The Eighth IMO Sofia, Bulgaria, July 3–13, 1966

3.8 The Eighth IMO Sofia, Bulgaria, July 3–13, 1966

3.8.1 Contest Problems

First Day

1. (USS) Three problems A, B, and C were given on a mathematics olympiad. All

25 students solved at least one of these problems. The number of students who solved B and not A is twice the number of students who solved C and not A. The number of students who solved only A is greater by 1 than the number of students who along with A solved at least one other problem. Among the students who solved only one problem, half solved A. How many students solved only B?

2. (HUN) If a, b, and c are the sides and α , β , and γ the respective angles of the triangle for which a + b = tan γ 2 (a tan α + b tan β ), prove that the triangle is isosceles.

3. (BGR) Prove that the sum of distances from the center of the circumsphere of the regular tetrahedron to its four vertices is less than the sum of distances from any other point to the four vertices.

Second Day

4. (YUG) Prove the following equality:

sin 2x sin 4x sin 8x + ··· + sin 2 n x = cot x − cot2

where n ∈ N and x / ∈ π 2 k Z for every k ∈ N.

5. (CZS) Solve the following system of equations: |a 1 −a 2 |x 2 + |a 1 −a 3 |x 3 + |a 1 −a 4 |x 4 = 1,

|a 2 −a 1 |x 1 + |a 2 −a 3 |x 3 + |a 2 −a 4 |x 4 = 1, |a 3 −a 1 |x 1 + |a 3 −a 2 |x 2 + |a 3 −a 4 |x 4 = 1,

|a 4 −a 1 |x 1 + |a 4 −a 2 |x 2 + |a 4 −a 3 |x 3 = 1, where a 1 ,a 2 ,a 3 , and a 4 are mutually distinct real numbers.

6. (POL) Let M, K, and L be points on (AB), (BC), and (CA), respectively. Prove that the area of at least one of the three triangles △MAL, △KBM, and △LCK is less than or equal to one-fourth the area of △ABC.

3.8.2 Some Longlisted Problems 1959–1966

1. (CZS) We are given n > 3 points in the plane, no three of which lie on a line. Does there necessarily exist a circle that passes through at least three of the given points and contains none of the other given points in its interior?

36 3 Problems

2. (GDR) Given n positive real numbers a 1 ,a 2 ,...,a n such that a 1 a 2 ··· a n = 1, prove that (1 + a 1 )(1 + a 2 ) ···(1 + a n )≥2 n .

3. (BGR) A regular triangular prism has height h and a base of side length a. Both bases have small holes in the centers, and the inside of the three vertical walls has a mirror surface. Light enters through the small hole in the top base, strikes each vertical wall once and leaves through the hole in the bottom. Find the angle at which the light enters and the length of its path inside the prism.

4. (POL) Five points in the plane are given, no three of which are collinear. Show that some four of them form a convex quadrilateral.

5. (USS) Prove the inequality

π sin x

π cos x

4 sin α

4 cos α

for any x , α with 0 ≤x≤ π /2 and π /6 < α < π /3.

6. (USS) A convex planar polygon M with perimeter l and area S is given. Let M (R) be the set of all points in space that lie a distance at most R from a point of M . Show that the volume V (R) of this set equals

V (R) = π R 3 + lR 2 + 2SR.

7. (USS) For which arrangements of two infinite circular cylinders does their in- tersection lie in a plane?

8. (USS) We are given a bag of sugar, a two-pan balance, and a weight of 1 gram. How do we obtain 1 kilogram of sugar in the smallest possible number of weigh- ings?

9. (ROU) Find x such that

sin 3x cos (60 ◦ − 4x) + 1

= 0, sin (60 ◦ − 7x) − cos(30 ◦ + x) + m

where m is a fixed real number.

10. (GDR) How many real solutions are there to the equation x = 1964 sinx − 189?

11. (CZS) Does there exist an integer z that can be written in two different ways as z = x! + y!, where x, y are natural numbers with x ≤ y?

12. (BGR) Find digits x , y, z such that the equality

√ xx | {z } ···x − yy···y = zz ···z | {z }

| {z }

2n

holds for at least two values of n ∈ N, and in that case find all n for which this equality is true.

3.8 IMO 1966 37

13. (YUG) Let a 1 ,a 2 ,...,a n

be positive real numbers. Prove the inequality ! 2

2 ∑ a i ≥4 a j i ∑ <j i <j a i +a j

and find the conditions on the numbers a i for equality to hold.

14. (POL) Compute the largest number of regions into which one can divide a disk by joining n points on its circumference.

15. (POL) Points A , B,C, D lie on a circle such that AB is a diameter and CD is not. If the tangents at C and D meet at P while AC and BD meet at Q, show that PQ is perpendicular to AB.

16. (CZS) We are given a circle K with center S and radius 1 and a square Q with center M and side 2. Let XY be the hypotenuse of an isosceles right triangle XY Z . Describe the locus of points Z as X varies along K and Y varies along the boundary of Q.

17. (ROU) Suppose ABCD and A ′ B ′ C ′ D ′ are two parallelograms arbitrarily ar- ranged in space, and let points M , N, P, Q divide the segments AA ′ , BB ′ ,CC ′ , DD ′ respectively in equal ratios.

(a) Show that MNPQ is a parallelogram; (b) Find the locus of MNPQ as M varies along the segment AA ′ .

18. (HUN) Solve the equation 1 sin x + 1 cos x = 1 p , where p is a real parameter. Discuss for which values of p the equation has at least one real solution and determine the number of solutions in [0, 2 π ) for a given p.

19. (HUN) Construct a triangle given the three exradii.

20. (HUN) We are given three equal rectangles with the same center in three mutu- ally perpendicular planes, with the long sides also mutually perpendicular. Con- sider the polyhedron with vertices at the vertices of these rectangles.

(a) Find the volume of this polyhedron; (b) can this polyhedron be regular, and under what conditions?

21. (BGR) Prove that the volume V and the lateral area S of a right circular cone

satisfy the inequality 6V π 2S ≤ π √ 3 . When does equality occur?

22. (BGR) Assume that two parallelograms P ,P ′ of equal areas have sides a , b and

a ′ ,b ′ respectively such that a ′ ≤a≤b≤b ′ and a segment of length b ′ can be placed inside P. Prove that P and P ′ can be partitioned into four pairwise con- gruent parts.

23. (BGR) Three faces of a tetrahedron are right triangles, while the fourth is not an obtuse triangle.

(a) Prove that a necessary and sufficient condition for the fourth face to be a right triangle is that at some vertex exactly two angles are right.

38 3 Problems (b) Prove that if all the faces are right triangles, then the volume of the tetrahe-

dron equals one -sixth the product of the three smallest edges not belonging to the same face.

24. (POL) There are n ≥ 2 people in a room. Prove that there exist two among them having equal numbers of friends in that room. (Friendship is always mutual.)

25. (GDR) Show that tan 7 30 = 6 + 2 − 3 − 2.

26. (CZS) (a) Prove that (a 1 +a 2 + ··· + a k ) 2 ≤ k(a 2 1 + ··· + a 2 k ), where k ≥ 1 is a natural number and a 1 ,...,a k are arbitrary real numbers. (b) If real numbers a 1 ,...,a n satisfy

a 1 +a 2 + ··· + a n ≥ (n − 1)(a 2 1 + ··· + a 2 n ), show that they are all nonnegative.

27. (GDR) We are given a circle K and a point P lying on a line g. Construct a circle that passes through P and touches K and g.

28. (CZS) Let there be given a circle with center S and radius 1 in the plane, and let ABC

be an arbitrary triangle circumscribed about the circle such that SA ≤ SB ≤ SC . Find the loci of the vertices A , B,C.

29. (ROU) (a) Find the number of ways 500 can be represented as a sum of con- secutive integers. (b) Find the number of such representations for N =2 α 3 β 5 γ , α , β , γ ∈ N. Which of these representations consist only of natural numbers? (c) Determine the number of such representations for an arbitrary natural num- ber N.

30. (ROU) If n is a natural number, prove that

(a) log 10 (n + 1) > 3 10n + log 10 n ; (b) log n! > 3n 1 + 1 10 1 2 3 + ··· + n −1 .

31. (ROU) Solve the equation |x 2 −1|+ |x 2 − 4| = mx as a function of the parameter m . Which pairs (x, m) of integers satisfy this equation?

32. (BGR) The sides a , b, c of a triangle ABC form an arithmetic progression; the sides of another triangle A 1 B 1 C 1 also form an arithmetic progression. Suppose that ∠A = ∠A 1 . Prove that the triangles ABC and A 1 B 1 C 1 are similar.

33. (BGR) Two circles touch each other from inside, and an equilateral triangle is inscribed in the larger circle. From the vertices of the triangle one draws seg- ments tangent to the smaller circle. Prove that the length of one of these segments equals the sum of the lengths of the other two.

34. (BGR) Determine all pairs of positive integers (x, y) satisfying the equation

2 x =3 y + 5.

35. (POL) If a , b, c, d are integers such that ad is odd and bc is even, prove that at least one root of the polynomial ax 3 + bx 2 + cx + d is irrational.

3.8 IMO 1966 39

36. (POL) Let ABCD be a cyclic quadrilateral. Show that the centroids of the trian- gles ABC, CDA, BCD, DAB lie on a circle.

37. (POL) Prove that the perpendiculars drawn from the midpoints of the sides of

a cyclic quadrilateral to the opposite sides meet at one point.

38. (ROU) Two concentric circles have radii R and r respectively. Determine the greatest possible number of circles that are tangent to both these circles and √

mutually nonintersecting. Prove that this number lies between 3 R + √ 2 r · √ R √ − r − 1 and

20 · R +r R −r .

39. (ROU) In a plane, a circle with center O and radius R and two points A , B are given.

(a) Draw a chord CD parallel to AB so that AC and BD intersect at a point P on the circle. (b) Prove that there are two possible positions of point P, say P 1 ,P 2 , and find the distance between them if OA = a, OB = b, AB = d.

40. (CZS) For a positive real number p, find all real solutions to the equation p

x 2 + 2px − p 2 − x 2 − 2px − p 2 = 1.

41. (CZS) If A 1 A 2 ...A n is a regular n-gon (n ≥ 3), how many different obtuse triangles A i A j A k exist?

42. (CZS) Let a 1 ,a 2 ,...,a n (n ≥ 2) be a sequence of integers. Show that there is

a subsequence a k 1 ,a k 2 ,...,a k m , where 1 ≤k 1 <k 2 < ··· < k m ≤ n, such that

a 2 k 2 1 +a 2 k 2 + ··· + a k m is divisible by n.

43. (CZS) Five points in a plane are given, no three of which are collinear. Every two of them are joined by a segment, colored either red or gray, so that no three segments form a triangle colored in one color.

(a) Prove that (1) every point is a vertex of exactly two red and two gray seg- ments, and (2) the red segments form a closed path that passes through each point.

(b) Give an example of such a coloring.

44. (YUG) What is the greatest number of balls of radius 1 /2 that can be placed within a rectangular box of size 10 × 10 × 1?

45. (YUG) An alphabet consists of n letters. What is the maximal length of a word, if

(i) two neighboring letters in a word are always different, and (ii) no word abab (a 6= b) can be obtained by omitting letters from the given word?

46. (YUG) Let

f (a, b, c) = |b − a| b +a 2 b + 2 |b − a| +a − + + .

|ab|

ab c |ab|

ab c

40 3 Problems Prove that f (a, b, c) = 4 max{1/a,1/b,1/c}.

47. (ROU) Find the number of lines dividing a given triangle into two parts of equal area which determine the segment of minimum possible length inside the trian- gle. Compute this minimum length in terms of the sides a , b, c of the triangle.

48. (USS) Find all positive numbers p for which the equation x 2 + px + 3p = 0 has integral roots.

49. (USS) Two mirror walls are placed to form an angle of measure α . There is a candle inside the angle. How many reflections of the candle can an observer see?

50. (USS) Given a quadrangle of sides a , b, c, d and area S, show that S ≤ a +c 2 · b +d 2 .

51. (USS) In a school, n children numbered 1 to n are initially arranged in the order

1 , 2, . . . , n. At a command, every child can either exchange its position with any other child or not move. Can they rearrange into the order n , 1, 2, . . . , n − 1 after two commands?

52. (USS) A figure of area 1 is cut out from a sheet of paper and divided into 10 parts, each of which is colored in one of 10 colors. Then the figure is turned to the other side and again divided into 10 parts (not necessarily in the same way). Show that it is possible to color these parts in the 10 colors so that the total area of the portions of the figure both of whose sides are of the same color is at least

53. (USS, 1966) Prove that in every convex hexagon of area S one can draw a diagonal that cuts off a triangle of area not exceeding 1 6 S .

54. (USS, 1966) Find the last two digits of a sum of eighth powers of 100 consecu- tive integers.

55. (USS, 1966) Given the vertex A and the centroid M of a triangle ABC, find the locus of vertices B such that all the angles of the triangle lie in the interval [40 ◦ , 70 ◦ ].

56. (USS, 1966) Let ABCD be a tetrahedron such that AB ⊥ CD, AC ⊥ BD, and AD ⊥ BC. Prove that the midpoints of the edges of the tetrahedron lie on a sphere.

57. (USS, 1966) Is it possible to choose a set of 100 (or 200) points on the boundary of a cube such that this set is fixed under each isometry of the cube into itself? Justify your answer.

3.9 IMO 1967 41