The Seventeenth IMO Burgas–Sofia, Bulgaria, 1975

3.17 The Seventeenth IMO Burgas–Sofia, Bulgaria, 1975

3.17.1 Contest Problems

First Day (July 7)

be two n-tuples of numbers. Prove that

1. Let x 1 ≥x 2 ≥ ··· ≥ x n and y 1 ≥y 2 ≥ ··· ≥ y n

∑ 2 (x

i −y i ) 2 ≤ ∑ (x i −z i )

i =1

i =1

is true when z 1 ,z 2 ,...,z n denote y 1 ,y 2 ,...,y n taken in another order.

2. Let a 1 ,a 2 ,a 3 , . . . be any infinite increasing sequence of positive integers. (For every integer i > 0, a i +1 >a i .) Prove that there are infinitely many m for which positive integers x , y, h, k can be found such that 0 < h < k < m and a m = xa h + ya k .

3. On the sides of an arbitrary triangle ABC, triangles BPC, CQA, and ARB are externally erected such that

∡PBC = ∡CAQ = 45 ◦ , ∡BCP = ∡QCA = 30 ◦ , ∡ABR = ∡BAR = 15 ◦ .

Prove that ∡QRP = 90 ◦ and QR = RP. Second Day (July 8)

4. Let A be the sum of the digits of the number 4444 4444 and B the sum of the digits of the number A. Find the sum of the digits of the number B.

5. Is it possible to plot 1975 points on a circle with radius 1 so that the distance between any two of them is a rational number (distances have to be measured by chords)?

6. The function f (x, y) is a homogeneous polynomial of the nth degree in x and y. If f (1, 0) = 1 and for all a, b, c,

f (a + b, c) + f (b + c, a) + f (c + a, b) = 0, prove that f (x, y) = (x − 2y)(x + y) n −1 .

3.17.2 Shortlisted Problems

1. (FRA) There are six ports on a lake. Is it possible to organize a series of routes satisfying the following conditions:

(i) Every route includes exactly three ports; (ii) No two routes contain the same three ports;

98 3 Problems (iii) The series offers exactly two routes to each tourist who desires to visit two

different arbitrary ports?

be two n-tuples of numbers. Prove that

2. (CZS) IMO 1 Let x 1 ≥x 2 ≥ ··· ≥ x n and y 1 ≥y 2 ≥ ··· ≥ y n

∑ (x i −y i ) 2 ≤ 2 (x i −z i )

i =1

i =1

is true when z 1 ,z 2 ,...,z n denote y 1 ,y 2 ,...,y n taken in another order.

3. (USA) Find the integer represented by ∑ 10 n 9 =1 n −2/3 . Here [x] denotes the great-

√ est integer less than or equal to x (e.g. [ 2 ] = 1).

4. (SWE) Let a 1 ,a 2 ,...,a n , . . . be a sequence of real numbers such that 0 ≤ a n ≤1 and a n − 2a n +1 +a n +2 ≥ 0 for n = 1,2,3,.... Prove that

0 ≤ (n + 1)(a n −a n +1 )≤2 for n = 1, 2, 3, . . . .

5. (SWE) Let M be the set of all positive integers that do not contain the digit 9 (base 10). If x 1 ,...,x n are arbitrary but distinct elements in M, prove that

j =1 x j

6. (USS) IMO 4 Let A be the sum of the digits of the number 16 16 and B the sum of the digits of the number A. Find the sum of the digits of the number B without calculating 16 16 .

7. (GDR) Prove that from x + y = 1 (x, y ∈ R) it follows that

x m +1

n ∑ +i

8. (NLD) IMO 3 On the sides of an arbitrary triangle ABC, triangles BPC, CQA, and ARB are externally erected such that ∡PBC = ∡CAQ = 45 ◦ , ∡BCP = ∡QCA = 30 ◦ , ∡ABR = ∡BAR = 15 ◦ .

Prove that ∡QRP = 90 ◦ and QR = RP.

9. (NLD) Let f (x) be a continuous function defined on the closed interval 0 ≤ x ≤

1. Let G ( f ) denote the graph of f (x): G( f ) = {(x,y) ∈ R 2 | 0 ≤ x ≤ 1, y = f (x)}. Let G a ( f ) denote the graph of the translated function f (x − a) (translated over

a distance a), defined by G a ( f ) = {(x,y) ∈ R 2 | a ≤ x ≤ a + 1, y = f (x − a)}. Is it possible to find for every a, 0 < a < 1, a continuous function f (x), defined on 0 ≤ x ≤ 1, such that f (0) = f (1) = 0 and G( f ) and G a ( f ) are disjoint point sets?

3.17 IMO 1975 99

10. (UNK) IMO 6 The function f (x, y) is a homogeneous polynomial of the nth degree in x and y. If f (1, 0) = 1 and for all a, b, c,

f (a + b, c) + f (b + c, a) + f (c + a, b) = 0,

prove that f (x, y) = (x − 2y)(x + y) n −1 .

11. (UNK) IMO 2 Let a 1 ,a 2 ,a 3 , . . . be any infinite increasing sequence of positive in- tegers. (For every integer i > 0, a i +1 >a i .) Prove that there are infinitely many m for which positive integers x , y, h, k can be found such that 0 < h < k < m and

a m = xa h + ya k .

12. (HEL) Consider on the first quadrant of the trigonometric circle the arcs AM 1 = x 1 , AM 2 =x 2 , AM 3 =x 3 , . . . , AM ν =x ν , such that x 1 <x 2 <x 3 < ··· < x ν . Prove that

∑ sin 2x i − ∑ sin (x i −x i +1 )< + ∑ sin (x i +x +1 ).

2 i i =0

i =0 i =0

be points in a plane such that (i) A

13. (ROU) Let A 0 ,A 1 ,...,A n

0 A 1 1 ≤ 2 A 1 A 2 ≤ ··· ≤ 2 n −1 A n −1 A n and

(ii) 0 < ∡A 0 A 1 A 2 < ∡A 1 A 2 A 3 < ··· < ∡A n −2 A n −1 A n < 180 ◦ , where all these angles have the same orientation. Prove that the segments

A k A k +1 ,A m A m +1 do not intersect for each k and m such that 0 ≤k≤m−2< n − 2.

14. (YUG) Let x 0 = 5 and x n +1 =x n + 1 x n (n = 0, 1, 2, . . .). Prove that 45 < x 1000 <

15. (USS) IMO 5 Is it possible to plot 1975 points on a circle with radius 1 so that the distance between any two of them is a rational number (distances have to be measured by chords)?

100 3 Problems