The Thirty-Second IMO Sigtuna, Sweden, July 12–23, 1991

3.32 The Thirty-Second IMO Sigtuna, Sweden, July 12–23, 1991

3.32.1 Contest Problems

First Day (July 17)

1. Prove for each triangle ABC the inequality

1 IA · IB · IC

A l B l C 27 where I is the incenter and l A ,l B ,l C are the lengths of the angle bisectors of ABC.

be all natural numbers that are less than n and relatively prime to n. Show that if a 1 ,a 2 ,...,a k is an arithmetic progression, then n is a prime number or a natural power of two.

2. Let n > 6 and let a 1 <a 2 <...<a k

3. Let S = {1,2,3,...,280}. Find the minimal natural number n such that in any n -element subset of S there are five numbers that are pairwise relatively prime.

Second Day (July 18)

4. Suppose G is a connected graph with n edges. Prove that it is possible to label the edges of G from 1 to n in such a way that in every vertex v of G with two or more incident edges, the set of numbers labeling those edges has no common divisor greater than 1.

5. Let ABC be a triangle and M an interior point in ABC. Show that at least one of the angles ∡MAB , ∡MBC, and ∡MCA is less than or equal to 30 ◦ .

6. Given a real number a > 1, construct an infinite and bounded sequence x 0 ,x 1 , x 2 , . . . such that for all natural numbers i and j, i 6= j, the following inequality holds:

|x a

i −x j ||i − j| ≥ 1.

3.32.2 Shortlisted Problems

1. (PHI 3) Let ABC be any triangle and P any point in its interior. Let P 1 ,P 2 be the feet of the perpendiculars from P to the two sides AC and BC. Draw AP and BP , and from C drop perpendiculars to AP and BP. Let Q 1 and Q 2 be the feet of these perpendiculars. Prove that the lines Q 1 P 2 ,Q 2 P 1 , and AB are concurrent.

2. (JPN 5) For an acute triangle ABC, M is the midpoint of the segment BC, P is

a point on the segment AM such that PM = BM, H is the foot of the perpendic- ular line from P to BC, Q is the point of intersection of segment AB and the line passing through H that is perpendicular to PB, and finally, R is the point of inter- section of the segment AC and the line passing through H that is perpendicular to PC. Show that the circumcircle of △QHR is tangent to the side BC at point H.

250 3 Problems

3. (PRK 1) Let S be any point on the circumscribed circle of △PQR. Then the feet of the perpendiculars from S to the three sides of the triangle lie on the same straight line. Denote this line by l (S, PQR). Suppose that the hexagon ABCDEF is inscribed in a circle. Show that the four lines l (A, BDF), l(B, ACE), l(D, ABF), and l (E, ABC) intersect at one point if and only if CDEF is a rectangle.

4. (FRA 2) IMO 5 Let ABC be a triangle and M an interior point in ABC. Show that at

least one of the angles ∡MAB , ∡MBC, and ∡MCA is less than or equal to 30 ◦ .

5. (ESP 4) In the triangle ABC, with ∡A = 60 ◦ , a parallel IF to AC is drawn through the incenter I of the triangle, where F lies on the side AB. The point P on the side BC is such that 3BP = BC. Show that ∡BFP = ∡B/2.

6. (USS 4) IMO 1 Prove for each triangle ABC the inequality

· IB · IC

1 IA 8

4 l A l B l C 27 where I is the incenter and l A ,l B ,l C are the lengths of the angle bisectors of ABC.

7. (CHN 2) Let O be the center of the circumsphere of a tetrahedron ABCD. Let L , M, N be the midpoints of BC,CA, AB respectively, and assume that AB + BC =

AD +CD, BC +CA = BD + AD, and CA + AB = CD + BD. Prove that ∠LOM = ∠MNG = ∠NOL.

8. (NLD 1) Let S be a set of n points in the plane. No three points of S are collinear. Prove that there exists a set P containing 2n − 5 points satisfying the following condition: In the interior of every triangle whose three vertices are elements of S lies a point that is an element of P.

9. (FRA 3) In the plane we are given a set E of 1991 points, and certain pairs of these points are joined with a path. We suppose that for every point of E, there exist at least 1593 other points of E to which it is joined by a path. Show that there exist six points of E every pair of which are joined by a path.

Alternative version. Is it possible to find a set E of 1991 points in the plane and paths joining certain pairs of the points in E such that every point of E is joined with a path to at least 1592 other points of E, and in every subset of six points of

E there exist at least two points that are not joined?

10. (USA 5) IMO 4 Suppose G is a connected graph with n edges. Prove that it is possible to label the edges of G from 1 to n in such a way that in every vertex v of G with two or more incident edges, the set of numbers labeling those edges has no common divisor greater than 1.

11. (AUS 4) Prove that

m =0 1991 −m

3.32 IMO 1991 251

12. (CHN 3) IMO 3 Let S = {1,2,3,...,280}. Find the minimal natural number n such that in any n-element subset of S there are five numbers that are pairwise rela- tively prime.

13. (POL 4) Given any integer n ≥ 2, assume that the integers a 1 ,a 2 ,...,a n are not divisible by n and, moreover, that n does not divide a 1 +a 2 + ···+ a n . Prove that there exist at least n different sequences (e 1 ,e 2 , ··· ,e n ) consisting of zeros or ones such that e 1 a 1 +e 2 a 2 + ··· + e n a n is divisible by n.

14. (POL 3) Let a , b, c be integers and p an odd prime number. Prove that if f (x) = ax 2 + bx + c is a perfect square for 2p − 1 consecutive integer values of x, then p divides b 2 − 4ac.

15. (USS 2) Let a n

be the last nonzero digit in the decimal representation of the number n!. Does the sequence a 1 ,a 2 ,...,a n , . . . become periodic after a finite number of terms?

be all natural numbers that are less than n and relatively prime to n. Show that if a 1 ,a 2 ,...,a k is an arithmetic progression, then n is a prime number or a natural power of two.

16. (ROU 1) IMO 2 Let n > 6 and a 1 <a 2 < ··· < a k

17. (HKG 4) Find all positive integer solutions x , y, z of the equation 3 x +4 y =5 z .

18. (BGR 1) Find the highest degree k of 1991 for which 1991 k divides the number

19. (IRL 5) Let a be a rational number with 0 < a < 1 and suppose that

cos 3 π a + 2 cos2 π a = 0.

(Angle measurements are in radians.) Prove that a = 2/3.

20. (IRL 3) Let α

be the positive root of the equation x 2 = 1991x + 1. For natural numbers m , n define

m ∗ n = mn + [ α m ][ α n ],

where [x] is the greatest integer not exceeding x. Prove that for all natural num- bers p , q, r,

(p ∗ q) ∗ r = p ∗ (q ∗ r).

21. (HKG 6) Let f (x) be a monic polynomial of degree 1991 with integer coeffi- cients. Define g (x) = f 2 (x) − 9. Show that the number of distinct integer solu- tions of g (x) = 0 cannot exceed 1995.

22. (USA 4) Real constants a , b, c are such that there is exactly one square all of whose vertices lie on the cubic curve y √ =x 3 + ax 2 + bx + c. Prove that the square has sides of length 4 72.

23. (IND 2) Let f and g be two integer-valued functions defined on the set of all integers such that

(i) f (m + f ( f (n))) = − f ( f (m + 1) − n for all integers m and n;

252 3 Problems (ii) g is a polynomial function with integer coefficients and g (n) = g( f (n)) for

all integers n. Determine f (1991) and the most general form of g.

24. (IND 1) An odd integer n ≥ 3 is said to be “nice” if there is at least one permu- tation a 1 ,a 2 ,...,a n of 1 , 2, . . . , n such that the n sums a 1 −a 2 +a 3 − ··· − a n −1 +

a n ,a 2 −a 3 +a 4 − ··· − a n +a 1 ,a 3 −a 4 +a 5 − ··· − a 1 +a 2 ,...,a n −a 1 +a 2 −

··· − a n −2 +a n −1 are all positive. Determine the set of all “nice” integers.

25. (USA 1) Suppose that n ≥ 2 and x 1 ,x 2 ,...,x n are real numbers between 0 and 1

(inclusive). Prove that for some index i between 1 and n − 1 the inequality

26. (CZS 1) Let n ≥ 2 be a natural number and let the real numbers p, a 1 ,a 2 , . . ., a n ,b 1 ,b 2 , . . ., b n satisfy 1 /2 ≤ p ≤ 1, 0 ≤ a i ,0 ≤b i ≤ p, i = 1,...,n, and ∑ n i =1 a i n =∑ i =1 b i = 1. Prove the inequality

27. (POL 2) Determine the maximum value of the sum ∑ i <j x i x j (x i +x j ) over all n -tuples (x 1 ,...,x n ), satisfying x i ≥ 0 and ∑ n i =1 x i = 1.

28. (NLD 2) IMO 6 Given a real number a > 1, construct an infinite and bounded se- quence x 0 ,x 1 ,x 2 , . . . such that for all natural numbers i and j, i 6= j, the following

inequality holds: |x i −x j ||i − j| a ≥ 1.

29. (FIN 2) We call a set S on the real line R superinvariant if for any stretching A of the set by the transformation taking x to A (x) = x 0 + a(x − x 0 ) there exists a translation B, B (x) = x + b, such that the images of S under A and B agree; i.e., for any x ∈ S there is a y ∈ S such that A(x) = B(y) and for any t ∈ S there is a u ∈ S such that B(t) = A(u). Determine all superinvariant sets.

Remark. It is assumed that a > 0.

30. (BGR 3) Two students A and B are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student

A : “Can you tell the integer written by the other student?” If A answers “no,” the referee puts the same question to student B. If B answers “no,” the referee puts the question back to A, and so on. Assume that both students are intelligent and truthful. Prove that after a finite number of questions, one of the students will answer “yes.”

3.33 IMO 1992 253