The Thirty-Eighth IMO Mar del Plata, Argentina, July 18–31, 1997

3.38 The Thirty-Eighth IMO Mar del Plata, Argentina, July 18–31, 1997

3.38.1 Contest Problems

First Day (July 24)

1. An infinite square grid is colored in the chessboard pattern. For any pair of pos- itive integers m , n consider a right-angled triangle whose vertices are grid points

and whose legs, of lengths m and n, run along the lines of the grid. Let S b be the total area of the black part of the triangle and S w the total area of its white part.

Define the function f (m, n) = |S b −S w |.

(a) Calculate f (m, n) for all m, n that have the same parity. (b) Prove that f

(m, n) ≤ 1

2 max (m, n).

(c) Show that f (m, n) is not bounded from above.

2. In triangle ABC the angle at A is the smallest. A line through A meets the circum- circle again at the point U lying on the arc BC opposite to A. The perpendicular bisectors of CA and AB meet AU at V and W , respectively, and the lines CV , BW meet at T . Show that AU = T B + TC.

3. Let x 1 ,x 2 ,...,x n

be real numbers satisfying the conditions

Show that there exists a permutation y 1 ,...,y n of the sequence x 1 ,...,x n such that

n +1

|y 1 + 2y 2 + ··· + ny n |≤

Second Day (July 25)

4. An n × n matrix with entries from {1,2,...,2n − 1} is called a silver matrix if for each i the union of the ith row and the ith column contains 2n − 1 distinct entries. Show that:

(a) There exist no silver matrices for n = 1997. (b) Silver matrices exist for infinitely many values of n.

5. Find all pairs of integers x , y ≥ 1 satisfying the equation x y 2 =y x .

6. For a positive integer n, let f (n) denote the number of ways to represent n as the sum of powers of 2 with nonnegative integer exponents. Representations that differ only in the ordering in their summands are not considered to be distinct. (For instance, f (4) = 4 because the number 4 can be represented in the following four ways: 4; 2+2; 2+1+1; 1+1+1+1.) Prove that the inequality

2 n 2 /4 < f (2 n )<2 n 2 /2

holds for any integer n ≥ 3.

3.38 IMO 1997 287

3.38.2 Shortlisted Problems

1. (BLR) IMO 1 An infinite square grid is colored in the chessboard pattern. For any pair of positive integers m , n consider a right-angled triangle whose vertices are grid points and whose legs, of lengths m and n, run along the lines of the grid.

Let S b be the total area of the black part of the triangle and S w the total area of

its white part. Define the function f (m, n) = |S b −S w |.

(a) Calculate f (m, n) for all m, n that have the same parity. (b) Prove that f

(m, n) ≤ 1

2 max (m, n).

(c) Show that f (m, n) is not bounded from above.

2. (CAN) Let R 1 ,R 2 , . . . be the family of finite sequences of positive integers de- fined by the following rules: R 1 = (1), and if R n −1 = (x 1 ,...,x s ), then

R n = (1, 2, . . . , x 1 , 1, 2, . . . , x 2 , . . . , 1, 2, . . . , x s , n). For example, R 2 = (1, 2), R 3 = (1, 1, 2, 3), R 4 = (1, 1, 1, 2, 1, 2, 3, 4).

Prove that if n > 1, then the kth term from the left in R n is equal to 1 if and only if the kth term from the right in R n is different from 1.

3. (GER) For each finite set U of nonzero vectors in the plane we define l (U) to

be the length of the vector that is the sum of all vectors in U . Given a finite set

V of nonzero vectors in the plane, a subset B of V is said to be maximal if l (B) is greater than or equal to l (A) for each nonempty subset A of V . (a) Construct sets of 4 and 5 vectors that have 8 and 10 maximal subsets re- spectively. (b) Show that for any set V consisting of n ≥ 1 vectors, the number of maximal subsets is less than or equal to 2n.

4. (IRN) IMO 4 An n × n matrix with entries from {1,2,...,2n − 1} is called a cov- eralls matrix if for each i the union of the ith row and the ith column contains 2n − 1 distinct entries. Show that:

(a) There exist no coveralls matrices for n = 1997. (b) Coveralls matrices exist for infinitely many values of n.

5. (ROU) Let ABCD be a regular tetrahedron and M , N distinct points in the planes ABC and ADC respectively. Show that the segments MN , BN, MD are the sides of a triangle.

6. (IRL) (a) Let n be a positive integer. Prove that there exist distinct positive integers x , y, z such that

x n −1 +y n =z n +1 .

(b) Let a , b, c be positive integers such that a and b are relatively prime and c is relatively prime either to a or to b. Prove that there exist infinitely many triples (x, y, z) of distinct positive integers x, y, z such that

x a +y b =z c .

288 3 Problems Original formulation: Let a , b, c, n be positive integers such that n is odd and ac

is relatively prime to 2b. Prove that there exist distinct positive integers x , y, z such that

(i) x a +y b =z c , and (ii) xyz is relatively prime to n.

7. (RUS) Let ABCDEF be a convex hexagon such that AB = BC, CD = DE, EF = FA . Prove that

BC DE FA 3

BE DA FC ≥ 2

When does equality occur?

8. (UNK) IMO 2 In triangle ABC the angle at A is the smallest. A line through A meets the circumcircle again at the point U lying on the arc BC opposite to A. The perpendicular bisectors of CA and AB meet AU at V and W , respectively, and the lines CV , BW meet at T . Show that AU = T B + TC. Original formulation. Four different points A , B,C, D are chosen on a circle Γ such that the triangle BCD is not right-angled. Prove that:

(a) The perpendicular bisectors of AB and AC meet the line AD at certain points W and V , respectively, and that the lines CV and BW meet at a certain point T .

(b) The length of one of the line segments AD, BT , and CT is the sum of the lengths of the other two.

9. (USA) Let A 1 A 2 A 3 be a nonisosceles triangle with incenter I. Let C i ,i = 1, 2, 3,

be the smaller circle through I tangent to A i A i +1 and A i A i +2 (the addition of indices being mod 3). Let B i ,i = 1, 2, 3, be the second point of intersection of

C i +1 and C i +2 . Prove that the circumcenters of the triangles A 1 B 1 I ,A 2 B 2 I ,A 3 B 3 I are collinear.

10. (CZE) Find all positive integers k for which the following statement is true: If F (x) is a polynomial with integer coefficients satisfying the condition

0 ≤ F(c) ≤ k for each c ∈ {0,1,...,k + 1}, then F (0) = F(1) = ··· = F(k + 1).

11. (NLD) Let P (x) be a polynomial with real coefficients such that P(x) > 0 for all x ≥ 0. Prove that there exists a positive integer n such that (1 + x) n P (x) is a polynomial with nonnegative coefficients.

12. (ITA) Let p be a prime number and let f (x) be a polynomial of degree d with integer coefficients such that:

(i) f (0) = 0, f (1) = 1; (ii) for every positive integer n, the remainder of the division of f (n) by p is either 0 or 1. Prove that d ≥ p − 1.

13. (IND) In town A, there are n girls and n boys, and each girl knows each boy. In town B, there are n girls g 1 ,g 2 ,...,g n and 2n − 1 boys b 1 ,b 2 , . . ., b 2n −1 . The

3.38 IMO 1997 289 girl g i ,i = 1, 2, . . . , n, knows the boys b 1 ,b 2 ,...,b 2i −1 , and no others. For all

r = 1, 2, . . . , n, denote by A(r), B(r) the number of different ways in which r girls from town A, respectively town B, can dance with r boys from their own town, forming r pairs, each girl with a boy she knows. Prove that A (r) = B(r) for each r = 1, 2, . . . , n.

14. (IND) Let b , m, n be positive integers such that b > 1 and m 6= n. Prove that if

b m − 1 and b n − 1 have the same prime divisors, then b + 1 is a power of 2.

15. (RUS) An infinite arithmetic progression whose terms are positive integers con- tains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

16. (BLR) In an acute-angled triangle ABC, let AD , BE be altitudes and AP, BQ internal bisectors. Denote by I and O the incenter and the circumcenter of the triangle, respectively. Prove that the points D, E, and I are collinear if and only if the points P, Q, and O are collinear.

17. (CZE) IMO 5

Find all pairs of integers x 2 , y ≥ 1 satisfying the equation x y =y x .

18. (UNK) The altitudes through the vertices A , B,C of an acute-angled triangle ABC meet the opposite sides at D , E, F, respectively. The line through D parallel to EF meets the lines AC and AB at Q and R, respectively. The line EF meets BC at P. Prove that the circumcircle of the triangle PQR passes through the midpoint of BC.

19. (IRL) Let a 1 ≥ ··· ≥ a n ≥a n +1 = 0 be a sequence of real numbers. Prove that s n

a k ≤ ∑ k ( a k − a k +1 ).

k =1

k =1

20. (IRL) Let D be an internal point on the side BC of a triangle ABC. The line AD meets the circumcircle of ABC again at X . Let P and Q be the feet of the

be the circle with diameter X D. Prove that the line PQ is tangent to γ if and only if AB = AC.

perpendiculars from X to AB and AC, respectively, and let γ

21. (RUS) IMO 3 Let x 1 ,x 2 ,...,x n

be real numbers satisfying the conditions

Show that there exists a permutation y 1 ,...,y n of the sequence x 1 ,...,x n such that

n +1

|y 1 + 2y 2 + ··· + ny n |≤

22. (UKR) (a) Do there exist functions f : R → R and g : R → R such that

g ( f (x)) = x 3 for all x ∈ R? (b) Do there exist functions f : R → R and g : R → R such that

f (g(x)) = x 2 and

f (g(x)) = x 2 and

g ( f (x)) = x 4 for all x ∈ R?

290 3 Problems

23. (UNK) Let ABCD be a convex quadrilateral and O the intersection of its diago- nals AC and BD. If

OA sin ∠A + OC sin ∠C = OB sin ∠B + OD sin ∠D, prove that ABCD is cyclic.

24. (LTU) IMO 6 For a positive integer n, let f (n) denote the number of ways to rep- resent n as the sum of powers of 2 with nonnegative integer exponents. Repre- sentations that differ only in the ordering in their summands are not considered

to be distinct. (For instance, f (4) = 4 because the number 4 can be represented in the following four ways: 4; 2+2; 2+1+1; 1+1+1+1.) Prove that the inequality

2 n 2 /4 < f (2 n )<2 n 2 /2

holds for any integer n ≥ 3.

25. (POL) The bisectors of angles A , B,C of a triangle ABC meet its circumcircle again at the points K , L, M, respectively. Let R be an internal point on the side

AB . The points P and Q are defined by the following conditions: RP is parallel to AK , and BP is perpendicular to BL; RQ is parallel to BL, and AQ is perpendicular to AK. Show that the lines KP , LQ, MR have a point in common.

26. (ITA) For every integer n ≥ 2 determine the minimum value that the sum

a 0 +a 1 + ··· + a n can take for nonnegative numbers a 0 ,a 1 ,...,a n satisfying the condition

a 0 = 1,

a i ≤a i +1 +a i +2

for i = 0, . . . , n − 2.

3.39 IMO 1998 291