The Thirty-Fifth IMO Hong Kong, July 9–22, 1994

3.35 The Thirty-Fifth IMO Hong Kong, July 9–22, 1994

3.35.1 Contest Problems

First Day (July 13)

1. Let m and n be positive integers. The set A = {a 1 ,a 2 ,...,a m } is a subset of

1 , 2, . . . , n. Whenever a i +a j ≤ n, 1 ≤ i ≤ j ≤ m, a i +a j also belongs to A. Prove that

a 1 +a 2 + ··· + a m

n +1

2. N is an arbitrary point on the bisector of ∠BAC. P and O are points on the lines AB and AN, respectively, such that ∡ANP = 90 ◦ = ∡APO. Q is an arbitrary point on NP, and an arbitrary line through Q meets the lines AB and AC at E and F respectively. Prove that ∡OQE = 90 ◦ if and only if QE = QF.

3. For any positive integer k, A k is the subset of {k + 1,k + 2,...,2k} consisting of all elements whose digits in base 2 contain exactly three 1’s. Let f (k) denote the number of elements in A k .

(a) Prove that for any positive integer m, f (k) = m has at least one solution. (b) Determine all positive integers m for which f (k) = m has a unique solution.

Second Day (July 14)

4. Determine all pairs (m, n) of positive integers such that n 3 mn +1 −1 is an integer.

5. Let S be the set of real numbers greater than −1. Find all functions f : S → S such that

f (x + f (y) + x f (y)) = y + f (x) + y f (x) for all x and y in S, and f (x)/x is strictly increasing for −1 < x < 0 and for 0 < x.

6. Find a set A of positive integers such that for any infinite set P of prime num- bers, there exist positive integers m ∈ A and n 6∈ A, both the product of the same number (at least two) of distinct elements of P.

3.35.2 Shortlisted Problems

1. A1 (USA) Let a

0 = 1994 and a n +1 = a n +1 for each nonnegative integer n. Prove

that 1994 − n is the greatest integer less than or equal to a n ,0 ≤ n ≤ 998.

2. A2 (FRA) IMO 1 Let m and n be positive integers. The set A = {a 1 ,a 2 ,...,a m } is a subset of {1,2,...,n}. Whenever a i +a j ≤ n, 1 ≤ i ≤ j ≤ m, a i +a j also belongs to A. Prove that

a 1 +a 2 + ··· + a m

n +1

272 3 Problems

3. A3 (UNK) IMO 5 Let S be the set of real numbers greater than −1. Find all func- tions f : S → S such that

f (x + f (y) + x f (y)) = y + f (x) + y f (x) for all x and y in S, and f (x)/x is strictly increasing for −1 < x < 0 and for 0 < x.

4. A4 (MNG) Let R denote the set of all real numbers and R + the subset of all positive ones. Let α and β

be given elements in R, not necessarily distinct. Find all functions f : R + → R such that

f (x) f (y) = y α f +x β f for all x and y in R + .

5. A5 (POL) Let f (x) = x 2 +1

for x 6= 0. Define f (x) = x and f (x) =

(0) (n)

2x

f (f (n−1) (x)) for all positive integers n and x 6= 0. Prove that for all nonnega- tive integers n and x 6= −1,0, or 1,

f (n) (x)

f (n+1)

(x)

f x +1 2 x n −1

6. C1 (UKR) On a 5 × 5 board, two players alternately mark numbers on empty cells. The first player always marks 1’s, the second 0’s. One number is marked per turn, until the board is filled. For each of the nine 3 × 3 squares the sum of the nine numbers on its cells is computed. Denote by A the maximum of these sums. How large can the first player make A, regardless of the responses of the second player?

7. C2 (COL) In a certain city, age is reckoned in terms of real numbers rather than integers. Every two citizens x and x ′ either know each other or do not know each other. Moreover, if they do not, then there exists a chain of citizens x =

x 0 ,x 1 ,...,x n =x ′ for some integer n ≥ 2 such that x i −1 and x i know each other. In a census, all male citizens declare their ages, and there is at least one male citizen. Each female citizen provides only the information that her age is the average of the ages of all the citizens she knows. Prove that this is enough to determine uniquely the ages of all the female citizens.

8. C3 (MKD) Peter has three accounts in a bank, each with an integral number of dollars. He is only allowed to transfer money from one account to another so that the amount of money in the latter is doubled.

(a) Prove that Peter can always transfer all his money into two accounts. (b) Can Peter always transfer all his money into one account?

9. C4 (EST) There are n + 1 fixed positions in a row, labeled 0 to n in increasing order from right to left. Cards numbered 0 to n are shuffled and dealt, one in each position. The object of the game is to have card i in the ith position for 0 ≤ i ≤ n. If this has not been achieved, the following move is executed. Determine the smallest k such that the kth position is occupied by a card l > k. Remove this card, slide all cards from the (k + 1)st to the lth position one place to the right, and replace the card l in the lth position.

3.35 IMO 1994 273 (a) Prove that the game lasts at most 2 n − 1 moves.

(b) Prove that there exists a unique initial configuration for which the game lasts exactly 2 n − 1 moves.

10. C5 (SWE) At a round table are 1994 girls, playing a game with a deck of n cards. Initially, one girl holds all the cards. In each turn, if at least one girl holds at least two cards, one of these girls must pass a card to each of her two neighbors. The game ends when and only when each girl is holding at most one card.

(a) Prove that if n ≥ 1994, then the game cannot end. (b) Prove that if n < 1994, then the game must end.

11. C6 (FIN) On an infinite square grid, two players alternately mark symbols on empty cells. The first player always marks X ’s, the second O’s. One symbol is marked per turn. The first player wins if there are 11 consecutive X ’s in a row, column, or diagonal. Prove that the second player can prevent the first from winning.

12. C7 (BRA) Prove that for any integer n ≥ 2, there exists a set of 2 n −1 points in the plane such that no 3 lie on a line and no 2n are the vertices of a convex

2n-gon.

13. G1 (FRA) A semicircle Γ is drawn on one side of a straight line l. C and D are points on Γ . The tangents to Γ at C and D meet l at B and A respectively, with the center of the semicircle between them. Let E be the point of intersection of

AC and BD, and F the point on l such that EF is perpendicular to l. Prove that EF bisects ∠CFD.

14. G2 (UKR) ABCD is a quadrilateral with BC parallel to AD. M is the midpoint of CD, P that of MA and Q that of MB. The lines DP and CQ meet at N. Prove that N is not outside triangle ABM. 9

15. G3 (RUS) A circle ω is tangent to two parallel lines l 1 and l 2 . A second circle ω 1 is tangent to l 1 at A and to ω externally at C. A third circle ω 2 is tangent to l 2 at B, to ω externally at D, and to ω 1 externally at E. AD intersects BC at Q. Prove that Q is the circumcenter of triangle CDE.

16. G4 (AUS-ARM) IMO 2 N is an arbitrary point on the bisector of ∠BAC. P and O are points on the lines AB and AN, respectively, such that ∡ANP = 90 ◦ = ∡APO. Q is an arbitrary point on NP, and an arbitrary line through Q meets the lines AB and AC at E and F respectively. Prove that ∡OQE = 90 ◦ if and only if QE = QF.

17. G5 (CYP) A line l does not meet a circle ω with center O. E is the point on l such that OE is perpendicular to l. M is any point on l other than E. The tangents from M to ω touch it at A and B. C is the point on MA such that EC is perpendicular to MA. D is the point on MB such that ED is perpendicular to MB.

9 This problem is false. However, it is true if “not outside ABM” is replaced by “not outside ABCD ”.

274 3 Problems The line CD cuts OE at F. Prove that the location of F is independent of that of

18. N1 (BGR) M is a subset of {1,2,3,...,15} such that the product of any three distinct elements of M is not a square. Determine the maximum number of ele- ments in M.

19. N2 (AUS) 3 IMO 4 Determine all pairs (m, n) of positive integers such that n +1 mn −1 is an integer.

20. N3 (FIN) IMO 6 Find a set A of positive integers such that for any infinite set P of prime numbers, there exist positive integers m ∈ A and n 6∈ A, both the product of the same number of distinct elements of P.

21. N4 (FRA) For any positive integer x 0 , three sequences {x n },{y n }, and {z n } are defined as follows: (i) y 0 = 4 and z 0 = 1; (ii) if x

≥ 0, x n n +1 = 2 ,y n +1 = 2y n , and z n +1 =z n ; (iii) if x is odd for n

n is even for n

n − 2 −z n ,y n +1 =y n , and z n +1 =y n +z n . The integer x 0 is said to be good if x n = 0 for some n ≥ 1. Find the number of

≥ 0, x n

+1 =x

good integers less than or equal to 1994.

22. N5 (ROU) IMO 3 For any positive integer k, A k is the subset of {k+1,k+2,...,2k} consisting of all elements whose digits in base 2 contain exactly three 1’s. Let

f (k) denote the number of elements in A k . (a) Prove that for any positive integer m, f (k) = m has at least one solution.

(b) Determine all positive integers m for which f (k) = m has a unique solution.

23. N6 (LVA) Let x 1 and x 2 be relatively prime positive integers. For n ≥ 2, define x n +1 =x n x n −1 + 1. (a) Prove that for every i > 1, there exists j > i such that x i i divides x j j .

(b) Is it true that x

1 must divide x j for some j > 1?

24. N7 (UNK) A wobbly number is a positive integer whose digits in base 10 are al- ternately nonzero and zero, the units digit being nonzero. Determine all positive integers that do not divide any wobbly number.

3.36 IMO 1995 275