The Thirty-Third IMO Moscow, Russia, July 10–21, 1992

3.33 The Thirty-Third IMO Moscow, Russia, July 10–21, 1992

3.33.1 Contest Problems

First Day (July 15)

1. Find all integer triples (p, q, r) such that 1 < p < q < r and (p − 1)(q − 1)(r − 1) is a divisor of (pqr − 1).

2. Find all functions f : R → R such that

f (x 2 + f (y)) = y + f (x) 2 for all x , y in R.

3. Given nine points in space, no four of which are coplanar, find the minimal natural number n such that for any coloring with red or blue of n edges drawn between these nine points there always exists a triangle having all edges of the same color.

Second Day (July 16)

4. In the plane, let there be given a circle C, a line l tangent to C, and a point M on l. Find the locus of points P that has the following property: There exist two points Q and R on l such that M is the midpoint of QR and C is the incircle of PQR .

5. Let V be a finite subset of Euclidean space consisting of points (x, y, z) with in- teger coordinates. Let S 1 ,S 2 ,S 3 be the projections of V onto the yz , xz, xy planes, respectively. Prove that

|V | 2 ≤ |S

1 ||S 2 ||S 3 |

( |X| denotes the number of elements of X).

6. For each positive integer n, denote by s (n) the greatest integer such that for all positive integer k ≤ s(n), n 2 can be expressed as a sum of squares of k positive integers.

(a) Prove that s (n) ≤ n 2 − 14 for all n ≥ 4. (b) Find a number n such that s (n) = n 2 − 14.

(c) Prove that there exist infinitely many positive integers n such that s (n) = n 2 − 14.

3.33.2 Longlisted Problems

1. (AUS 1) Points D and E are chosen on the sides AB and AC of the triangle ABC in such a way that if F is the intersection point of BE and CD, then AE + EF =

AD + DF. Prove that AC + CF = AB + BF.

254 3 Problems

2. (AUS 2) (SL92-1). Original formulation. Let m be a positive integer and x 0 ,y 0 integers such that x 0 ,y 0 are relatively prime, y 0 divides x 2 0 2 + m, and x 0 divides y 0 + m.

Prove that there exist positive integers x and y such that x and y are relatively prime, y divides x 2 + m, x divides y 2 + m, and x + y ≤ m + 1.

3. (AUS 3) Let ABC be a triangle, O its circumcenter, S its centroid, and H its orthocenter. Denote by A 1 ,B 1 , and C 1 the centers of the circles circumscribed about the triangles CHB, CHA, and AHB, respectively. Prove that the triangle ABC is congruent to the triangle A 1 B 1 C 1 and that the nine-point circle of △ABC

is also the nine-point circle of △A 1 B 1 C 1 .

4. (CAN 1) Let p, q, and r be the angles of a triangle, and let a = sin 2p, b = sin 2q, and c = sin 2r. If s = (a + b + c)/2, show that

s (s − a)(s − b)(s − c) ≥ 0.

When does equality hold?

5. (CAN 2) Let I , H, O be the incenter, centroid, and circumcenter of the non- isosceles triangle ABC. Prove that AI kHO if and only if ∡BAC = 120 ◦ .

6. (CAN 3) Suppose that n numbers x 1 ,x 2 ,...,x n are chosen randomly from the set

1 +x 2 + ··· + x n ≡ 0 (mod 5) is at least 1 /5.

2 2 {1,2,3,4,5}. Prove that the probability that x 2

7. (CAN 4) Let X be a bounded, nonempty set of points in the Cartesian plane. Let

f (X) be the set of all points that are at a distance of at most 1 from some point in

X . Let f n (X) = f ( f (. . . ( f (X )). . . )) (n times). Show that f n (X ) becomes “more circular” as n gets larger. In other words, if r n = sup{radii of circles contained in f n (X )} and R n = inf{radii of circles containing f n (X )}, then show that R n /r n gets arbitrarily close to 1 as n becomes arbitrarily large.

8. (CHN 1) (SL92-2).

9. (CHN 2) (SL92-3).

10. (CHN 3) (SL92-4).

11. (COL 1) Let φ (n, m), m 6= 1, be the number of positive integers less than or equal to n that are coprime with m. Clearly, φ (m, m) = φ (m), where φ (m) is Euler’s phi function. Find all integers m that satisfy the following inequality:

φ (n, m) φ (m) ≥

for every positive integer n.

12. (COL 2) Given a triangle ABC such that the circumcenter is in the interior of the incircle, prove that the triangle ABC is acute-angled.

13. (COL 3) (SL92-5).

3.33 IMO 1992 255

14. (FIN 1) Integers a 1 ,a 2 ,...,a n satisfy |a k | = 1 and

∑ a k a k +1 a k +2 a k +3 = 2,

k =1

where a n +j =a j . Prove that n 6= 1992.

15. (FIN 2) Prove that there exist 78 lines in the plane such that they have exactly 1992 points of intersection.

16. (FIN 3) Find all triples (x, y, z) of integers such that

17. (FRA 1) (SL92-20).

18. (FRG 1) Fibonacci numbers are defined as follows: F 1 =F 2 = 1, F n +2 =F n +1 +

be the number of words that consist of n letters 0 or 1 and contain no two letters 1 at distance two from each other. Express a n in terms of Fibonacci numbers.

F n ,n ≥ 1. Let a n

19. (FRG 2) Denote by a n the greatest number that is not divisible by 3 and that divides n. Consider the sequence s 0 = 0, s n =a 1 +a 2 + ··· + a n ,n ∈ N. Denote by A (n) the number of all sums s k (0 ≤k≤3 n ,k∈N 0 ) that are divisible by 3. Prove the formula

A (n) = 3 n −1 +2·3 (n/2)−1 cos (n π /6), n ∈N 0 .

20. (FRG 3) Let X and Y be two sets of points in the plane and M be a set of seg- ments connecting points from X and Y . Let k be a natural number. Prove that the segments from M can be painted using k colors in such a way that for any point x ∈ X ∪Y and two colors α and β ( α 6= β ), the difference between the number of α -colored segments and the number of β -colored segments originating in X is less than or equal to 1.

21. (UNK 1) Prove that if x , y, z > 1 and 1 x + 1 y + 1 z = 2, then √

x +y+z≥ x −1+ y − 1+ z − 1.

22. (UNK 2) (SL92-21).

23. (HKG 1) An Egyptian number is a positive integer that can be expressed as

a sum of positive integers, not necessarily distinct, such that the sum of their reciprocals is 1. For example, 32 = 2 + 3 + 9 + 18 is Egyptian because 1 2 + 1 3 +

9 + 1 18 = 1. Prove that all integers greater than 23 are Egyptian.

24. (ISL 1) Let Q + denote the set of nonnegative rational numbers. Show that there exists exactly one function f : Q + →Q + satisfying the following conditions:

(i) if 0 <q< 1 2 , then f (q) = 1 + f

1 −2q ;

256 3 Problems (ii) if 1 < q ≤ 2, then f (q) = 1 + f (q + 1);

(iii) f (q) f (1/q) = 1 for all q ∈ Q + . Find the smallest rational number q ∈Q + such that f (q) = 19/92.

25. (IND 1) (a) Show that the set N of all natural numbers can be partitioned into three disjoint subsets A, B, and C satisfying the following conditions:

where HK stands for {hk | h ∈ H, k ∈ K} for any two subsets H, K of N, and H 2 denotes HH. (b) Show that for every such partition of N, min{n ∈ N | n ∈ A and n + 1 ∈ A} is less than or equal to 77.

26. (IND 2) (SL92-6).

27. (IND 3) Let ABC be an arbitrary scalene triangle. Define Σ to be the set of all circles y that have the following properties:

(i) y meets each side of △ABC in two (possibly coincident) points; (ii) if the points of intersection of y with the sides of the triangle are labeled by P, Q, R, S, T , U, with the points occurring on the sides in orders

B (B, P, Q,C), B(C, R, S, A), B(A, T,U, B), then the following relations of parallelism hold: T S kBC; PUkCA; RQkAB. (In the limiting cases, some of the conditions of parallelism will hold vacuously; e.g., if A lies on the circle y , then T , S both coincide with A and the relation T S kBC holds vacuously.)

(a) Under what circumstances is Σ nonempty? (b) Assuming that Σ is nonempty, show how to construct the locus of centers

of the circles in the set Σ . (c) Given that the set Σ has just one element, deduce the size of the largest angle of △ABC. (d) Show how to construct the circles in Σ that have, respectively, the largest and the smallest radii.

28. (IND 4) (SL92-7). Alternative formulation. Two circles G 1 and G 2 are inscribed in a segment of a circle G and touch each other externally at a point W . Let A be a point of inter- section of a common internal tangent to G 1 and G 2 with the arc of the segment, and let B and C be the endpoints of the chord. Prove that W is the incenter of the triangle ABC.

29. (IND 5) (SL92-8).

30. (IND 6) Let P n = (19 + 92)(19 2 + 92 2 ) ···(19 n + 92 n ) for each positive integer n . Determine, with proof, the least positive integer m, if it exists, for which P m is divisible by 33 33 .

31. (IRL 1) (SL92-19).

3.33 IMO 1992 257

32. (IRL 2) Let S n = {1,2,...,n} and f n :S n →S n

be defined inductively as follows:

f 1 (1) = 1, f n (2 j) = j ( j = 1, 2, . . . , [n/2]) and (i) if n = 2k (k ≥ 1), then f n (2 j − 1) = f k ( j) + k ( j = 1, 2, . . . , k); (ii) if n = 2k + 1 (k ≥ 1), then f n (2k + 1) = k + f k +1 (1), f n (2 j − 1) = k +

f k +1 ( j + 1) ( j = 1, 2, . . . , k). Prove that f n (x) = x if and only if x is an integer of the form

(2n + 1)(2 d − 1)

2 d +1 −1

for some positive integer d.

33. (IRL 3) Let a , b, c be positive real numbers and p, q, r complex numbers. Let S

be the set of all solutions (x, y, z) in C of the system of simultaneous equations ax + by + cz = p,

ax 2 + by 2 + cz 2 = q, ax 3 + bx 3 + cx 3 = r.

Prove that S has at most six elements.

34. (IRL 4) Let a , b, c be integers. Prove that there are integers p 1 ,q 1 ,r 1 ,p 2 ,q 2 ,r 2 such that

a =q 1 r 2 −q 2 r 1 ,b=r 1 p 2 −r 2 p 1 ,c=p 1 q 2 −p 2 q 1 .

35. (IRN 1) (SL92-9).

36. (IRN 2) Find all rational solutions of

a 2 +c 2 + 17(b 2 +d 2 ) = 21,

ab + cd = 2.

37. (IRN 3) Let the circles C 1 ,C 2 , and C 3 be orthogonal to the circle C and intersect each other inside C forming acute angles of measures A, B, and C. Show that

A +B+C< π .

38. (ITA 1) (SL92-10).

39. (ITA 2) Let n ≥ 2 be an integer. Find the minimum k for which there exists

a partition of {1,2,...,k} into n subsets X 1 ,X 2 ,...,X n such that the following condition holds: for any i , j, 1 ≤ i < j ≤ n, there exist x 1 ∈X 1 ,x 2 ∈X 2 such that |x i −x j | = 1.

40. (ITA 3) The colonizers of a spherical planet have decided to build N towns, each having area 1 /1000 of the total area of the planet. They also decided that any two points belonging to different towns will have different latitude and different longitude. What is the maximal value of N?

41. (JPN 1) Let S be a set of positive integers n 1 ,n 2 ,...,n 6 and let n ( f ) de- note the number n 1 n f (1) +n 2 n f (2) + ··· + n 6 n f (6) , where f is a permutation of {1,2,...,6}. Let

258 3 Problems

Ω = {n( f ) | f is a permutation of {1,2,...,6}}. Give an example of positive integers n 1 ,...,n 6 such that Ω contains as many

elements as possible and determine the number of elements of Ω .

42. (JPN 2) (SL92-11).

43. (KOR 1) Find the number of positive integers n satisfying φ (n) | n such that

n ∑ −1

m =1

What is the largest number among them? As usual, φ (n) is the number of positive integers less than or equal to n and relatively prime to n. 7

44. (KOR 2) (SL92-16).

45. (KOR 3) Let n be a positive integer. Prove that the number of ways to express n as a sum of distinct positive integers (up to order) and the number of ways to express n as a sum of odd positive integers (up to order) are the same.

46. (KOR 4) Prove that the sequence 5 , 12, 19, 26, 33, . . . contains no term of the form 2 n − 1.

47. (KOR 5) Find the largest integer not exceeding ∏ +2 1992 3n n =1 3n +1 .

48. (MNG 1) Find all the functions f : R + → R satisfying the identity

2 ·f 2 , x, y ∈ R , where α , β are given real numbers.

f (x) f (y) = y + α β ·f +x

49. (MNG 2) Given real numbers x i (i = 1, 2, . . . , 4x + 2) such that

4x +2

i ∑ +1 (−1) x i x i +1 = 4m (x 1 =x 4k +3 ),

i =1

prove that it is possible to choose numbers x k 1 ,...,x k 6 such that

6 ∑ i (−1) x

k i x k i +1 > m (x k 1 =x k 7 ).

i =1

50. (MNG 3) Let N be a point inside the triangle ABC. Through the midpoints of the segments AN, BN, and CN the lines parallel to the opposite sides of △ABC are constructed. Let A N ,B N , and C N

be the intersection points of these lines. If N is the orthocenter of the triangle ABC, prove that the nine-point circles of △ABC and △A N B N C N coincide.

Remark. The statement of the original problem was that the nine-point circles of the triangles A N B N C N and A M B M C M coincide, where N and M are the orthocen- ter and the centroid of △ABC. This statement is false.

7 The problem in this formulation is senseless. The correct formulation could be, “Find ... such that ∑ ∞

n m =1

m − n −1 m = 1992 . . . .”

3.33 IMO 1992 259

51. (NLD 1) (SL92-12).

52. (NLD 2) Let n be an integer > 1. In a circular arrangement of n lamps L 0 ,...,L n −1 , each one of which can be either ON or OFF, we start with the situa- tion that all lamps are ON, and then carry out a sequence of steps, Step 0 , Step 1 ,... . If L j −1 ( j is taken mod n) is ON, then Step j changes the status of L j (it goes from ON to OFF or from OFF to ON) but does not change the status of any of the other lamps. If L j −1 is OFF, then Step j does not change anything at all. Show that:

(a) There is a positive integer M (n) such that after M(n) steps all lamps are

ON again. (b) If n has the form 2 k , then all lamps are ON after n 2 − 1 steps. (c) If n has the form 2 k + 1, then all lamps are ON after n 2 − n + 1 steps.

53. (NZL 1) (SL92-13).

54. (POL 1) Suppose that n > m ≥ 1 are integers such that the string of digits 143 occurs somewhere in the decimal representation of the fraction m /n. Prove that n > 125

55. (POL 2) (SL92-14).

56. (POL 3) A directed graph (any two distinct vertices joined by at most one directed line) has the following property: If x, u, and v are three distinct vertices such that x → u and x → v, then u → w and v → w for some vertex w. Suppose that x → u → y → ··· → z is a path of length n, that cannot be extended to the right (no arrow goes away from z). Prove that every path beginning at x arrives after n steps at z.

57. (POL 4) For positive numbers a , b, c define A = (a + b + c)/3, G = (abc) 1 /3 ,

H = 3/(a −1 +b −1 +c −1 ). Prove that

for every a , b, c > 0.

58. (POR 1) Let ABC be a triangle. Denote by a, b, and c the lengths of the sides opposite to the angles A, B, and C, respectively. Prove that 8

bc sin A + sin B + sinC

a +b+c cos (A/2) sin(B/2) sin(C/2)

59. (PRK 1) Let a regular 7-gon A 0 A 1 A 2 A 3 A 4 A 5 A 6 be inscribed in a circle. Prove that for any two points P, Q on the arc A 0 A 6 the following equality holds:

∑ i (−1) PA

i = ∑ (−1) QA i .

i =0

i =0

8 The statement of the problem is obviously wrong, and the authors couldn’t determine a suitable alteration of the formulation which would make the problem correct. We put it

here only for completeness of the problem set.

260 3 Problems

60. (PRK 2) (SL92-15).

61. (PRK 3) There are a board with 2n · 2n (= 4n 2 ) squares and 4n 2 − 1 cards numbered with different natural numbers. These cards are put one by one on each of the squares. One square is empty. We can move a card to an empty square from one of the adjacent squares (two squares are adjacent if they have a common edge). Is it possible to exchange two cards on two adjacent squares of

a column (or a row) in a finite number of movements?

62. (ROU 1) Let c 1 ,...,c n (n ≥ 2) be real numbers such that 0 ≤ ∑c i ≤ n. Prove that there exist integers k 1 ,...,k n such that ∑ k i = 0 and 1 − n ≤ c i + nk i ≤ n for every i = 1, . . . , n.

63. (ROU 2) Let a and b be integers. Prove that 2a 2 b −1 2 +2 is not an integer.

64. (ROU 3) For any positive integer n consider all representations n =a 1 +···+a k , where a 1 >a 2 > ··· > a k > 0 are integers such that for all i ∈ {1,2,...,k − 1}, the number a i is divisible by a i +1 . Find the longest such representation of the number 1992.

65. (SAF 1) If A, B, C, and D are four distinct points in space, prove that there is a plane P on which the orthogonal projections of A, B, C, and D form a parallelo- gram (possibly degenerate).

66. (ESP 1) A circle of radius ρ is tangent to the sides AB and AC of the triangle ABC , and its center K is at a distance p from BC.

(a) Prove that a (p− ρ ) = 2s(r − ρ ), where r is the inradius and 2s the perimeter of ABC. (b) Prove that if the circle intersect BC at D and E, then

4 rr 1 ( ρ − r)(r 1 − ρ )

DE =

(r 1 − r) where r 1 is the exradius corresponding to the vertex A.

67. (ESP 2) In a triangle, a symmedian is a line through a vertex that is symmetric to the median with respect to the internal bisector (all relative to the same vertex).

In the triangle ABC, the median m a meets BC at A ′ and the circumcircle again at

A 1 . The symmedian s a meets BC at M and the circumcircle again at A 2 . Given that the line A 1 A 2 contains the circumcenter O of the triangle, prove that: AA ′

68. (ESP 3) Show that the numbers tan (r π /15), where r is a positive integer less than 15 and relatively prime to 15, satisfy

x 8 − 92x 6 + 134x 4 − 28x 2 + 1 = 0.

69. (SWE 1) (SL92-17).

3.33 IMO 1992 261

70. (THA 1) Let two circles A and B with unequal radii r and R, respectively, be tangent internally at the point A 0 . If there exists a sequence of distinct circles (C n ) such that each circle is tangent to both A and B, and each circle C n +1 touches circle C n at the point A n , prove that

4 π Rr

∑ |A n +1 A n |<

n =1

R +r

71. (THA 2) Let P 1 (x, y) and P 2 (x, y) be two relatively prime polynomials with complex coefficients. Let Q (x, y) and R(x, y) be polynomials with complex co- efficients and each of degree not exceeding d. Prove that there exist two integers

A 1 ,A 2 not simultaneously zero with |A i | ≤ d + 1 (i = 1,2) and such that the polynomial A 1 P 1 (x, y) + A 2 P 2 (x, y) is coprime to Q(x, y) and R(x, y).

72. (TUR 1) In a school six different courses are taught: mathematics, physics, biology, music, history, geography. The students were required to rank these courses according to their preferences, where equal preferences were allowed. It turned out that:

(i) mathematics was ranked among the most preferred courses by all students; (ii) no student ranked music among the least preferred ones; (iii) all students preferred history to geography and physics to biology; and (iv) no two rankings were the same.

Find the greatest possible value for the number of students in this school.

73. (TUR 2) Let {A n | n = 1,2,...} be a set of points in the plane such that for each n , the disk with center A n and radius 2 n contains no other point A j . For any given positive real numbers a < b and R, show that there is a subset G of the plane satisfying:

(i) the area of G is greater than or equal to R; (ii) for each point P in G, a <∑ ∞

1 n =1 |A n P | < b.

74. (TUR 3) Let S = 1992 m | n,m ∈ Z . Show that every real number x ≥ 0 is an accumulation point of S.

75. (TWN 1) A sequence {a n } of positive integers is defined by

2 , n ∈ N.

Determine the positive integers that occur in the sequence.

76. (TWN 2) Given any triangle ABC and any positive integer n, we say that n is

a decomposable number for triangle ABC if there exists a decomposition of the triangle ABC into n subtriangles with each subtriangle similar to △ABC. Deter- mine the positive integers that are decomposable numbers for every triangle.

77. (TWN 3) Show that if 994 integers are chosen from 1 , 2, . . . , 1992 and one of the chosen integers is less than 64, then there exist two among the chosen integers such that one of them is a factor of the other.

262 3 Problems

78. (USA 1) Let F n

be the nth Fibonacci number, defined by F 1 =F 2 = 1 and F n =

F n −1 +F n −2 for n > 2. Let A 0 ,A 1 ,A 2 , . . . be a sequence of points on a circle of

radius 1 such that the minor arc from A k −1 to A k runs clockwise and such that

4F +1

µ (A

for k ≥ 1, where µ (XY ) denotes the radian measure of the arc XY in the clock- wise direction. What is the limit of the radian measure of arc A 0 A n as n ap- proaches infinity?

79. (USA 2) (SL92-18).

80. (USA 3) Given a graph with n vertices and a positive integer m that is less than n , prove that the graph contains a set of m + 1 vertices in which the difference between the largest degree of any vertex in the set and the smallest degree of any vertex in the set is at most m − 1.

81. (USA 4) Suppose that points X ,Y, Z are located on sides BC, CA, and AB, re- spectively, of △ABC in such a way that △XY Z is similar to △ABC. Prove that the orthocenter of △XY Z is the circumcenter of △ABC.

82. (VNM 1) Let f (x) = x m +a 1 x m −1 + ··· + a m −1 x +a m and g (x) = x n +b 1 x n −1 + ··· + b n −1 +b n

be two polynomials with real coefficients such that for each real number x, f (x) is the square of an integer if and only if so is g(x). Prove that if n + m > 0, then there exists a polynomial h(x) with real coefficients such that

f (x) · g(x) = (h(x)) 2 .

3.33.3 Shortlisted Problems

1. (AUS 2) Prove that for any positive integer m there exists an infinite number of pairs of integers (x, y) such that (i) x and y are relatively prime; (ii) y divides

x 2 + m; (iii) x divides y 2 + m.

be the set of all nonnegative real numbers. Given two pos- itive real numbers a and b, suppose that a mapping f : R + →R + satisfies the functional equation

2. (CHN 1) Let R +

f ( f (x)) + a f (x) = b(a + b)x. Prove that there exists a unique solution of this equation.

3. (CHN 2) The diagonals of a quadrilateral ABCD are perpendicular: AC ⊥BD. Four squares, ABEF , BCGH,CDIJ, DAKL, are erected externally on its sides. The intersection points of the pairs of straight lines CL , DF; DF, AH; AH, BJ;

BJ ,CL are denoted by P 1 ,Q 1 ,R 1 ,S 1 , respectively, and the intersection points of the pairs of straight lines AI , BK; BK,CE; CE, DG; DG, AI are denoted by P 2 ,Q 2 ,R 2 ,S 2 , respectively. Prove that P 1 Q 1 R 1 S 1 ∼ =P 2 Q 2 R 2 S 2 .

4. (CHN 3) IMO 3 Given nine points in space, no four of which are coplanar, find the minimal natural number n such that for any coloring with red or blue of n edges

3.33 IMO 1992 263 drawn between these nine points there always exists a triangle having all edges

of the same color.

5. (COL 3) Let ABCD be a convex quadrilateral such that AC = BD. Equilateral triangles are constructed on the sides of the quadrilateral. Let O 1 ,O 2 ,O 3 ,O 4 be the centers of the triangles constructed on AB , BC,CD, DA respectively. Show

that O 1 O 3 is perpendicular to O 2 O 4 .

6. (IND 2) IMO 2 Find all functions f : R → R such that

f (x 2 + f (y)) = y + f (x) 2 for all x , y in R.

7. (IND 4) Circles G ,G 1 ,G 2 are three circles related to each other as follows: Circles G 1 and G 2 are externally tangent to one another at a point W and both these circles are internally tangent to the circle G. Points A , B,C are located on the circle G as follows: Line BC is a direct common tangent to the pair of circles

G 1 and G 2 , and line WA is the transverse common tangent at W to G 1 and G 2 , with W and A lying on the same side of the line BC. Prove that W is the incenter of the triangle ABC.

8. (IND 5) Show that in the plane there exists a convex polygon of 1992 sides satisfying the following conditions:

(i) its side lengths are 1 , 2, 3, . . . , 1992 in some order; (ii) the polygon is circumscribable about a circle.

Alternative formulation. Does there exist a 1992-gon with side lengths 1, 2, 3, . . ., 1992 circumscribed about a circle? Answer the same question for a 1990- gon.

9. (IRN 1) Let f (x) be a polynomial with rational coefficients and α

be a real number such that α 3 − α =f( α ) 3 −f( α ) = 33 1992 . Prove that for each n ≥ 1,

(f (n) ( α )) 3 −f (n) ( α ) = 33 1992 , where f (n) (x) = f ( f (. . . f (x))), and n is a positive integer.

10. (ITA 1) IMO 5 Let V be a finite subset of Euclidean space consisting of points (x, y, z) with integer coordinates. Let S 1 ,S 2 ,S 3 be the projections of V onto the yz , xz, xy planes, respectively. Prove that

|V | 2 ≤ |S

1 ||S 2 ||S 3 |

( |X| denotes the number of elements of X).

11. (JPN 2) In a triangle ABC, let D and E be the intersections of the bisectors of ∠ABC and ∠ACB with the sides AC , AB, respectively. Determine the angles ∠A , ∠B, ∠C if

∡BDE = 24 ◦ , ∡CED = 18 ◦ .

12. (NLD 1) Let f , g, and a be polynomials with real coefficients, f and g in one variable and a in two variables. Suppose

264 3 Problems

f (x) − f (y) = a(x,y)(g(x) − g(y)) for all x , y ∈ R. Prove that there exists a polynomial h with f (x) = h(g(x)) for all x ∈ R.

13. (NZL 1) IMO 1 Find all integer triples (p, q, r) such that 1 < p < q < r and (p −

1 )(q − 1)(r − 1) is a divisor of (pqr − 1).

14. (POL 2) For any positive integer x define

g (x) = greatest odd divisor of x,

x /2 + x/g(x), if x is even;

f (x) =

2 (x+1)/2 , if x is odd. Construct the sequence x 1 = 1, x n +1 = f (x n ). Show that the number 1992 ap-

pears in this sequence, determine the least n such that x n = 1992, and determine whether n is unique.

15. (PRK 2) Does there exist a set M with the following properties? (i) The set M consists of 1992 natural numbers. (ii) Every element in M and the sum of any number of elements have the form m k (m, k ∈ N, k ≥ 2).

16. (KOR 2) Prove that N = 5 125 5 −1 25 −1 is a composite number.

17. (SWE 1) Let α (n) be the number of digits equal to one in the binary represen- tation of a positive integer n. Prove that:

(a) the inequality α (n 2 )≤ 1 2 α (n)( α (n) + 1) holds; (b) the above inequality is an equality for infinitely many positive integers;

(c) there exists a sequence (n i ) ∞ 1 such that α (n 2 i )/ α (n i ) → 0 as i → ∞. Alternative parts: Prove that there exists a sequence (n i ) ∞ 1 such that α (n 2 i )/ α (n i ) tends to

(d) ∞; (e) an arbitrary real number γ ∈ (0,1);

(f) an arbitrary real number γ ≥ 0.

18. (USA 2) Let [x] denote the greatest integer less than or equal to x. Pick any x 1 in [0, 1) and define the sequence x 1 ,x 2 ,x 3 , . . . by x n +1 = 0 if x n = 0 and x n +1 =

1 /x n − [1/x n ] otherwise. Prove that

1 +x 2 + ··· + x n <

F F + ··· +

2 3 F n +1 where F 1 =F 2 = 1 and F n +2 =F n +1 +F n for n ≥ 1.

19. (IRL 1) Let f (x) = x 8 + 4x 6 + 2x 4 + 28x 2 + 1. Let p > 3 be a prime and suppose there exists an integer z such that p divides f (z). Prove that there exist integers z 1 ,z 2 ,...,z 8 such that if

g (x) = (x − z 1 )(x − z 2 ) ···(x − z 8 ), then all coefficients of f (x) − g(x) are divisible by p.

3.33 IMO 1992 265

20. (FRA 1) IMO 4 In the plane, let there be given a circle C, a line l tangent to C, and

a point M on l. Find the locus of points P that have the following property: There exist two points Q and R on l such that M is the midpoint of QR and C is the incircle of PQR.

21. (UNK 2) IMO 6 For each positive integer n, denote by s (n) the greatest integer such that for all positive integers k ≤ s(n), n 2 can be expressed as a sum of squares of k positive integers.

(a) Prove that s (n) ≤ n 2 − 14 for all n ≥ 4. (b) Find a number n such that s (n) = n 2 − 14.

(c) Prove that there exist infinitely many positive integers n such that s (n) = n 2 − 14.

266 3 Problems