The Forty-Fourth IMO Tokyo, Japan, July 7–19, 2003

3.44 The Forty-Fourth IMO Tokyo, Japan, July 7–19, 2003

3.44.1 Contest Problems

First Day (July 13)

1. Let A be a 101-element subset of the set S = {1,2,...,1000000}. Prove that there exist numbers t 1 ,t 2 , . . . ,t 100 in S such that the sets

j = 1, 2, . . . , 100, are pairwise disjoint.

A j = {x + t j |x ∈ A},

2. Determine all pairs (a, b) of positive integers such that

a 2 2ab 2 −b 3 +1

is a positive integer.

3. Each pair of opposite sides of a convex hexagon has the following property: The √ distance between their midpoints is equal to 3 /2 times the sum of their lengths. Prove that all the angles of the hexagon are equal.

Second Day (July 14)

4. Let ABCD be a cyclic quadrilateral. Let P , Q, R be the feet of the perpendiculars from D to the lines BC ,CA, AB, respectively. Show that PQ = QR if and only if the bisectors of ∠ABC and ∠ADC are concurrent with AC.

be real numbers. (a) Prove that

5. Let n be a positive integer and let x 1 ≤x 2 ≤ ··· ≤ x n

2 (n 2 − 1) n

3 ∑ i −x j i ) i , j=1

∑ 2 |x

(b) Show that equality holds if and only if x 1 ,...,x n is an arithmetic progres- sion.

6. Let p be a prime number. Prove that there exists a prime number q such that for every integer n, the number n p − p is not divisible by q.

3.44.2 Shortlisted Problems

1. A1 (USA) Let a ij ,i = 1, 2, 3, j = 1, 2, 3, be real numbers such that a ij is positive for i = j and negative for i 6= j.

Prove that there exist positive real numbers c 1 ,c 2 ,c 3 such that the numbers

a 31 c 1 +a 32 c 2 +a 33 c 3 are all negative, all positive, or all zero.

a 11 c 1 +a 12 c 2 +a 13 c 3 ,

a 21 c 1 +a 22 c 2 +a 23 c 3 ,

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2. A2 (AUS) Find all nondecreasing functions f : R → R such that (i) f (0) = 0, f (1) = 1; (ii) f (a) + f (b) = f (a) f (b) + f (a + b − ab) for all real numbers a,b such that

a < 1 < b.

3. A3 (GEO) Consider pairs of sequences of positive real numbers a 1 ≥a 2 ≥a 3 ≥ ···, b 1 ≥b 2 ≥b 3 ≥ ··· and the sums A n =a 1 + ··· + a n ,B n =b 1 + ··· + b n ,n =

1 , 2, . . . . For any pair define c i = min{a i ,b i } and C n =c 1 + ···+ c n ,n = 1, 2, . . . . (a) Does there exist a pair (a i ) i ≥1 , (b i ) i ≥1 such that the sequences (A n ) n ≥1 and (B n ) n ≥1 are unbounded while the sequence (C n ) n ≥1 is bounded? (b) Does the answer to question (1) change by assuming additionally that b i =

1 /i, i = 1, 2, . . .? Justify your answer.

be real numbers. (a) Prove that n

4. A4 (IRL) IMO 5 Let n be a positive integer and let x 1 ≤x 2 ≤ ··· ≤ x n

2 (n 2 − 1) n

∑ |x i −x j | ≤ ∑ (x i −x j ) 2 .

i , j=1

3 i , j=1 (b) Show that equality holds if and only if x 1 ,...,x n is an arithmetic progres-

sion.

5. A5 (KOR) Let R +

be the set of all positive real numbers. Find all functions

f : R + →R + that satisfy the following conditions: √

(i) f (xyz) + f (x) + f (y) + f (z) = f (√xy) f (√yz) f ( zx ) for all x, y, z ∈ R + . (ii) f (x) < f (y) for all 1 ≤ x < y.

6. A6 (USA) Let n be a positive integer and let (x 1 ,...,x n ), (y 1 ,...,y n ) be two sequences of positive real numbers. Suppose (z 2 ,z 3 ,...,z 2n ) is a sequence of positive real numbers such that

z 2 i +j ≥x i y j for all 1 ≤ i, j ≤ n. Let M = max{z 2 ,...,z 2n }. Prove that M 2 +z

7. C1 (BRA) IMO 1 Let A be a 101-element subset of the set S = {1,2,..., 1000000}. Prove that there exist numbers t 1 ,t 2 , . . . ,t 100 in S such that the sets

A j = {x + t j | x ∈ A}, j = 1,2,...,100,

are pairwise disjoint.

be closed disks in the plane. (A closed disk is a region bounded by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most 2003 disks D i . Prove that there exists a disk D k that intersects at most 7 · 2003 − 1 other disks D i .

8. C2 (GEO) Let D 1 ,...,D n

314 3 Problems

9. C3 (LTU) Let n ≥ 5 be a given integer. Determine the largest integer k for which there exists a polygon with n vertices (convex or not, with non-self-intersecting boundary) having k internal right angles.

be real numbers. Let A = (a ij ) 1 ≤i, j≤n be the matrix with entries

10. C4 (IRN) Let x 1 ,...,x n and y 1 ,...,y n

a , if x i +y ij j = ≥ 0;

0 , if x i +y j < 0.

Suppose that B is an n × n matrix whose entries are 0, 1 such that the sum of the elements in each row and each column of B is equal to the corresponding sum for the matrix A. Prove that A = B.

11. C5 (ROU) Every point with integer coordinates in the plane is the center of a disk with radius 1 /1000.

(a) Prove that there exists an equilateral triangle whose vertices lie in different disks. (b) Prove that every equilateral triangle with vertices in different disks has side length greater than 96.

12. C6 (SAF) Let f (k) be the number of integers n that satisfy the following con- ditions:

(i) 0 ≤ n < 10 k , so n has exactly k digits (in decimal notation), with leading zeros allowed; (ii) the digits of n can be permuted in such a way that they yield an integer divisible by 11. Prove that f (2m) = 10 f (2m − 1) for every positive integer m.

13. G1 (FIN) IMO 4 Let ABCD be a cyclic quadrilateral. Let P , Q, R be the feet of the perpendiculars from D to the lines BC ,CA, AB, respectively. Show that PQ = QR if and only if the bisectors of ∠ABC and ∠ADC are concurrent with AC.

14. G2 (HEL) Three distinct points A , B,C are fixed on a line in this order. Let Γ

be a circle passing through A and C whose center does not lie on the line AC. Denote by P the intersection of the tangents to Γ at A and C. Suppose Γ meets the segment PB at Q. Prove that the intersection of the bisector of ∠AQC and the line AC does not depend on the choice of Γ .

15. G3 (IND) Let ABC be a triangle and let P be a point in its interior. Denote by D , E, F the feet of the perpendiculars from P to the lines BC, CA, and AB, respectively. Suppose that

AP 2 + PD 2 = BP 2 + PE 2 = CP 2 + PF 2 . Denote by I A ,I B ,I C the excenters of the triangle ABC. Prove that P is the circum-

center of the triangle I A I B I C .

16. G4 (ARM) Let Γ 1 , Γ 2 , Γ 3 , Γ 4 be distinct circles such that Γ 1 , Γ 3 are externally tangent at P, and Γ 2 , Γ 4 are externally tangent at the same point P. Suppose that

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Γ 1 and Γ 2 ; Γ 2 and Γ 3 ; Γ 3 and Γ 4 ; Γ 4 and Γ 1 meet at A , B,C, D, respectively, and that all these points are different from P. Prove that

17. G5 (KOR) Let ABC be an isosceles triangle with AC = BC, whose incenter is I. Let P be a point on the circumcircle of the triangle AIB lying inside the triangle ABC. The lines through P parallel to CA and CB meet AB at D and E, respectively. The line through P parallel to AB meets CA and CB at F and G, respectively. Prove that the lines DF and EG intersect on the circumcircle of the triangle ABC.

18. G6 (POL) IMO 3 Each pair of opposite sides of a convex hexagon has the follow- √ ing property: The distance between their midpoints is equal to 3 /2 times the sum of their lengths. Prove that all the angles of the hexagon are equal.

19. G7 (SAF) Let ABC be a triangle with semiperimeter s and inradius r. The semi- circles with diameters BC ,CA, AB are drawn outside of the triangle ABC. The circle tangent to all three semicircles has radius t. Prove that

<t≤ + 2 1 2 −

20. N1 (POL) Let m be a fixed integer greater than 1. The sequence x 0 ,x 1 ,x 2 , . . . is defined as follows:

Find the greatest k for which the sequence contains k consecutive terms divisible by m.

21. N2 (USA) Each positive integer a undergoes the following procedure in order to obtain the number d = d(a):

(1) move the last digit of a to the first position to obtain the number b; (2) square b to obtain the number c; (3) move the first digit of c to the end to obtain the number d.

(All the numbers in the problem are considered to be represented in base 10.) For example, for a = 2003, we have b = 3200, c = 10240000, and d = 02400001 = 2400001 = d(2003).

Find all numbers a for which d (a) = a 2 .

22. N3 (BGR) IMO 2 Determine all pairs (a, b) of positive integers such that

a 2 2ab 2 −b 3 +1

is a positive integer.

316 3 Problems

23. N4 (ROU) Let b be an integer greater than 5. For each positive integer n, con- sider the number

x n = 11 . . . 1 22 ...2 5 | {z } , | {z }

n −1

written in base b. Prove that the following condition holds if and only if b = 10: There exists a positive integer M such that for every integer n greater than M, the number x n is a perfect square.

24. N5 (KOR) An integer n is said to be good if |n| is not the square of an integer. Determine all integers m with the following property: m can be represented in infinitely many ways as a sum of three distinct good integers whose product is the square of an odd integer.

25. N6 (FRA) IMO 6 Let p be a prime number. Prove that there exists a prime number q such that for every integer n, the number n p − p is not divisible by q.

26. N7 (BRA) The sequence a 0 ,a 1 ,a 2 , . . . is defined as follows:

a 0 = 2,

a k +1 = 2a 2 k −1

for k ≥ 0. Prove that if an odd prime p divides a n , then 2 n +3 divides p 2 − 1.

27. N8 (IRN) Let p be a prime number and let A be a set of positive integers that satisfies the following conditions:

(i) the set of prime divisors of the elements in A consists of p − 1 elements; (ii) for any nonempty subset of A, the product of its elements is not a perfect p th power. What is the largest possible number of elements in A?

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