The Thirty-Sixth IMO Toronto, Canada, July 13–25, 1995

3.36 The Thirty-Sixth IMO Toronto, Canada, July 13–25, 1995

3.36.1 Contest Problems

First Day (July 19)

1. Let A , B,C, and D be distinct points on a line, in that order. The circles with diameters AC and BD intersect at X and Y . O is an arbitrary point on the line XY but not on AD. CO intersects the circle with diameter AC again at M, and BO intersects the other circle again at N. Prove that the lines AM, DN, and XY are concurrent.

2. Let a, b, and c be positive real numbers such that abc = 1. Prove that

(b + c) b 3 (a + c) c 3 (a + b)

3. Determine all integers n > 3 such that there are n points A 1 ,A 2 ,...,A n in the plane that satisfy the following two conditions simultaneously: (a) No three lie on the same line. (b) There exist real numbers p 1 ,p 2 ,...,p n such that the area of △A i A j A k is equal to p i +p j +p k , for 1 ≤ i < j < k ≤ n.

Second Day (July 20)

4. The positive real numbers x 0 ,x 1 ,...,x 1995 satisfy x 0 =x 1995 and

for i = 1, 2, . . . , 1995. Find the maximum value that x 0 can have.

5. Let ABCDEF be a convex hexagon with AB = BC = CD, DE = EF = FA, and ∡BCD = ∡EFA = π /3 (that is, 60 ◦ ). Let G and H be two points interior to the hexagon, such that angles AGB and DHE are both 2 π /3 (that is, 120 ◦ ). Prove that AG + GB + GH + DH + HE ≥ CF.

6. Let p be an odd prime. Find the number of p-element subsets A of {1,2,...,2p} such that the sum of all elements of A is divisible by p.

3.36.2 Shortlisted Problems

1. A1 (RUS) IMO 2 Let a, b, and c be positive real numbers such that abc = 1. Prove that

3 + 3 + 3 ≥ a . (b + c) b (a + c) c (a + b) 2

276 3 Problems

2. A2 (SWE) Let a and b be nonnegative integers such that ab ≥c 2 , where c is an integer. Prove that there is a number n and integers x 1 ,x 2 ,...,x n ,y 1 ,y 2 ,...,y n such that

x 2 ∑ 2 i = a,

∑ y i = b, and

be real numbers such that 2 ≤a i ≤ 3 for i = 1,2,...,n. If s = a 1 +a 2 + ··· + a n , prove that

3. A3 (UKR) Let n be an integer, n ≥ 3. Let a 1 ,a 2 ,...,a n

4. A4 (USA) Let a, b, and c be given positive real numbers. Determine all positive real numbers x, y, and z such that

x +y+z=a+b+c

and

4xyz

− (a 2 x +b 2 y +c 2 z ) = abc.

5. A5 (UKR) Let R be the set of real numbers. Does there exist a function f : R → R that simultaneously satisfies the following three conditions?

(a) There is a positive number M such that −M ≤ f (x) ≤ M for all x. (b) f (1) = 1.

(c) If x 6= 0, then

f x + 2 = f (x) + f .

be real numbers such that x i <x i +1 for 1 ≤ i ≤ n − 1. Prove that

6. A6 (JPN) Let n be an integer, n ≥ 3. Let x 1 ,x 2 ,...,x n

(n − 1) ∑ x

n −1

i x j > ∑ (n − i)x i ∑ ( j − 1)x j .

7. G1 (BGR) IMO 1 Let A , B,C, and D be distinct points on a line, in that order. The circles with diameters AC and BD intersect at X and Y . O is an arbitrary point on the line XY but not on AD. CO intersects the circle with diameter AC again at M, and BO intersects the other circle again at N. Prove that the lines AM , DN, and XY are concurrent.

8. G2 (GER) Let A , B, and C be noncollinear points. Prove that there is a unique point X in the plane of ABC such that X A 2 + XB 2 + AB 2 = XB 2 + XC 2 + BC 2 = XC 2 + XA 2 +CA 2 .

9. G3 (TUR) The incircle of ABC touches BC, CA, and AB at D, E, and F re- spectively. X is a point inside ABC such that the incircle of X BC touches BC at

D also, and touches CX and X B at Y and Z, respectively. Prove that EFZY is a cyclic quadrilateral.

3.36 IMO 1995 277

10. G4 (UKR) An acute triangle ABC is given. Points A 1 and A 2 are taken on the side BC (with A 2 between A 1 and C), B 1 and B 2 on the side AC (with B 2 between

B 1 and A), and C 1 and C 2 on the side AB (with C 2 between C 1 and B) such that ∠AA 1 A 2 = ∠AA 2 A 1 = ∠BB 1 B 2 = ∠BB 2 B 1 = ∠CC 1 C 2 = ∠CC 2 C 1 . The lines AA 1 , BB 1 , and CC 1 form a triangle, and the lines AA 2 , BB 2 , and CC 2

form a second triangle. Prove that all six vertices of these two triangles lie on a single circle.

11. G5 (NZL) IMO 5 Let ABCDEF be a convex hexagon with AB = BC = CD, DE = EF = FA, and ∡BCD = ∡EFA = π /3 (that is, 60 ◦ ). Let G and H be two points interior to the hexagon such that angles AGB and DHE are both 2 π /3 (that is, 120 ◦ ). Prove that AG + GB + GH + DH + HE ≥ CF.

12. G6 (USA) Let A 1 A 2 A 3 A 4 be a tetrahedron, G its centroid, and A ′ ,A ′ 1 2 ,A ′ 3 , and A ′ 4 the points where the circumsphere of A 1 A 2 A 3 A 4 intersects GA 1 , GA 2 , GA 3 , and GA 4 , respectively. Prove that

13. G7 (LVA) O is a point inside a convex quadrilateral ABCD of area S. K, L, M, and N are interior points of the sides AB, BC, CD, and DA respectively. If OKBL √ √ √

and OMDN are parallelograms, prove that S ≥ S 1 + S 2 , where S 1 and S 2 are the areas of ONAK and OLCM respectively.

14. G8 (COL) Let ABC be a triangle. A circle passing through B and C intersects the sides AB and AC again at C ′ and B ′ , respectively. Prove that BB ′ ,CC ′ , and

HH ′ are concurrent, where H and H ′ are the orthocenters of triangles ABC and AB ′ C ′ respectively.

15. N1 (ROU) Let k be a positive integer. Prove that there are infinitely many perfect squares of the form n2 k − 7, where n is a positive integer.

16. N2 (RUS) Let Z denote the set of all integers. Prove that for any integers A and B, one can find an integer C for which M 1 = {x 2 + Ax + B : x ∈ Z} and M 2 = {2x 2 + 2x + C : x ∈ Z} do not intersect.

17. N3 (CZE) IMO 3 Determine all integers n > 3 such that there are n points A 1 ,A 2 , ...,A n in the plane that satisfy the following two conditions simultaneously: (a) No three lie on the same line. (b) There exist real numbers p 1 ,p 2 ,...,p n such that the area of △A i A j A k is equal to p i +p j +p k , for 1 ≤ i < j < k ≤ n.

18. N4 (BGR) Find all positive integers x and y such that x +y 2 +z 3 = xyz, where z is the greatest common divisor of x and y.

278 3 Problems

19. N5 (IRL) At a meeting of 12k people, each person exchanges greetings with exactly 3k + 6 others. For any two people, the number who exchange greetings with both is the same. How many people are at the meeting?

20. N6 (POL) IMO 6 Let p be an odd prime. Find the number of p-element subsets A of {1,2,...,2p} such that the sum of all elements of A is divisible by p.

21. N7 (BLR) Does there exist an integer n > 1 that satisfies the following condi- tion? The set of positive integers can be partitioned into n nonempty subsets such that an arbitrary sum of n − 1 integers, one taken from each of any n − 1 of the subsets, lies in the remaining subset.

22. N8 (GER) Let p be an odd prime. Determine positive integers x and y for which √ x

≤ y and 2p − x − y is nonnegative and as small as possible.

23. S1 (UKR) Does there exist a sequence F (1), F(2), F(3), . . . of nonnegative in- tegers that simultaneously satisfies the following three conditions?

(a) Each of the integers 0 , 1, 2, . . . occurs in the sequence. (b) Each positive integer occurs in the sequence infinitely often.

(c) For any n ≥ 2,

F F n 163

= F(F(n)) + F(F(361)).

24. S2 (POL) IMO 4 The positive real numbers x 0 ,x 1 ,...,x 1995 satisfy x 0 =x 1995 and

for i = 1, 2, . . . , 1995. Find the maximum value that x 0 can have.

25. S3 (POL) For an integer x ≥ 1, let p(x) be the least prime that does not divide x , and define q (x) to be the product of all primes less than p(x). In particu- lar, p (1) = 2. For x such that p(x) = 2, define q(x) = 1. Consider the sequence

x 0 ,x 1 ,x 2 , . . . defined by x 0 = 1 and x n p (x n )

x n +1 = q (x n )

for n ≥ 0. Find all n such that x n = 1995.

26. S4 (NZL) Suppose that x 1 ,x 2 ,x 3 , . . . are positive real numbers for which

n −1

j =0

for n = 1, 2, 3, . . . . Prove that for all n,

2 − n −1 ≤x n <2− n .

3.36 IMO 1995 279

27. S5 (FIN) For positive integers n, the numbers f (n) are defined inductively as follows: f (1) = 1, and for every positive integer n, f (n + 1) is the greatest integer

m such that there is an arithmetic progression of positive integers a 1 <a 2 < ··· <

a m = n for which

f (a 1 ) = f (a 2 ) = ··· = f (a m ). Prove that there are positive integers a and b such that f (an + b) = n + 2 for

every positive integer n.

28. S6 (IND) Let N denote the set of all positive integers. Prove that there exists a unique function f : N → N satisfying

f (m + f (n)) = n + f (m + 95)

for all m and n in N. What is the value of ∑ 19

k =1 f (k)?

280 3 Problems