The Third-Seventh IMO Mumbai, India, July 5–17, 1996

3.37 The Third-Seventh IMO Mumbai, India, July 5–17, 1996

3.37.1 Contest Problems

First Day (July 10)

1. We are given a positive integer r and a rectangular board ABCD with dimensions |AB| = 20, |BC| = 12. The rectangle is divided into a grid of 20×12 unit squares. The following moves are permitted on the board: One can move from one square √ to another only if the distance between the centers of the two squares is r . The task is to find a sequence of moves leading from the square corresponding to vertex A to the square corresponding to vertex B.

(a) Show that the task cannot be done if r is divisible by 2 or 3. (b) Prove that the task is possible when r = 73.

(c) Is there a solution when r = 97?

2. Let P be a point inside △ABC such that

∠APB − ∠C = ∠APC − ∠B.

Let D , E be the incenters of △APB,△APC respectively. Show that AP, BD, and CE meet in a point.

3. Let N 0 denote the set of nonnegative integers. Find all functions f from N 0 into itself such that

f (m + f (n)) = f ( f (m)) + f (n), ∀m,n ∈ N 0 . Second Day (July 11)

4. The positive integers a and b are such that the numbers 15a + 16b and 16a − 15b are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

5. Let ABCDEF be a convex hexagon such that AB is parallel to DE, BC is parallel to EF, and CD is parallel to AF. Let R A ,R C ,R E be the circumradii of triangles FAB , BCD, DEF respectively, and let P denote the perimeter of the hexagon. Prove that

P R A +R C +R E ≥ .

6. Let p , q, n be three positive integers with p + q < n. Let (x 0 ,x 1 ,...,x n ) be an (n + 1)-tuple of integers satisfying the following conditions: (i) x 0 =x n = 0. (ii) For each i with 1 ≤ i ≤ n, either x i −x i −1 = p or x i −x i −1 = −q. Show that there exists a pair (i, j) of distinct indices with (i, j) 6= (0,n) such that x i =x j .

3.37 IMO 1996 281

3.37.2 Shortlisted Problems

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

ab bc ca

a 5 +b 5 + ab b 5 +c 5 + bc c 5 +a 5 + ca When does equality hold?

2. A2 (IRL) Let a 1 ≥a 2 ≥ ··· ≥ a n

be real numbers such that

a k +a k

1 2 + ··· + a n ≥0

for all integers k > 0. Let p = max{|a 1 |,... ,|a n |}. Prove that p = a 1 and that

) ···(x − a n n )≤x −a 1 for all x >a 1 .

(x − a 1 )(x − a 2 n

3. A3 (HEL) Let a > 2 be given, and define recursively

Show that for all k ∈ N, we have

2 +a− a 2 −4 .

be nonnegative real numbers, not all zero. (a) Prove that x n −a 1 x n −1 −···−a n −1 x −a n = 0 has precisely one positive real root.

4. A4 (KOR) Let a 1 ,a 2 ,...,a n

(b) Let A =∑ j =1 a j ,B = ∑ ja j , and let R be the positive real root of the

j =1

equation in part (a). Prove that

A A ≤R B .

5. A5 (ROU) Let P (x) be the real polynomial function P(x) = ax 3 + bx 2 + cx + d. Prove that if |P(x)| ≤ 1 for all x such that |x| ≤ 1, then

|a| + |b| + |c| + |d| ≤ 7.

6. A6 (IRL) Let n be an even positive integer. Prove that there exists a positive integer k such that

k = f (x)(x + 1) n + g(x)(x n + 1)

for some polynomials f (x), g(x) having integer coefficients. If k 0 denotes the

least such k, determine k 0 as a function of n.

A6 ′ Let n be an even positive integer. Prove that there exists a positive integer k such that

282 3 Problems k = f (x)(x + 1) n + g(x)(x n + 1)

for some polynomials f (x), g(x) having integer coefficients. If k 0 denotes the least such k, show that k 0 =2 q , where q is the odd integer determined by n = q2 r , r ∈ N. A6 ′′ Prove that for each positive integer n, there exist polynomials f (x), g(x) having integer coefficients such that

f (x)(x + 1) 2 n + g(x)(x 2 n + 1) = 2.

7. A7 (ARM) Let f be a function from the set of real numbers R into itself such that for all x ∈ R, we have | f (x)| ≤ 1 and

42 6 7 Prove that f is a periodic function (that is, there exists a nonzero real number c

such that f (x + c) = f (x) for all x ∈ R).

8. A8 (ROU) IMO 3 Let N 0 denote the set of nonnegative integers. Find all functions

f from N 0 into itself such that

f (m + f (n)) = f ( f (m)) + f (n), ∀m,n ∈ N 0 .

9. A9 (POL) Let the sequence a (n), n = 1, 2, 3, . . ., be generated as follows: a(1) =

0, and for n > 1,

(Here [t] = the greatest integer ≤ t.) (a) Determine the maximum and minimum value of a (n) over n ≤ 1996 and

n (n+1)

a (n) = a([n/2]) + (−1) 2 .

find all n ≤ 1996 for which these extreme values are attained. (b) How many terms a (n), n ≤ 1996, are equal to 0?

10. G1 (UNK) Let triangle ABC have orthocenter H, and let P be a point on its circumcircle, distinct from A , B,C. Let E be the foot of the altitude BH, let PAQB and PARC be parallelograms, and let AQ meet HR in X. Prove that EX is parallel to AP.

11. G2 (CAN) IMO 2 Let P be a point inside △ABC such that

∠APB − ∠C = ∠APC − ∠B.

Let D , E be the incenters of △APB,△APC respectively. Show that AP,BD and CE meet in a point.

12. G3 (UNK) Let ABC be an acute-angled triangle with BC > CA. Let O be the circumcenter, H its orthocenter, and F the foot of its altitude CH. Let the per- pendicular to OF at F meet the side CA at P. Prove that ∠FHP = ∠BAC. Possible second part: What happens if |BC| ≤ |CA| (the triangle still being acute- angled)?

3.37 IMO 1996 283

13. G4 (USA) Let △ABC be an equilateral triangle and let P be a point in its inte- rior. Let the lines AP , BP,CP meet the sides BC,CA, AB in the points A 1 ,B 1 ,C 1 respectively. Prove that

A 1 B 1 ·B 1 C 1 ·C 1 A 1 ≥A 1 B ·B 1 C ·C 1 A .

14. G5 (ARM) IMO 5 Let ABCDEF be a convex hexagon such that AB is parallel to DE , BC is parallel to EF, and CD is parallel to AF. Let R A ,R C ,R E be the circum- radii of triangles FAB , BCD, DEF respectively, and let P denote the perimeter of the hexagon. Prove that

P R A +R C +R E ≥ .

15. G6 (ARM) Let the sides of two rectangles be {a,b} and {c,d} with a < c ≤

d < b and ab < cd. Prove that the first rectangle can be placed within the second one if and only if

(b 2 −a 2 ) 2 2 ≤ (bd − ac) 2 + (bc − ad) .

16. G7 (UNK) Let ABC be an acute-angled triangle with circumcenter O and cir- cumradius R. Let AO meet the circle BOC again in A ′ , let BO meet the circle

COA again in B ′ , and let CO meet the circle AOB again in C ′ . Prove that

OA ′

· OB 3 ′ · OC ′ ≥ 8R .

When does equality hold?

17. G8 (RUS) Let ABCD be a convex quadrilateral, and let R A ,R B ,R C , and R D denote the circumradii of the triangles DAB, ABC, BCD, and CDA respectively.

Prove that R A +R C >R B +R D if and only if ∠A + ∠C > ∠B + ∠D.

18. G9 (UKR) In the plane are given a point O and a polygon F (not necessarily convex). Let P denote the perimeter of F , D the sum of the distances from O to the vertices of F , and H the sum of the distances from O to the lines containing the sides of F . Prove that

D 2 2 P −H 2 ≥ .

19. N1 (UKR) Four integers are marked on a circle. At each step we simultaneously replace each number by the difference between this number and the next number on the circle, in a given direction (that is, the numbers a , b, c, d are replaced by

a − b,b − c,c − d,d − a). Is it possible after 1996 such steps to have numbers

a , b, c, d such that the numbers |bc − ad|,|ac − bd|,|ab − cd| are primes?

20. N2 (RUS) IMO 4 The positive integers a and b are such that the numbers 15a + 16b and 16a − 15b are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

284 3 Problems

21. N3 (BGR) A finite sequence of integers a 0 ,a 1 ,...,a n is called quadratic if for each i ∈ {1,2,...,n} we have the equality |a i −a i −1 |=i 2 . (a) Prove that for any two integers b and c, there exist a natural number n and

a quadratic sequence with a 0 = b and a n = c.

(b) Find the smallest natural number n for which there exists a quadratic se- quence with a 0 = 0 and a n = 1996.

22. N4 (BGR) Find all positive integers a and b for which

where as usual, [t] refers to greatest integer that is less than or equal to t.

23. N5 (ROU) Let N 0 denote the set of nonnegative integers. Find a bijective func-

tion f from N 0 into N 0 such that for all m ,n∈N 0 ,

f (3mn + m + n) = 4 f (m) f (n) + f (m) + f (n).

24. C1 (FIN) IMO 1 We are given a positive integer r and a rectangular board ABCD with dimensions |AB| = 20, |BC| = 12. The rectangle is divided into a grid of

20 × 12 unit squares. The following moves are permitted on the board: One can move from one square to another only if the distance between the centers of the two squares is √ r . The task is to find a sequence of moves leading from the square corresponding to vertex A to the square corresponding to vertex B.

(a) Show that the task cannot be done if r is divisible by 2 or 3. (b) Prove that the task is possible when r = 73.

(c) Is there a solution when r = 97?

25. C2 (UKR) An (n − 1) × (n − 1) square is divided into (n − 1) 2 unit squares in the usual manner. Each of the n 2 vertices of these squares is to be colored red

or blue. Find the number of different colorings such that each unit square has exactly two red vertices. (Two coloring schemes are regarded as different if at least one vertex is colored differently in the two schemes.)

26. C3 (USA) Let k , m, n be integers such that 1 < n ≤ m − 1 ≤ k. Determine the maximum size of a subset S of the set {1,2,3,...,k} such that no n distinct elements of S add up to m.

27. C4 (FIN) Determine whether or not there exist two disjoint infinite sets A and

B of points in the plane satisfying the following conditions: (i) No three points in A ∪ B are collinear, and the distance between any two points in A ∪ B is at least 1. (ii) There is a point of A in any triangle whose vertices are in B, and there is

a point of B in any triangle whose vertices are in A .

28. C5 (FRA) IMO 6 Let p , q, n be three positive integers with p + q < n. Let (x 0 ,x 1 ,

...,x n ) be an (n + 1)-tuple of integers satisfying the following conditions:

(i) x 0 =x n = 0.

3.37 IMO 1996 285 (ii) For each i with 1 ≤ i ≤ n, either x i −x i −1 = p or x i −x i −1 = −q.

Show that there exists a pair (i, j) of distinct indices with (i, j) 6= (0,n) such that x i =x j .

29. C6 (CAN) A finite number of beans are placed on an infinite row of squares. A sequence of moves is performed as follows: At each stage a square containing more than one bean is chosen. Two beans are taken from this square; one of them is placed on the square immediately to the left, and the other is placed on the square immediately to the right of the chosen square. The sequence terminates if at some point there is at most one bean on each square. Given some initial configuration, show that any legal sequence of moves will terminate after the same number of steps and with the same final configuration.

30. C7 (IRL) Let U be a finite set and let f , g be bijective functions from U onto itself. Let

S = {w ∈ U : f ( f (w)) = g(g(w))}, T = {w ∈ U : f (g(w)) = g( f (w))}, and suppose that U = S ∪ T . Prove that for every w ∈ U, f (w) ∈ S if and only if

g (w) ∈ S.

286 3 Problems