The Thirty-Fourth IMO Istanbul, Turkey, July 13–24, 1993

3.34 The Thirty-Fourth IMO Istanbul, Turkey, July 13–24, 1993

3.34.1 Contest Problems

First Day (July 18)

1. Let n > 1 be an integer and let f (x) = x n + 5x n −1 + 3. Prove that there do not exist polynomials g (x), h(x), each having integer coefficients and degree at least one, such that f (x) = g(x)h(x).

2. A , B,C, D are four points in the plane, with C, D on the same side of the line AB, such that AC · BD = AD · BC and ∡ADB = 90 ◦ + ∡ACB. Find the ratio

AB ·CD , AC · BD

and prove that circles ACD , BCD are orthogonal. (Intersecting circles are said to

be orthogonal if at either common point their tangents are perpendicular.)

3. On an infinite chessboard, a solitaire game is played as follows: At the start, we have n 2 pieces occupying n 2 squares that form a square of side n. The only allowed move is a jump horizontally or vertically over an occupied square to an unoccupied one, and the piece that has been jumped over is removed. For what positive integers n can the game end with only one piece remaining on the board?

Second Day (July 19)

4. For three points A , B,C in the plane we define m(ABC) to be the smallest length of the three altitudes of the triangle ABC, where in the case of A , B,C collinear, m (ABC) = 0. Let A, B,C be given points in the plane. Prove that for any point X in the plane,

m (ABC) ≤ m(ABX) + m(AXC) + m(XBC).

5. Let N = {1,2,3,...}. Determine whether there exists a strictly increasing func- tion f : N → N with the following properties:

f (1) = 2;

f ( f (n)) = f (n) + n

(n ∈ N). (2)

6. Let n be an integer greater than 1. In a circular arrangement of n lamps L 0 , . . ., L n −1 , each one of that can be either on or off, we start with the situation where all lamps are on, and then carry out a sequence of steps, Step 0 , Step 1 , . . . . If L j −1 (indices are taken modulo 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:

3.34 IMO 1993 267 (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.

3.34.2 Shortlisted Problems

1. (BRA 1) Show that there exists a finite set A ⊂R 2 such that for every X ∈A there are points Y 1 ,Y 2 , . . . ,Y 1993 in A such that the distance between X and Y i is equal to 1, for every i.

2. (CAN 2) Let triangle ABC be such that its circumradius R is equal to 1. Let r

be the inradius of ABC and let p be the inradius of the orthic triangle A ′ B ′ C ′ of triangle ABC. Prove that p

3 (1 + r) 2 .

Remark. The orthic triangle is the triangle whose vertices are the feet of the altitudes of ABC.

3. (ESP 1) Consider the triangle ABC, its circumcircle k with center O and radius R , and its incircle with center I and radius r. Another circle k c is tangent to the sides CA ,CB at D, E, respectively, and it is internally tangent to k. Show that the incenter I is the midpoint of DE.

4. (ESP 2) In the triangle ABC, let D , E be points on the side BC such that ∠BAD = ∠CAE. If M , N are, respectively, the points of tangency with BC of the incircles of the triangles ABD and ACE, show that

5. (FIN 3) IMO 3 On an infinite chessboard, a solitaire game is played as follows: At the start, we have n 2 pieces occupying n 2 squares that form a square of side n . The only allowed move is a jump horizontally or vertically over an occu- pied square to an unoccupied one, and the piece that has been jumped over is removed. For what positive integers n can the game end with only one piece remaining on the board?

6. (GER 1) IMO 5 Let N = {1,2,3,...}. Determine whether there exists a strictly increasing function f : N → N with the following properties:

f (1) = 2;

f ( f (n)) = f (n) + n

(n ∈ N). (2)

7. (GEO 3) Let a , b, c be given integers a > 0, ac − b 2 =P=P 1 ··· P m where P 1 ,...,P m are (distinct) prime numbers. Let M (n) denote the number of pairs of integers (x, y) for which

ax 2 + 2bxy + cy 2 = n.

Prove that M (n) is finite and M(n) = M(P k · n) for every integer k ≥ 0.

268 3 Problems

8. (IND 1) Define a sequence h f (n)i ∞ n =1 of positive integers by f (1) = 1 and

f (n) = (n − 1) − n, if f (n − 1) > n;

f (n − 1) + n, if f (n − 1) ≤ n, for n ≥ 2. Let S = {n ∈ N | f (n) = 1993}.

(a) Prove that S is an infinite set. (b) Find the least positive integer in S.

(c) If all the elements of S are written in ascending order as n 1 <n 2 <n 3 < ···, show that

n lim i +1 = 3.

i →∞ n i

9. (IND 4) (a) Show that the set Q + of all positive rational numbers can be partitioned into three disjoint subsets A , B,C satisfying the following conditions:

BC = A, where HK stands for the set {hk | h ∈ H,k ∈ K} for any two subsets H,K

BA = B,

B 2 = C,

of Q + and H 2 stands for HH.

(b) Show that all positive rational cubes are in A for such a partition of Q + .

(c) Find such a partition Q + = A ∪ B ∪C with the property that for no positive integer n ≤ 34 are both n and n + 1 in A; that is,

min {n ∈ N | n ∈ A,n + 1 ∈ A} > 34.

10. (IND 5) A natural number n is said to have the property P if whenever n divides

2 − 1 for some integer a, n n also necessarily divides a − 1. (a) Show that every prime number has property P.

(b) Show that there are infinitely many composite numbers n that possess prop- erty P.

11. (IRL 1) IMO 1 Let n > 1 be an integer and let f (x) = x n + 5x n −1 + 3. Prove that there do not exist polynomials g (x), h(x), each having integer coefficients and degree at least one, such that f (x) = g(x)h(x).

12. (IRL 2) Let n , k be positive integers with k ≤ n and let S be a set containing n distinct real numbers. Let T be the set of all real numbers of the form x 1 +x 2 + ··· + x k , where x 1 ,x 2 ,...,x k are distinct elements of S. Prove that T contains at least k (n − k) + 1 distinct elements.

13. (IRL 3) Let S be the set of all pairs (m, n) of relatively prime positive integers m , n with n even and m < n. For s = (m, n) ∈ S write n = 2 k n 0 , where k ,n 0 are positive integers with n 0 odd and define f (s) = (n 0 ,m+n−n 0 ). Prove that f is a function from S to S and that for each s = (m, n) ∈ S, there exists

a positive integer t m ≤ +n+1

4 such that f t (s) = s, where

f t (s) = ( f ◦ f ◦ ··· ◦ f ) (s).

{z

t times

3.34 IMO 1993 269 If m + n is a prime number that does not divide 2 k − 1 for k = 1,2,...,m + n − 2,

prove that the smallest value of t that satisfies the above conditions is m +n+1 4 , where [x] denotes the greatest integer less than or equal to x.

14. (ISR 1) The vertices D , E, F of an equilateral triangle lie on the sides BC,CA, AB respectively of a triangle ABC. If a , b, c are the respective lengths of these sides, and S the area of ABC, prove that

15. (MKD 1) IMO 4 For three points A , B,C in the plane we define m(ABC) to be the smallest length of the three altitudes of the triangle ABC, where in the case of

A , B,C collinear, m(ABC) = 0. Let A, B,C be given points in the plane. Prove that for any point X in the plane,

m (ABC) ≤ m(ABX) + m(AXC) + m(XBC).

16. (MKD 3) Let n ∈ N, n ≥ 2, and A 0 = (a 01 ,a 02 ,...,a 0n ) be any n-tuple of natural numbers such that 0 ≤a 0i ≤ i − 1, for i = 1,...,n. The n-tuples A 1 = (a 11 ,a 12 ,...,a 1n ), A 2 = (a 21 ,a 22 ,...,a 2n ), . . . are defined by

a i +1, j = Card{a i ,l | 1 ≤ l ≤ j − 1,a i ,l ≥a i ,j }, for i ∈ N and j = 1,...,n. Prove that there exists k ∈ N, such that A k +2 =A k .

17. (NLD 2) IMO 6 Let n be an integer greater than 1. In a circular arrangement of n lamps L 0 , . . ., L n −1 , each one of that can be either on or off, we start with the situation where all lamps are on, and then carry out a sequence of steps, Step 0 , Step 1 , . . . . If L j −1 (indices are taken modulo 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.

18. (POL 1) Let S n

be the number of sequences (a 1 ,a 2 ,...,a n ), where a i ∈ {0,1}, in which no six consecutive blocks are equal. Prove that S n → ∞ as n → ∞.

19. (ROU 2) Let a , b, n be positive integers, b > 1 and b n − 1 | a. Show that the representation of the number a in the base b contains at least n digits different from zero.

n ∈ R (n ≥ 2) such that 0 ≤ ∑ i =1 c i ≤ n. Show that we can find integers k 1 ,...,k n such that ∑ n i =1 k i = 0 and

20. (ROU 3) Let c 1 ,...,c

1 −n≤c i + nk i ≤n for every i = 1, . . . , n.

270 3 Problems

21. (UNK 1) A circle S is said to cut a circle Σ diametrally if their common chord is a diameter of Σ .

Let S A ,S B ,S C be three circles with distinct centers A , B,C respectively. Prove that A , B,C are collinear if and only if there is no unique circle S that cuts each of S A ,S B ,S C diametrally. Prove further that if there exists more than one circle S that cuts each of S A ,S B ,S C diametrally, then all such circles pass through two fixed points. Locate these points in relation to the circles S A ,S B ,S C .

22. (UNK 2) IMO 2 A , B,C, D are four points in the plane, with C, D on the same side of the line AB, such that AC · BD = AD · BC and ∡ADB = 90 ◦ + ∡ACB. Find the ratio

AB ·CD ,

AC · BD

and prove that circles ACD , BCD are orthogonal. (Intersecting circles are said to

be orthogonal if at either common point their tangents are perpendicular.)

23. (UNK 3) A finite set of (distinct) positive integers is called a “DS-set” if each of the integers divides the sum of them all. Prove that every finite set of positive integers is a subset of some DS-set.

24. (USA 3) Prove that

3 for all positive real numbers a , b, c, d.

b + 2c + 3d

c + 2d + 3a d + 2a + 3b a + 2b + 3c

25. (VNM 1) Solve the following system of equations, in which a is a given number satisfying |a| > 1:

x 2 = ax 2 + 1, x 2 1 2 = ax 3 + 1,

··· x 2 999 = ax 1000 + 1, x 2 1000 = ax 1 + 1.

26. (VNM 2) Let a , b, c, d be four nonnegative numbers satisfying a + b + c + d = 1. Prove the inequality

1 176 abc + bcd + cda + dab ≤

abcd .

3.35 IMO 1994 271