The Ninth IMO Cetinje, Yugoslavia, July 2–13, 1967

3.9 The Ninth IMO Cetinje, Yugoslavia, July 2–13, 1967

3.9.1 Contest Problems

First Day (July 5)

1. ABCD is a parallelogram; AB = a, AD = 1, α is the size of ∠DAB, and the three angles of the triangle ABD are acute. Prove that the four circles K A ,K B , K C ,K D , each of radius 1, whose centers are the vertices A, B, C, D, cover the √ parallelogram if and only if a ≤ cos α +

3 sin α .

2. Exactly one side of a tetrahedron is of length greater than 1. Show that its volume is less than or equal to 1 /8.

3. Let k, m, and n be positive integers such that m + k + 1 is a prime number greater than n + 1. Write c s for s (s + 1). Prove that the product (c m +1 −c k )(c m +2 −

c k ) ···(c m +n −c k ) is divisible by the product c 1 c 2 ··· c n . Second Day (July 6)

4. The triangles A 0 B 0 C 0 and A ′ B ′ C ′ have all their angles acute. Describe how to construct one of the triangles ABC, similar to A ′ B ′ C ′ and circumscribing A 0 B 0 C 0 (so that A, B, C correspond to A ′ ,B ′ ,C ′ , and AB passes through C 0 , BC through

A 0 , and CA through B 0 ). Among these triangles ABC describe, and prove, how to construct the triangle with the maximum area.

5. Consider the sequence (c n ):

where a 1 ,a 2 ,...,a 8 are real numbers, not all equal to zero. Given that among the numbers of the sequence (c n ) there are infinitely many equal to zero, determine all the values of n for which c n = 0.

6. In a sports competition lasting n days there are m medals to be won. On the first day, one medal and 1 /7 of the remaining m − 1 medals are won. On the second day, 2 medals and 1 /7 of the remainder are won. And so on. On the nth day exactly n medals are won. How many days did the competition last and what was the total number of medals?

3.9.2 Longlisted Problems

1. (BGR 1) Prove that all numbers in the sequence

42 3 Problems 107811 110778111 111077781111

are perfect cubes.

2. (BGR 2) Prove that 1 n 2 + 3 1 2 n + 1 6 ≥ (n!) 2 /n (n is a positive integer) and that equality is possible only in the case n = 1.

2 + x 16 , where x ∈ (0, π /2).

3. (BGR 3) Prove the trigonometric inequality cos x x 2 <1− 4

4. (BGR 4) Suppose medians m a and m b of a triangle are orthogonal. Prove that: (a) The medians of that triangle correspond to the sides of a right-angled trian- gle. (b) The inequality

5 (a 2 +b 2 −c 2 ) ≥ 8ab

is valid, where a, b, and c are side lengths of the given triangle.

5. (BGR 5) Solve the system

x 2 + x − 1 = y, y 2 + y − 1 = z, z 2 + z − 1 = x.

6. (BGR 6) Solve the system

|x + y| + |1 − x| = 6, |x + y + 1| + |1 − y| = 4.

7. (CZS 1) Find all real solutions of the system of equations

8. (CZS 2) IMO 1 ABCD is a parallelogram; AB = a, AD = 1, α is the size of ∠DAB, and the three angles of the triangle ABD are acute. Prove that the four circles K A , K B ,K C ,K D , each of radius 1, whose centers are the vertices A, B, C, D, cover the √ parallelogram if and only if a ≤ cos α +

3 sin α .

9. (CZS 3) The circle k and its diameter AB are given. Find the locus of the cen- ters of circles inscribed in the triangles having one vertex on AB and two other vertices on k.

10. (CZS 4) The square ABCD is to be decomposed into n triangles (nonoverlap- ping) all of whose angles are acute. Find the smallest integer n for which there exists a solution to this problem and construct at least one decomposition for this n . Answer whether it is possible to ask additionally that (at least) one of these triangles has a perimeter less than an arbitrarily given positive number.

3.9 IMO 1967 43

11. (CZS 5) Let n be a positive integer. Find the maximal number of noncongruent triangles whose side lengths are integers less than or equal to n.

12. (CZS 6) Given a segment AB of the length 1, define the set M of points in the following way: it contains the two points A, B, and also all points obtained from

A , B by iterating the following rule: for every pair of points X, Y in M, the set M also contains the point Z of the segment XY for which Y Z = 3XZ. (a) Prove that the set M consists of points X from the segment AB for which the distance from the point A is either

, where n, k are nonnegative integers.

(b) Prove that the point X 0 for which AX 0 = 1/2 = X 0 B does not belong to the set M.

13. (GDR 1) Find whether among all quadrilaterals whose interiors lie inside a semicircle of radius r there exists one (or more) with maximal area. If so, deter- mine their shape and area.

14. (GDR 2) Which fraction p √ /q, where p, q are positive integers less than 100, is closest to 2? Find all digits after the decimal point in the decimal representa- √ tion of this fraction that coincide with digits in the decimal representation of 2 (without using any tables).

15. (GDR 3) Suppose tan α = p/q, where p and q are integers and q 6= 0. Prove that the number tan β for which tan 2 β = tan 3 α is rational only when p 2 +q 2 is the square of an integer.

16. (GDR 4) Prove the following statement: If r 1 and r 2 are real numbers whose quotient is irrational, then any real number x can be approximated arbitrarily well by numbers of the form z k 1 ,k 2 =k 1 r 1 +k 2 r 2 ,k 1 ,k 2 integers; i.e., for every real number x and every positive real number p two integers k 1 and k 2 can be

found such that |x − (k 1 r 1 +k 2 r 2 )| < p.

17. (UNK 1) IMO 3 Let k, m, and n be positive integers such that m + k + 1 is a prime number greater than n + 1. Write c s for s (s + 1). Prove that the product (c m +1 −

c k )(c m +2 −c k ) ···(c m +n −c k ) is divisible by the product c 1 c 2 ···c n .

18. (UNK 5) If x is a positive rational number, show that x can be uniquely expressed in the form

where a 1 ,a 2 , . . . are integers, 0 ≤ a n ≤ n − 1 for n > 1, and the series terminates. Show also that x can be expressed as the sum of reciprocals of different integers,

each of which is greater than 10 6 .

19. (UNK 6) The n points P 1 ,P 2 ,...,P n are placed inside or on the boundary of a disk of radius 1 in such a way that the minimum distance d n between any two

44 3 Problems of these points has its largest possible value D n . Calculate D n for n = 2 to 7 and

justify your answer.

20. (HUN 1) In space, n points (n ≥ 3) are given. Every pair of points determines some distance. Suppose all distances are different. Connect every point with the nearest point. Prove that it is impossible to obtain a polygonal line in such a way.

21. (HUN 2) Without using any tables, find the exact value of the product π

7 π P = cos

22. (HUN 3) The distance between the centers of the circles k 1 and k 2 with radii r is equal to r. Points A and B are on the circle k 1 , symmetric with respect to the line connecting the centers of the circles. Point P is an arbitrary point on k 2 . Prove that

PA 2 + PB 2 ≥ 2r 2 .

When does equality hold?

23. (HUN 4) Prove that for an arbitrary pair of vectors f and g in the plane, the inequality

af 2 + b f g + cg 2 ≥0

holds if and only if the following conditions are fulfilled: a ≥ 0, c ≥ 0, 4ac ≥ b 2 .

24. (HUN 5) IMO 6 Father has left to his children several identical gold coins. Ac- cording to his will, the oldest child receives one coin and one-seventh of the remaining coins, the next child receives two coins and one-seventh of the re- maining coins, the third child receives three coins and one-seventh of the re- maining coins, and so on through the youngest child. If every child inherits an integer number of coins, find the number of children and the number of coins.

25. (HUN 6) Three disks of diameter d are touching a sphere at their centers. More- over, each disk touches the other two disks. How do we choose the radius R of the sphere so that the axis of the whole figure makes an angle of 60 ◦ with the line connecting the center of the sphere with the point on the disks that is at the largest distance from the axis? (The axis of the figure is the line having the property that rotation of the figure through 120 ◦ about that line brings the figure to its initial position. The disks are all on one side of the plane, pass through the center of the sphere, and are orthogonal to the axes.)

26. (ITA 1) Let ABCD be a regular tetrahedron. To an arbitrary point M on one edge, say CD, corresponds the point P = P(M), which is the intersection of two lines AH and BK, drawn from A orthogonally to BM and from B orthogonally to AM . What is the locus of P as M varies?

1 The statement so formulated is false. It would be trivially true under the additional assump- tion that the polygonal line is closed. However, from the offered solution, which is not clear,

it does not seem that the proposer had this in mind.

3.9 IMO 1967 45

27. (ITA 2) Which regular polygons can be obtained (and how) by cutting a cube with a plane?

28. (ITA 3) Find values of the parameter u for which the expression

y (x − u) + tanx + tan(x + u)

tan

tan (x − u)tanxtan(x + u)

does not depend on x.

29. (ITA 4) IMO 4 The triangles A 0 B 0 C 0 and A ′ B ′ C ′ have all their angles acute. De- scribe how to construct one of the triangles ABC, similar to A ′ B ′ C ′ and circum- scribing A 0 B 0 C 0 (so that A, B, C correspond to A ′ ,B ′ ,C ′ , and AB passes through

C 0 , BC through A 0 , and CA through B 0 ). Among these triangles ABC, describe, and prove, how to construct the triangle with the maximum area.

30. (MNG 1) Given m +n numbers a i (i = 1, 2, . . . , m), b j (j = 1, 2, . . . , n), determine the number of pairs (a i ,b j ) for which |i− j| ≥ k, where k is a nonnegative integer.

31. (MNG 2) An urn contains balls of k different colors; there are n i balls of the ith color. Balls are drawn at random from the urn, one by one, without replacement. Find the smallest number of draws necessary for getting m balls of the same color.

32. (MNG 3) Determine the volume of the body obtained by cutting the ball of radius R by the trihedron with vertex in the center of that ball if its dihedral angles are α , β , γ .

33. (MNG 4) In what case does the system

x + y + mz = a, x + my + z = b, mx + y + z = c,

have a solution? Find the conditions under which the unique solution of the above system is an arithmetic progression.

34. (MNG 5) The faces of a convex polyhedron are six squares and eight equilateral triangles, and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge are equal. Prove that it is possible to circumscribe a sphere around this polyhedron and compute the ratio of the squares of the volumes of the polyhedron and of the ball whose boundary is the circumscribed sphere.

35. (MNG 6) Prove the identity

n x 2k

∑ n tan 1 +2 = sec + sec

(x/2)) k

36. (POL 1) Prove that the center of the sphere circumscribed around a tetrahedron ABCD coincides with the center of a sphere inscribed in that tetrahedron if and only if AB = CD, AC = BD, and AD = BC.

46 3 Problems

37. (POL 2) Prove that for arbitrary positive numbers the following inequality holds:

1 1 1 a 8 +b 8 +c 8

38. (POL 3) Does there exist an integer such that its cube is equal to 3n 2 + 3n + 7, where n is integer?

39. (POL 4) Show that the triangle whose angles satisfy the equality

sin 2 A + sin 2 B + sin 2 C cos 2 A + cos 2 B + cos 2

is a right-angled triangle.

40. (POL 5) IMO 2 Exactly one side of a tetrahedron is of length greater than 1. Show that its volume is less than or equal to 1 /8.

41. (POL 6) A line l is drawn through the intersection point H of the altitudes of an acute-angled triangle. Prove that the symmetric images l a ,l b ,l c of l with respect to sides BC, CA, AB have one point in common, which lies on the circumcircle of ABC.

42. (ROU 1) Decompose into real factors the expression 1 − sin 5 x − cos 5 x .

43. (ROU 2) The equation x 5 +5 λ x 4 −x 3 +( λα

− 4)x 2 − (8 λ + 3)x + λα −2=0 is given.

(a) Determine α such that the given equation has exactly one root independent of λ . (b) Determine α such that the given equation has exactly two roots indepen- dent of λ .

44. (ROU 3) Suppose p and q are two different positive integers and x is a real number. Form the product (x + p)(x + q).

(a) Find the sum S (x, n) = ∑(x + p)(x + q), where p and q take values from 1 to n. (b) Do there exist integer values of x for which S (x, n) = 0?

45. (ROU 4) (a) Solve the equation

3 sin x + sin

+x + sin 3 +x + cos 2x = 0.

3 3 4 (b) Suppose the solutions are in the form of arcs AB of the trigonometric circle

(where A is the beginning of arcs of the trigonometric circle), and P is a regular n-gon inscribed in the circle with one vertex at A. (1) Find the subset of arcs with the endpoint B at a vertex of the regular

dodecagon.

3.9 IMO 1967 47 (2) Prove that the endpoint B cannot be at a vertex of P if 2 , 3 ∤ n or n is

prime.

46. (ROU 5) If x, y, z are real numbers satisfying the relations x + y + z = 1 and arctan x + arctany + arctanz = π /4, prove that

x 2n +1 +y 2n +1 +z 2n +1 =1

for all positive integers n.

47. (ROU 6) Prove the inequality x 1 x 2 ··· x k x n −1 1 +x n −1

≤x 1 +x 2 + ··· + x k , where x i > 0 (i = 1, 2, . . . , k), k ∈ N, n ∈ N.

2 + ··· + x k

n −1

n +k−1

n +k−1 n +k−1

48. (SWE 1) Determine all positive roots of the equation x x = 1/ 2.

49. (SWE 2) Let n and k be positive integers such that 1 ≤ n ≤ N + 1, 1 ≤ k ≤ N +1. Show that

min

|sinn − sink| < .

6=k

50. (SWE 3) The function ϕ (x, y, z), defined for all triples (x, y, z) of real numbers, is such that there are two functions f and g defined for all pairs of real numbers such that

ϕ (x, y, z) = f (x + y, z) = g(x, y + z)

for all real x, y, and z. Show that there is a function h of one real variable such that

ϕ (x, y, z) = h(x + y + z)

for all real x, y, and z.

51. (SWE 4) A subset S of the set of integers 0 , . . . , 99 is said to have property A if it is impossible to fill a crossword puzzle with 2 rows and 2 columns with numbers in S (0 is written as 00, 1 as 01, and so on). Determine the maximal number of elements in sets S with property A.

52. (SWE 5) In the plane a point O and a sequence of points P 1 ,P 2 ,P 3 , . . . are given. The distances OP 1 , OP 2 , OP 3 , . . . are r 1 ,r 2 ,r 3 , . . . , where r 1 ≤r 2 ≤r 3 ≤ ··· . Let α satisfy 0 < α < 1. Suppose that for every n the distance from the point P n to any other point of the sequence is greater than or equal to r α n . Determine the exponent β , as large as possible, such that for some C independent of n, 2

n ≥ Cn , n = 1, 2, . . . .

2 This problem is not elementary. The solution offered by the proposer, which is not quite clear and complete, only shows that if such a β exists, then β ≥ 1 2 (1− α ) .

48 3 Problems

53. (SWE 6) In making Euclidean constructions in geometry it is permitted to use

a straightedge and compass. In the constructions considered in this question, no compasses are permitted, but the straightedge is assumed to have two parallel edges, which can be used for constructing two parallel lines through two given points whose distance is at least equal to the breadth of the ruler. Then the dis- tance between the parallel lines is equal to the breadth of the straightedge. Carry through the following constructions with such a straightedge. Construct:

(a) The bisector of a given angle. (b) The midpoint of a given rectilinear segment.

(c) The center of a circle through three given noncollinear points. (d) A line through a given point parallel to a given line.

54. (USS 1) Is it possible to put 100 (or 200) points on a wooden cube such that by all rotations of the cube the points map into themselves? Justify your answer.

55. (USS 2) Find all x for which for all n, √

3 sin x + sin 2x + sin 3x + ···+ sinnx ≤

56. (USS 3) In a group of interpreters each one speaks one or several foreign lan- guages; 24 of them speak Japanese, 24 Malay, 24 Farsi. Prove that it is possible to select a subgroup in which exactly 12 interpreters speak Japanese, exactly 12 speak Malay, and exactly 12 speak Farsi.

57. (USS 4) IMO 5 Consider the sequence (c n ):

where a 1 ,a 2 ,...,a 8 are real numbers, not all equal to zero. Given that among the numbers of the sequence (c n ) there are infinitely many equal to zero, determine all the values of n for which c n = 0.

58. (USS 5) A linear binomial l (z) = Az + B with complex coefficients A and B is given. It is known that the maximal value of |l(z)| on the segment −1 ≤ x ≤ 1 (y = 0) of the real line in the complex plane (z = x + iy) is equal to M. Prove that for every z

|l(z)| ≤ M ρ ,

where ρ is the sum of distances from the point P = z to the points Q 1 :z = 1 and Q 3 :z = −1.

59. (USS 6) On the circle with center O and radius 1 the point A 0 is fixed and points A 1 ,A 2 ,...,A 999 ,A 1000 are distributed in such a way that ∠A 0 OA k = k (in

radians). Cut the circle at points A 0 ,A 1 ,...,A 1000 . How many arcs with different lengths are obtained?

3.10 IMO 1968 49