathematician has some friends (friendship is symmetrical relation). Prove that there exists a mathematician, such that the average of the numbers of friends of all his friends is not less than the average of the number of friends of all the mathematicians.

236 3 Problems

12. (CZS 5) For any permutation p of the set {1,2,...,n} define d(p) = |p(1)−1|+ |p(2) − 2| + ···+ |p(n) − n|. Denote by i(p) the number of integer pairs (i, j) in the permutation p such that 1 ≤ i < j ≤ n and p(i) > p( j). Find all real numbers

c such that the inequality i (p) ≤ cd(p) holds for any positive integer n and any permutation p.

13. (FIN 1) Six cities A, B, C, D, E, and F are located at the vertices of a regu- lar hexagon in that order. Let G be the center of the hexagon. The sides of the hexagon are the roads connecting these cities. Furthermore, there are roads con- necting the cities B, C, E, F, and G. Because of raining, one or more of the roads may be destroyed. The probability of each road remaining undestroyed is equal to p. Determine the probability that it is possible to travel between the cities A and D.

14. (FIN 2) We call a set S ⊆ R superinvariant, if for any stretching A of the set S by the transformation taking x to A (x) = x 0 + a(x−x 0 ) (here a > 0 is a real number), there exists a transformation B, B (x) = x + b such that the images of S under A and B agree: i.e. for any x ∈ S, there exists y ∈ S such that A(x) = B(y) and for any t ∈ S there is u ∈ S such that B(t) = A(u). Determine all superinvariant sets.

15. (FRA 1) (SL90-5)

16. (FRA 2) We say that an integer k ≥ 1 has property P, if there exists at least one integer m ≥ 1 which cannot be expressed in the form m = ε 1 z 1 + ε 2 z k 2 + ε 2k z k 2k , where z i is nonnegative integer and ε i = ±1, i = 1,2,...,2k. Prove that there are infinitely many integers k having the property P.

17. (FRA 3) 1990 mathematicians attend a meeting. Every mathematician has at least 1327 friends (friendship is symmetric relation). Prove that it is possible to find four mathematicians such that any two of them are friends.

18. (FRG 1) Find, with proof, the least positive integer n having the following prop- erty: All the binary representations of 1, 2, . . . , 1990 appear after the decimal point in the binary expression of 1 /n.

19. (FRG 2) (SL90-6)

20. (FRG 3) Is it possible to express the three-dimensional space as a union of dis- joint circles?

21. (HEL 1) Point O is in the interior of △ABC. Three lines through O parallel to BC , CA, and AB intersect the sides AB and AC at D and E; the sides BC and BA at F and G; and the sides CA and CB at H and I, respectively.

22. (HEL 2) (SL90-7)

23. (HUN 1) (SL90-8)

24. (HUN 2) Find the real number t such that the following system of equations has

a unique real number solution (x, y, z, v):

3.31 IMO 1990 237 x +y+z+v=0

(xy + yz + zv) + t(xz + xv + yv) = 0.

25. (HUN 3) (SL90-9)

26. (ISL 1) Prove that there exist infinitely many positive integers n such that

is a perfect square. Obviously, 1 is the least integer having this prop- erty. Find the next two least integers with this property.

1 2 +2 2 +···+n 2 n

27. (ISL 2) (SL90-10)

28. (IND 1) Let ABC be an acute-angled triangle. Assume that the circle Γ satisfies the following two conditions:

(i) Γ intersects all three sides of △ABC. (ii) These points form a hexagon whose three pairs of opposite sides are par- allel. (The hexagon may be degenerate if two or more vertices coincide. In this case opposite sides being parallel is defined through limit behavior.)

Construct the locus of the centers of such circles Γ .

29. (IND 2) Function f (n), n ∈ N is defined as follows: Let A(n) and B(n) be co- prime positive integers such that

n ! (n + 1000)!

If B (n) = 1 then f (n) = 1; if B(n) 6= 1 then f (n) is the largest prime factor of

B (n). Prove that the set of values of f (n) is finite and find the maximum value for f (n).

30. (IND 3) (SL90-11)

31. (IND 4) Let S = {1,2,...,1990}. A 31-element subset of S is called good if the sum of its elements is divisible by 5. Find the number of good subsets of S.

32. (IRN 1) Using the following five figures is it possible to construct a paral- lelepiped whose side lengths are all integers greater than 1 and whose volume is 1990? In the following figure, every square represents a unit cube.

33. (IRN 2) Let S be a set with 1990 elements. Let P be a set of ordered sequences of 100 elements from S. If x = (. . . , a, . . . , b, . . .) ∈ P, for a,b ∈ S, then we say that the ordered pair (a, b) appears in x. Assume that any ordered pair from S appears in at most one element of P. Prove that P has at most 800 elements.

34. (IRN 3) There are n non-coplanar points in the space. Prove that there exists a circle that passes through exactly three of those points.

238 3 Problems

35. (IRN 4) Prove that if |x| < 1, then x

36. (IRL 1) (SL90-12)

37. (IRL 2) (SL90-13)

38. (IRL 3) Let α

be a positive solution of the quadratic equation x 2 = 1990x + 1. For every m , n ∈ N define the operation m⋆n = mn+[ α m ][ α n ], where [x] denotes the largest integer not exceeding x. Prove that (p ⋆ q) ⋆ r = p ⋆ (q ⋆ r) holds for all p , q, r ∈ N.

39. (IRL 4) Let a, b, c be integers. Prove that there are integers p 1 ,q 1 ,r 1 ,p 2 ,q 2 , and r 2 satisfying a =q 1 r 2 −q 2 r 1 ,b =r 1 p 2 −r 2 p 1 , and c =p 1 q 2 −p 2 q 1 .

40. (ISR 1) Given three letters X , Y , Z, we can construct letter sequences such as XZ , ZZY XYY , X XY ZX X , etc. For any given sequence, one can perform the following operations:

T 1 If the right-most letter is Y , we add Y Z after it, for example: T 1 (XY ZZXY ) = XY ZZXYY Z ; T 2 If the sequence contains YYY , this can be replaced by Z as in the following

example: T 2 (XXYY ZYYY X ) = XY ZZX; T 3 Xp can be replaced by X pX where p is any subsequence of letters: Exam- ple: T 3 (XXY Z) = XXY ZX ; T 4 In a sequence that contains one or more letters Z, we can replace the first Z by XY . Example: T 4 (XXYY ZZX ) = XY XY ZX ; T 5 We can replace any of X X , YY , ZZ by X , for example: T 5 (ZZY XYY ) = XY X X , or T 5 (ZZY XYY ) = XY XYY , or T 5 (ZZY XYY ) = ZZY XX.

Using the above operations is it possible to obtain XY ZZ starting from XY Z?

41. (ISR 2) Given a positive integer n, calculate S n =∑ n r =0 2 r −2n 2n · −r n .

42. (ITA 1) Find n points P 1 , ...,P n on the circumference of a unit circle such that ∑ 1 ≤i< j≤n P i P j is maximal.

43. (ITA 2) Let V be a finite set of points in the three-dimensional space. Let S 1 ,S 2 , S 3 be the sets consisting of the orthogonal projections of the points of V onto the

planes Oyz, Ozx, and Oxy respectively. Prove that |V | 2 ≤ |S 1 | · |S 2 | · |S 3 |, where |A| denotes the number of elements in the set A.

44. (ITA 3) Prove that for any positive integer n, the number of odd integers among the binomial coefficients n k (0 ≤ k ≤ n) is a power of 2.

45. (ITA 4) A tourist is looking for a treasure on an island. The treasure is hidden behind the series of doors each of which is colored with one of n possible colors. The tourist has n keys, all of different colors. Each key can open any door, how- ever, a key gets destroyed when it opens the door of the same color as the key itself (if it opens a door of some other color, it remains intact). Once the tourist

3.31 IMO 1990 239 starts using a particular key, she must continue using only that key until it gets

destroyed. Find the least number of doors to ensure that no tourist can get the treasure, no matter how he chooses the order of keys.

46. (JPN 1) Let P be an interior point of triangle ABC. Let Q, R, S be the in- tersections of AP, BP, CP with sides BC, CA, AB respectively. Prove that S

QRS ≤ 4 S ABC .

47. (JPN 2) (SL90-14) √

48. (JPN 3) Prove that 2 + 5 + 1990 is irrational.

49. (LUX 1) Let AB and AC be two chords of a circle with center O. The diameter perpendicular to BC intersects AB and AC at F and G respectively (F is inside the circle). Let T be the point on tangency of the circle and the tangent from G. Prove that F is the projection of T on OG.

50. (MEX 1) During the duration of the class, n children sit in a circle and play the following game: The teacher goes around the children in the clockwise direction and hands out candies according to the following rules: The teacher selects a child, gives him/her a candy as well to the child child next to him; then the teacher skips one child and gives a candy to the next one; then the teacher skips two children, gives a candy to the next; then skips over three children, ... Find the value of n such that the teacher ends up giving at least one candy to each of the children after finitely many steps.

51. (MEX 2) (SL90-15)

52. (MNG 1) Let a > 0 be a real number. Assume that real numbers a 1 , ...,a n satisfy 0 <a i ≤ a for i = 1,2,...,n. Prove that: (a) If n = 4, then

1 4 a 1 a 2 +a 2 a 3 +a 3 a 4 +a 4 a

a 2 =1 ≤ 2.

(b) If n = 6, then

53. (MNG 2) Find all real solutions to the system of equations:

x 3 +y 3 = 1, x 5 +y 5 = 1.

54. (MNG 3) Given a set M = {1,2,...,n}, let φ :M → M be a bijection. (a) Prove that there are bijections φ 1 , φ 2 :M → M such that φ 2 ◦ φ 1 = φ and φ 2 1 = φ 2 2 = id, where id is the identity mapping.

240 3 Problems (b) Prove that the conclusion in (a) still holds if M is the set of all positive

integers.

55. (MNG 4) Given points A, M, M 1 , and a rational number λ 6= −1, construct a triangle ABC such that: M ∈ BC, M 1 ∈B 1 C 1 , where B 1 and C 1 are the projections of B, C to AC and AB respectively, and

BM MC

56. (MAR 1) For positive integers n, p, n ≥ p, define real number K n ,p as follows: K n ,0 = 1 n +1 ,K n ,p =K n −1,p−1 −K n ,p−1 for 1 ≤ p ≤ n. (a) If S n

=∑ n p =0 K n ,p ,n = 0, 1, 2, . . ., find lim n →∞ S n . (b) Find T n

=∑ p p =0 (−1) K n ,p ,n = 0, 1, 2, . . ..

57. (MAR 2) The sequence {u n } is defined by u 1 = 1, u 2 = 1, u n =u n −1 + 2u n −2 , for n ≥ 3. (a) Prove that for any positive integers n, p (p > 1), u n +p =u n +1 u p + 2u n u p −1 . (b) Find the greatest common divisor of u n and u n +3 .

58. (NLD 1) (SL90-16)

59. (NLD 2) Given eight real numbers a 1 ≤a 2 ≤ ··· ≤ a 7 ≤a 8 , let x = a 1 +···+a 8 ,

a 1 2 +···+a 8 2

8 . Prove that

2 y −x 2

≤a 8 −a 1 ≤4 y −x 2 .

60. (NLD 3) (SL90-17)

61. (NLD 4) Prove that we can fill in the three dimensional space with regular tetra- hedrons and regular octahedrons all of which have the same edge-lengths. Find the ratio of the number of regular tetrahedrons used to the number of regular octahedrons.

62. (NOR 1) (SL90-18)

63. (POL 1) (SL90-19)

64. (POL 2) Given an m-element set M and its k-element subset K ⊆ M, we say that

a function f : K → M has a path, if there exists an element x 0 ∈ K such that

f (x 0 )=x 0 , or there exists a chain x 0 ,x 1 ,...,x j =x 0 ∈ K such that x i = f (x i −1 ),

for i = 1, 2, . . . , j. Find the number of functions f : K → M that have paths.

65. (POL 3) (SL90-20)

66. (POL 4) Find all continuous bounded functions f : R → R such that ( f (x)) 2 − ( f (y)) 2 = f (x + y) f (x − y), for all x,y,∈ R.

67. (PRK 1) Let a + bi and c + di be two roots of the equation x n = 1990 (n ≥ 3 is an integer). Assume that f (2, 1) = (1, 2) where f is the linear transformation:

Denote by r the distance between the image of (2, 2) and the origin. Find the range for the values of r.

68. (PRK 2) A mobile point M starts from the origin O (0, 0) and moves along the line l with slope k, where k is an irrational number.

(a) Prove that the point O (0, 0) is the only rational point (i.e. with both rational coordinates) on the line l. (b) Prove that for any number ε > 0 there are integers m and n such that the distance between l and the point (m, n) is less than ε .

69. (PRK 3) Consider the set of cuboids: three edges a, b, c from a common vertex satisfy the condition: 2 a

(a) Prove that there are 100 pairs of cuboids in this set with equal volumes in each pair. (b) For each pair of the above cuboids, find the ratio of the sum of their edges.

70. (PRK 4) Let M be a point on the side BC of a triangle ABC. (a) Prove that if M is the midpoint of BC, then AB 2 + AC 2 = 2(AM 2 + BM 2 ). (b) If there exists a point N

∈ BC different than M satisfying AB 2 + AC 2 =

2 (AN 2 + BN 2 ), find the region that the point A might occupy.

71. (PRK 5) Given a point P = (p 1 ,...,p n ), find the point X = (x 1 ,...,x n ) satisfying x 1 ≤x 2 ≤ ··· ≤ x n such that X minimizes the expression q (x 1 −p 1 ) 2 + ··· + (x n −p n ) 2 .

be integers such that the ordered (a i ,b i ) are distinct for i = 1, 2, . . . , n and |a 1 b 2 −a 2 b 1 |= |a 2 b 3 −a 3 b 2 | = ··· = |a n −1 b n −a n b n −1 | = 1. Prove that there exists a pair of indices i and j that satisfy 2 ≤ |i − j| ≤ n − 2 and |a i b j −a j b i | = 1. Alternative formulation. Let n ≥ 5 be a positive integers and let P 1 , ...,P n be the points with integral coordinates in the coordinate system with the origin O. The areas of the triangles OP 1 P 2 , OP 2 P 3 , . . . , OP n −1 P n are equal to 1 2 . Prove that there exists a pair of integers i, j, such that 2 ≤ |i − j| ≤ n − 2 for which the area of △OP i P j is equal to 1 2 .

72. (KOR 1) Let n ≥ 5 be a positive integer. Let a 1 ,b 1 ,a 2 ,b 2 ,...,a n ,b n

73. (KOR 2) A function f : Q → R satisfies the following conditions: (i) f (0) = 0 and for every nonzero a ∈ Q, f (a) > 0; (ii) f ( α + β )=f( α )f( β ); (iii) f ( α + β ) ≤ max{ f ( α ), f ( β )}. Let x be an integer for which f

(x) 6= 1. Prove that f (1 + x + ··· + x n ) = 1 for every positive integer n.

74. (KOR 3) Let L be a subset of the plane defined by L = {(41x + 2y,59x + 15y) : x , y ∈ Z}. Prove that every parallelogram with center at the origin and area of 1990 contains at least two points of L.

242 3 Problems

75. (ROU 1) (SL90-21)

76. (ROU 2) Prove that there are at least two non-congruent cyclic quadrilaterals with equal areas and perimeters.

77. (ROU 3) Let a , b, c ∈ R. Prove that (a 2 + ab + b 2 )(b 2 + bc + c 2 )(c 2 + ca + a 2 ) ≥ (ab + bc + ca) 3 . When does the equality hold?

78. (ROU 4) (SL90-22)

79. (ROU 5) (SL90-23)

80. (ESP 1) Function f : N × N → Q satisfies the following conditions: (i) f (1, 1) = 1; (ii) f (p + 1, q) + f (p, q + 1) = f (p, q) for all p, q ∈ N; (iii) q f (p + 1, q) = p f (p, q + 1) for all p, q ∈ N. Find f (1990, 31).

81. (ESP 2) Circle k (K, ρ ) tangents the sides AB and BC of △ABC and intersects the side BC at points D and E. Let p be the distance from K to the side BC.

(a) Prove that a (p − ρ ) = 2s(r − ρ ), where r is the inradius, s the semi- perimeter of △ABC and a the length of the side BC. (b) Prove that

rr 1 ( ρ − r)(r 1 − ρ )

DE =

r 1 −r

where r 1 is the radius of the excircle of △ABC opposite to A.

82. (ESP 3) Define the symmedian S a of triangle ABC as the line symmetric to the median from A with respect to the bisector of ∠CAB. Assume that the median m a intersects BC at A ′ and the circumcircle at A 1 . Assume that the symmedian S a intersects BC at M and the circumcircle at A 2 . Denote by O the circumcenter

of △ABC. If A 1 , O, and A 2 are collinear, prove that:

83. (SWE 1) Point D lies on the hypothenuse BC of the right triangle ABC. The inradii of the triangles ADB and ADC are equal. Prove that S ABC = AD 2 .

84. (SWE 2) Let a 1 ,a 2 ,...,a n ∈ (0,2n) be n distinct integers (n ≥ 4). Prove that there exists a subset of the set {a 1 ,...,a n } whose sum of elements is divisible by 2n.

85. (SWE 3) Let A 1 ,A 2 ,...,A n (n ≥ 4) be n convex sets in plane. Given that every three of these sets have a common point, prove that there exists a point belonging to all the sets.

86. (SWE 4) Given a function f (x) = sin x + sin( π x ) and a positive number d, prove that for every n ∈ N there exists a real number p such that p > n and | f (x + p) −

f (x)| < d holds for all real numbers x.

3.31 IMO 1990 243

87. (THA 1) Let m be a positive odd integer not divisible by 3. Prove that

112 | 4 m

88. (THA 2) (SL90-24)

be pairwise non-intersecting sets such that S k has exactly k elements (k = 1, 2, . . . , n). Denote S = S 1 ∪S 2 ∪ ··· ∪ S n . A function f : S → S maps all elements of S k to a fixed element of S k for k = 1, 2, . . . , n. Find the number of functions g : S → S satisfying f ◦ g ◦ f = f (i.e. f (g( f (x))) = f (x) for all x).

89. (THA 3) Let n be a positive integer. Let S 1 , ...,S n

90. (TUR 1) Let P be a variable point on the circumference of a quarter-circle with radii OA, OB (∠AOB = 90 ◦ ). Let H be the projection of P on OA. Find the locus of the incenters of △HPO.

91. (TUR 2) Quadrilateral ABCD is circumscribed around the circle with center O. If AB = CD and M and K are the midpoints of BC and AD respectively prove that OM = OK.

92. (TUR 3) Let n be a positive integer and m (n+1)(n+2) = 2 . There are n distinct lines L 1 ,L 2 , ...,L n in coordinate plane and m distinct points A 1 ,A 2 , ...,A m satisfying the following two conditions: (i) Any two of the lines are non-parallel. (ii) Any three lines are non-concurrent. (iii) Only A 1 does not line on any line L k and there are exactly k + 1 among the points A 1 ,...,A m that lie on line L k (k = 1, 2, . . . , n). Prove that there exists a unique polynomial p (x, y) of degree n satisfying p(A 1 )=

1 and p (A j ) = 0 for j = 2, 3, . . . , m.

93. (TUR 4) (SL90-25)

94. (USA 1) Given an integer n > 1 and a real number t ≥ 1 let P be a parallelogram with vertices (0, 0), (0,t), (tF 2n +1 ,tF 2n ), (tF 2n +1 ,tF 2n + t), where F n is the n-th

term of the Fibonacci sequence defined by F 0 = 1, F 1 = 1, and F m +1 =F m +

F m −1 for m ≥ 1. Let L be the number of integral points (i.e. points whose all coordinates are integers) in the interior of P, and let M be the area of P. (a) Prove that for any integral point (a, b) there exists a unique pair of integers ( j, k) such that j(F √ n +1 ,F n ) + k(F √ n ,F n −1 ) = (a, b). (b) Prove that

95. (USA 2) (SL90-26)

96. (USA 3) Given a triangle ABC, points X , Y , Z are on the sides BC, CA, AB, respectively, such that △XY Z ∼ △ABC with ∠X = ∠A, ∠Y = ∠B, and ∠Z = ∠C. Prove that the orthocenter of triangle XY Z is the circumcenter of the triangle ABC .

97. (USA 4) Given a convex hexagon ABCDEF, assume that ∠BCA = ∠DEC = ∠AFB = ∠CBD = ∠EDG. Prove that AB = CD = EF.

244 3 Problems

98. (USS 1) (SL90-27)

99. (USS 2) Given a 10 × 10 chessboard colored in a standard way, prove that for any 46 unit squares without common edges one can choose 30 squares of the same color.

100. (USS 3) (SL90-28) 101. (USS 4) The side lengths of two equilateral triangles ABC and KLM are 1 and

4 respectively. The triangle KLM is located in the interior of the triangle ABC. Denote by Σ the sum of the distances from A to the lines KL, LM, and MK. Find

the position of KLM that maximizes Σ . 102. (USS 5) We call a point (x, y) a lattice point of the coordinate plane if both x and

y are integers. Knowing that the vertices of triangle ABC are all lattice points, and that there exists exactly one lattice point in the interior of △ABC (excluding the sides), prove that S

ABC ≤ 2 .

103. (VNM 1) Find the minimal value of the function

f (x) = 15 − 12cosx + 4 −2

3 sin x

3 sin x − 6cosx. 104. (VNM 2) Let x , y, z ∈ R such that x ≥ y ≥ z > 0. Prove that

105. (YUG 1) Let S and T respectively be the circumcenter and the centroid of the tri- angle ABC. If M is a point in the plane of △ABC such that 90 ◦ ≤ ∠SMT < 180 ◦ , denote by A 1 ,B 1 , and C 1 the intersections of AM, BM, CM with the cir- cumcircle of the triangle ABC respectively. Prove that MA 1 + MB 1 + MC 1 ≥ MA + MB + MC.

106. (YUG 2) Let S be the incenter of the triangle ABC. Let A 1 ,B 1 ,C 1 be the in- tersections of AS, BS, and CS respectively with the circumcircle of the triangle

ABC . Prove that SA 1 + SB 1 + SC 1 ≥ SA + SB + SC.

107. (YUG 3) Let a, b, c, and P be the side lengths and the area of a triangle, respec- tively. Prove that

a 2 +b 2 +c 2 −4 2 3P ·a +b 2 +c 2 ≥2 a 2 2 2 2 2 (b − c) 2 +b (c − a) +c (a − b) . 108. (YUG 4) Let (a 1 ,a 2 ,...a n ) be a permutation of the set {1,2,...,n}. Prove that

a 1 a 2 a n −1 + + ··· +

1 2 n −1

a 2 a 3 + ··· + a n

3.31 IMO 1990 245

3.31.3 Shortlisted Problems

1. (AUS 3) The integer 9 can be written as a sum of two consecutive integers:

9 = 4 + 5. Moreover, it can be written as a sum of (more than one) consecutive positive integers in exactly two ways: 9 = 4 + 5 = 2 + 3 + 4. Is there an integer that can be written as a sum of 1990 consecutive integers and that can be written as a sum of (more than one) consecutive positive integers in exactly 1990 ways?

2. (CAN 1) Given n countries with three representatives each, m committees

A (1), A(2), . . . A(m) are called a cycle if (i) each committee has n members, one from each country;

(ii) no two committees have the same membership; (iii) for i = 1, 2, . . . , m, committee A(i) and committee A(i + 1) have no member in common, where A (m + 1) denotes A(1); (iv) if 1 < |i − j| < m − 1, then committees A(i) and A( j) have at least one member in common. Is it possible to have a cycle of 1990 committees with 11 countries?

3. (CZS 1) IMO 2 On a circle, 2n − 1 (n ≥ 3) different points are given. Find the minimal natural number N with the property that whenever N of the given points are colored black, there exist two black points such that the interior of one of the corresponding arcs contains exactly n of the given 2n − 1 points.

4. (CZS 2) Assume that the set of all positive integers is decomposed into r (dis- joint) subsets A 1 ∪A 2 ∪··· A r = N. Prove that one of them, say A i , has the follow- ing property: There exists a positive m such that for any k one can find numbers

a 1 ,a 2 ,...,a k in A i with 0 <a j +1 −a j ≤ m (1 ≤ j ≤ k − 1).

5. (FRA 1) Given △ABC with no side equal to another side, let G, K, and H be its centroid, incenter, and orthocenter, respectively. Prove that ∠GKH > 90 ◦ .

6. (FRG 2) IMO 5 Two players A and B play a game in which they choose numbers alternately according to the following rule: At the beginning, an initial natural number n 0 > 1 is given. Knowing n 2k , player A may choose any n 2k +1 ∈ N such that

2k ≤n 2k +1 ≤n 2k .

Then player B chooses a number n 2k +2 ∈ N such that

where p is a prime number and r ∈ N. It is stipulated that player A wins the game if he (she) succeeds in choosing the number 1990, and player B wins if he (she) succeeds in choosing 1. For which

natural numbers n 0 can player A manage to win the game, for which n 0 can player B manage to win, and for which n 0 can players A and B each force a tie?

7. (HEL 2) Let f (0) = f (1) = 0 and

246 3 Problems n +2

n = 0, 1, 2, 3, . . .. Show that the numbers f (1989), f (1990), f (1991) are divisible by 13.

n +1

n f 2 (n + 2) = 4 f (n + 1) − 16 f (n) + n · 2 ,

8. (HUN 1) For a given positive integer k denote the square of the sum of its digits

by f 1 (k) and let f n +1 (k) = f 1 (f n (k)).

Determine the value of f 1991 (2 1990 ).

9. (HUN 3) The incenter of the triangle ABC is K. The midpoint of AB is C 1 and that of AC is B 1 . The lines C 1 K and AC meet at B 2 , the lines B 1 K and AB at C 2 . If the areas of the triangles AB 2 C 2 and ABC are equal, what is the measure of angle ∠CAB?

10. (ISL 2) A plane cuts a right circular cone into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the ratio of the volume of the smaller part to the volume of the whole cone.

11. (IND 3 ′ ) IMO 1 Given a circle with two chords AB ,CD that meet at E, let M be

a point of chord AB other than E. Draw the circle through D, E, and M. The tangent line to the circle DEM at E meets the lines BC , AC at F, G, respectively.

Given AM AB = λ , find GE EF .

12. (IRL 1) Let ABC be a triangle and L the line through C parallel to the side AB. Let the internal bisector of the angle at A meet the side BC at D and the line L at E and let the internal bisector of the angle at B meet the side AC at F and the line L at G. If GF = DE, prove that AC = BC.

13. (IRL 2) An eccentric mathematician has a ladder with n rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers a rungs of the ladder, and when he descends, each step he takes covers b rungs of the ladder, where a and b are fixed positive integers. By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again. Find, with proof, the minimum value of n, expressed in terms of a and b.

14. (JPN 2) In the coordinate plane a rectangle with vertices (0, 0), (m, 0), (0, n), (m, n) is given where both m and n are odd integers. The rectangle is partitioned into triangles in such a way that

(i) each triangle in the partition has at least one side (to be called a “good” side) that lies on a line of the form x = j or y = k, where j and k are integers, and the altitude on this side has length 1;

(ii) each “bad” side (i.e., a side of any triangle in the partition that is not a

“good” one) is a common side of two triangles in the partition. Prove that there exist at least two triangles in the partition each of which has two good sides.

15. (MEX 2) Determine for which positive integers k the set

X = {1990,1990 + 1,1990 + 2,...,1990 + k}

3.31 IMO 1990 247 can be partitioned into two disjoint subsets A and B such that the sum of the

elements of A is equal to the sum of the elements of B.

16. (NLD 1) IMO 6 Is there a 1990-gon with the following properties (i) and (ii)? (i) All angles are equal; (ii) The lengths of the 1990 sides form a permutation of the numbers 1 2 ,2 2 , . . ., 1989 2 , 1990 2 .

17. (NLD 3) Unit cubes are made into beads by drilling a hole through them along

a diagonal. The beads are put on a string in such a way that they can move freely in space under the restriction that the vertices of two neighboring cubes are touching. Let A be the beginning vertex and B be the end vertex. Let there be p × q × r cubes on the string (p,q,r ≥ 1).

(a) Determine for which values of p, q, and r it is possible to build a block with dimensions p, q, and r. Give reasons for your answers. (b) The same question as (a) with the extra condition that A = B.

18. (NOR) Let a

2 . Define the function f : Z → Z by

a , b be natural numbers with 1 ≤ a ≤ b, and M = +b

if n ≥ M. Let f 1 (n) = f (n), f i +1 (n) = f ( f i (n)), i = 1, 2, . . . . Find the smallest natural

n − b,

number k such that f k (0) = 0.

19. (POL 1) Let P be a point inside a regular tetrahedron T of unit volume. The four planes passing through P and parallel to the faces of T partition T into 14 pieces. Let f (P) be the joint volume of those pieces that are neither a tetrahedron nor a parallelepiped (i.e., pieces adjacent to an edge but not to a vertex). Find the exact bounds for f (P) as P varies over T .

20. (POL 3) Prove that every integer k greater than 1 has a multiple that is less than

k 4 and can be written in the decimal system with at most four different digits.

21. (ROU 1 ′ ) Let n be a composite natural number and p a proper divisor of n . Find the binary representation of the smallest natural number N such that

(1+2 p +2 n −p )N−1 2 n

is an integer.

22. (ROU 4) Ten localities are served by two international airlines such that there exists a direct service (without stops) between any two of these localities and all airline schedules offer round-trip service between the cities they serve. Prove that at least one of the airlines can offer two disjoint round trips each containing an odd number of landings.

23. (ROU 5) IMO 3 Find all positive integers n having the property that 2 n +1 n 2 is an integer.

24. (THA 2) Let a , b, c, d be nonnegative real numbers such that ab+ bc+ cd +da =

1. Show that

248 3 Problems

b +c+d a +c+d a +b+d a +b+c ≥ 3

be the set of positive rational numbers. Construct a func- tion f : Q +

25. (TUR 4) IMO 4 Let Q +

→Q + such that

f (x)

f (x f (y)) =

for all x , y in Q + .

26. (USA 2) Let P be a cubic polynomial with rational coefficients, and let q 1 ,q 2 ,q 3 , . . . be a sequence of rational numbers such that q n = P(q n +1 ) for all n ≥ 1. Prove that there exists k ≥ 1 such that for all n ≥ 1, q n +k =q n .

27. (USS 1) Find all natural numbers n for which every natural number whose decimal representation has n − 1 digits 1 and one digit 7 is prime.

28. (USS 3) Prove that in the coordinate plane it is impossible to draw a closed broken line satisfying the following conditions:

(i) the coordinates of each vertex are rational; (ii) the length of each of its edges is 1; (iii) the line has an odd number of vertices.

3.32 IMO 1991 249