The Tenth IMO Moscow–Leningrad, Soviet Union, July 5–18, 1968

3.10 The Tenth IMO Moscow–Leningrad, Soviet Union, July 5–18, 1968

3.10.1 Contest Problems

First Day

1. Prove that there exists a unique triangle whose side lengths are consecutive nat- ural numbers and one of whose angles is twice the measure of one of the others.

2. Find all positive integers x for which p (x) = x 2 − 10x − 22, where p(x) denotes the product of the digits of x.

3. Let a , b, c be real numbers. Prove that the system of equations

  2 ax 

1 + bx 1 +c=x 2 ,

ax 2 2 + bx 2 +c=x 3 , ······ ··· ···

 2    ax n −1 + bx n −1 +c=x n , 

ax 2 n + bx n +c=x 1 ,

(a) has no real solutions if (b − 1) 2 − 4ac < 0; (b) has a unique real solution if (b − 1) 2 − 4ac = 0;

(c) has more than one real solution if (b − 1) 2 − 4ac > 0. Second Day

4. Prove that in any tetrahedron there is a vertex such that the lengths of its sides through that vertex are sides of a triangle.

5. Let a > 0 be a real number and f (x) a real function defined on all of R, satisfying for all x ∈ R,

2 (x) − f (x) 2 . (a) Prove that the function f is periodic; i.e., there exists b > 0 such that for all x ,f (x + b) = f (x). (b) Give an example of such a nonconstant function for a = 1.

f (x + a) = + f

6. Let [x] denote the integer part of x, i.e., the greatest integer not exceeding x. If n is a positive integer, express as a simple function of n the sum

3.10.2 Shortlisted Problems

1. (SWE 2) Two ships sail on the sea with constant speeds and fixed directions. It is known that at 9:00 the distance between them was 20 miles; at 9:35, 15 miles; and at 9:55, 13 miles. At what moment were the ships the smallest distance from each other, and what was that distance?

50 3 Problems

2. (ROU 5) IMO 1 Prove that there exists a unique triangle whose side lengths are consecutive natural numbers and one of whose angles is twice the measure of one of the others.

3. (POL 4) IMO 4 Prove that in any tetrahedron there is a vertex such that the lengths of its sides through that vertex are sides of a triangle.

4. (BGR 2) IMO 3 Let a , b, c be real numbers. Prove that the system of equations

ax  2 1 + bx 1 +c=x 2 ,

ax  2  2 + bx 2 +c=x 3 , ······ ··· ···

  ax 2  n + bx n −1 +c=x n ,  

ax 2 n + bx n +c=x 1 , has a unique real solution if and only if (b − 1) 2 − 4ac = 0.

Remark. It is assumed that a 6= 0.

5. (BGR 5) Let h n

be the apothem (distance from the center to one of the sides) of

a regular n-gon (n ≥ 3) inscribed in a circle of radius r. Prove the inequality (n + 1)h n +1 − nh n > r.

Also prove that if r on the right side is replaced with a greater number, the inequality will not remain true for all n ≥ 3.

6. (HUN 1) If a i (i = 1, 2, . . . , n) are distinct non-zero real numbers, prove that the equation

a a 1 + ··· + −x 2 −x a n −x

=n

has at least n − 1 real roots.

7. (HUN 5) Prove that the product of the radii of three circles exscribed to a given triangle does not exceed 3

8 times the product of the side lengths of the triangle. When does equality hold?

8. (ROU 2) Given an oriented line ∆ and a fixed point A on it, consider all trape- zoids ABCD one of whose bases AB lies on ∆ , in the positive direction. Let E ,F

be the midpoints of AB and CD respectively. Find the loci of vertices B ,C, D of trapezoids that satisfy the following: (i) |AB| ≤ a

(a fixed); (ii) |EF| = l

(l fixed); (iii) the sum of squares of the nonparallel sides of the trapezoid is constant.

Remark. The constants are chosen so that such trapezoids exist.

9. (ROU 3) Let ABC be an arbitrary triangle and M a point inside it. Let d a ,d b ,d c

be the distances from M to sides BC ,CA, AB; a, b, c the lengths of the sides re- spectively, and S the area of the triangle ABC. Prove the inequality

3.10 IMO 1968 51 4S 2

abd a d b + bcd b d c + cad c d a ≤

3 Prove that the left-hand side attains its maximum when M is the centroid of the

triangle.

10. (ROU 4) Consider two segments of length a √ , b (a > b) and a segment of length

c = ab . (a) For what values of a /b can these segments be sides of a triangle? (b) For what values of a /b is this triangle right-angled, obtuse-angled, or acute-

angled?

11. (ROU 6) Find all solutions (x 1 ,x 2 ,...,x n ) of the equation

1 x 1 +1 (x 1 + 1)(x 2 + 1)

(x 1 + 1) ···(x n −1 + 1)

= 0. x 1 x 1 x 2 x 1 x 2 x 3 x 1 x 2 ··· x n

12. (POL 1) If a and b are arbitrary positive real numbers and m an integer, prove that

b a ≥2

m +1

13. (POL 5) Given two congruent triangles A 1 A 2 A 3 and B 1 B 2 B 3 (A i A k =B i B k ), prove that there exists a plane such that the orthogonal projections of these tri- angles onto it are congruent and equally oriented.

14. (BGR 5) A line in the plane of a triangle ABC intersects the sides AB and AC respectively at points X and Y such that BX = CY . Find the locus of the center of the circumcircle of triangle X AY .

15. (UNK 1) IMO 6 Let [x] denote the integer part of x, i.e., the greatest integer not exceeding x. If n is a positive integer, express as a simple function of n the sum

16. (UNK 3) A polynomial p (x) = a 0 x k +a 1 x k −1 + ···+a k with integer coefficients is said to be divisible by an integer m if p (x) is divisible by m for all integers x. Prove that if p (x) is divisible by m, then k!a 0 is also divisible by m. Also prove that if a 0 , k, m are nonnegative integers for which k!a 0 is divisible by m, there exists a polynomial p (x) = a 0 x k + ··· + a k divisible by m.

17. (UNK 4) Given a point O and lengths x , y, z, prove that there exists an equilateral triangle ABC for which OA = x, OB = y, OC = z, if and only if x+y ≥ z, y+z ≥ x, z + x ≥ y (the points O,A,B,C are coplanar).

18. (ITA 2) If an acute-angled triangle ABC is given, construct an equilateral trian- gle A ′ B ′ C ′ in space such that lines AA ′ , BB ′ ,CC ′ pass through a given point.

19. (ITA 5) We are given a fixed point on the circle of radius 1, and going from this point along the circumference in the positive direction on curved distances

52 3 Problems

0 , 1, 2, . . . from it we obtain points with abscissas n = 0, 1, 2, . . . respectively. How many points among them should we take to ensure that some two of them are less than the distance 1 /5 apart?

20. (CZS 1) Given n (n ≥ 3) points in space such that every three of them form a triangle with one angle greater than or equal to 120 ◦ , prove that these points can

be denoted by A 1 ,A 2 ,...,A n in such a way that for each i , j, k, 1 ≤ i < j < k ≤ n, angle A i A j A k is greater than or equal to 120 ◦ .

21. (CZS 2) Let a 0 ,a 1 ,...,a k (k ≥ 1) be positive integers. Find all positive integers y such that

a 0 | y; (a 0 +a 1 ) | (y + a 1 ); . . . ; (a 0 +a n ) | (y + a n ).

22. (CZS 3) IMO 2 Find all positive integers x for which p (x) = x 2 − 10x − 22, where p (x) denotes the product of the digits of x.

23. (CZS 4) Find all complex numbers m such that polynomial

x 3 +y 3 +z 3 + mxyz

can be represented as the product of three linear trinomials.

24. (MNG 1) Find the number of all n-digit numbers for which some fixed digit stands only in the ith (1 < i < n) place and the last j digits are distinct. 3

25. (MNG 2) Given k parallel lines and a few points on each of them, find the number of all possible triangles with vertices at these given points. 4

26. (GDR) IMO 5 Let a > 0 be a real number and f (x) a real function defined on all of R, satisfying for all x ∈ R,

2 (x) − f (x)

f (x + a) = + f 2 .

(a) Prove that the function f is periodic; i.e., there exists b > 0 such that for all x ,f (x + b) = f (x). (b) Give an example of such a nonconstant function for a = 1.

3 The problem is unclear. Presumably n, i, j, and the ith digit are fixed. 4 The problem is unclear. The correct formulation could be the following:

Given k parallel lines l 1 ,...,l k and n i points on the line l i ,i = 1, 2, . . . , k, find the maxi- mum possible number of triangles with vertices at these points.

3.11 IMO 1969 53