Shortlisted Problems for the 31st IMO

Beijing (China), 1990

Original HTML-version by Enrique Valeriano Cuba

Problem 1. (AUS 3)
The integer 9 can be written as a sum of two consecutive integers: 9 = 4 + 5; moreover it can be written as a sum of (more than one) consecutive positive integers in exactly two ways, namely 9 = 4 + 5 = 2 + 3 + 4. Is there an integer which can be written as a sum of 1990 consecutive positive integers and which can be written as a sum of (more than one) consecutive positive integers in exactly 1990 ways?.

Problem 2. (CAN 1)
Given n countries with 3 representatives each, m committees A(1), A(2), ..., A(m) are called a cycle if:

It is possible to have a cycle of 1990 committees with 11 countries?

Problem 3'. (CZE 1')
On a circle 2n - 1 (n ³ 3) different points are given. Find the minimum of natural number N with the property that whenever N of the given points are coloured black, there exist two black points such that the interior of one of the corresponding arcs contains exactly n of the given 2n - 1 points.

Problem 3. (CZE 1)
On a circle 2n - 1 (n ³ 3) different points are given, from which n are coloured black. Prove that one may find two black points such that one of the corresponding arcs contains exactly n of the given 2n - 1 points.

Problem 4. (CZE 2)
Assume that the set of all positive integers is descomposed into r (disjoint) subsets A1 U A2 U ... U Ar = N. Prove that one of them, say Ai, has the following property: There exist a positive m such that for any k one can find numbers a1, a2, ..., ak in Ai with 0 < aj+1 - aj £ m (1 £ j £ k - 1).

Problem 5. (FRA 1)
Given the triangle ABC with no side equal to another side, let G, K and H be its centroid, incenter and orthocenter, respectively. Prove that ÐGKH > 90°.

Problem 6. (FRG 2)
Two players A and B play a game in which they choose numbers alternately according to the following rule.

At the begining, an initial natural number n0 > 0 is given. Having known n2k, the player A may choose any n2k+1 Î N, such that

n2k £ n2k+1 £ (n2k)2

Then the player B chooses a number n2k+2 Î N such that

(n2k+1)/(n2k+2) = pr

where p is a prime number and r Î N.

It is stipulated that, player A wins the game if he (she) succeeds in choose the number 1990, and player B wins if he (she) succeeds in choosing 1.

For which initial n0 the player A can manage to win the game, for which n0 player B can manage to win, and for which n0 players A and B would go to a tie?.

Problem 7. (HEL 2)
Let f(0) = f(1) = 0 and f(n+2) = 4n+2.f(n+1) - 16n+1.f(n) + n.2n2, n = 0, 1, 2, 3, ... . Show that the numbers f(1989), f(1990), f(1991) are divisible by 13.

Problem 8. (HUN 1)
For a given positive integer k denote the square of the sum of its digits by f1(k) and let

fn+1(k) = f1 ( fn(k) ).

Determine the value fo f1991 (21990). Show that the numbers f(1989), f(1990), f(1991) are divisible by 13.

Problem 9. (HUN 3)
The incenter of the triangle ABC is K. The midpoint of AB is C1 and that of AC is B1. The lines C1K and AC meet at B2, the lines B1K and AB meet at C2. If the areas of the triangles AB2C2 and ABC are equal, what is the measure of angle Ð CAB?.

Problem 10. (ICE 2)
A plane cuts a right circular cone in two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the ratio of the volume of the smaller part to the volume of the whole cone.

Problem 11'. (IND 3')
Given a circle with two chords AB, CD which meet at E. Let M be a pint of chord AB other than E. Draw the circle through D, E and M. The tangent line to the circle DEM at E meets the lines BC, AC at F, G, respectively. Given (AM/AB) = l, find (GE/EF).

Problem 11. (IND 3)
Triangle ABC is acute-angled with circumcircle G and orthocenter H; also AB ¹AC. Let AH meet BC and G at D and E, respectively. The tangent to circle (DEF) at D meets the lines AB, AC at M, L, respectively. Prove that MD = DL.

Problem 12. (IRE 1)
Let ABC be a triangle and L the line through C parallel to the side AB. Let be the internal bisector of the angle at A meet the side BC at D and the line L at E and let the internal bisector of the angle at B meet the side AC at F and the line L at G. If GF = DE, prove that AC = BC.

Problem 13. (IRE 2)
An eccentric mathematician has a ladder with n rungs which he always ascends and descends in the following way: when he ascends each step he takes covers b rungs of the ladder, where a and b are fixed positive integers. By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again. Find, with proof, the minimum value of n, expressed in terms of a and b.

Problem 14. (JAP 2)
On the coordinate plane a rectangle with vertices (0,0), (m,0), (0,n), (m,n) is given where both m and n are odd integers.
The rectangle is partitioned into triangles in such a way that:
Prove that there exist at least two triangles in the partition each of which has two good sides.

Problem 15. (MEX 2)
Determine for which positive integers k, the set

X = {1990, 1990+1, ..., 1990 + k}

can be partitioned into two disjoint subsets A and B such that the sum of the elements of A is equal to the sum of the elements of B.

Problem 16. (NET 1)
Is there a 1990-gon with the following properties (1) and (2)?.
  1. All angles are equal;
  2. The lengths of the 1990 sides are a permutation of the numers 12, 22, ..., 19892, 19902.
Problem 17. (NET 3)
Unit cubes are made into beads by drilling a hole through them along a diagonal. The beads are put on a string in such a way that they can move freely in space under the restriction that the vertices of two neighbooring cubes are touching. Let A be the begin-vertex and B be the end-vertex. Let there be p x q x r cubes on the string (p, q, r ³ 1).
  1. (a) Determine for which values of p, q and r it is possible to build a block with dimensions p, q and r. Give reasons for your answer.
  2. (b) The same question as (a) with the extra condition that A = B.
Problem 18. (NOR)
Let a, b be natural numbers with 1 £ a £ b, and M = (a + b)/2. Define the function f: Z ® Z by

f(n) = n + a, if n < M
f(n) = n - b, if n ³ M

Let f1(n) = f(n), f i+1(n) = f(f i (n)); i = 1, 2, ... . Find the smallest natural number k such that f k(0) = 0

Problem 19. (POL 1)
Let P be a point inside a regular tetrahedron T of unit volume. The four planes passing through P and parallel to the faces of T partition T into 14 pieces. Let f (P) be the joint volume of those pieces which are neither a tetrahedron nor a parallelepiped (i.e., pieces adjacent to an edge but not to a vertex). Find the exact bounds for f (P) as P varies over T.

Problem 20. (POL 3)
Prove that every integer k (>1) has a positive multiple which is less than k4 and can be written in the decimal system with at most four different digits.

Problem 21. (ROM 1')
Let n be a composite natural number and p be a proper divisor of n. Find the binary representation of the smallest natural number N such that ((1 + 2p + 2n-p).N - 1)/ (2n) is an integer.

Problem 22. (ROM 4)
Ten localities are served by two international airlines such that there exist a direct service (without stops) between any two of these localities and all airline schedules are both ways. Prove that at least one of the airlines can offer two disjoint round trips containing each an odd number of landings.

Problem 23. (ROM 5)
FInd all positive integers n having the property that (2n + 1)/(n2) is an integer.

Problem 24. (THA 2)
Let a,b,c,d be nonnegative real numbers such that ab + bc + cd + da = 1. Show that

(a3)/(b + c + d) + (b3)/(a + c + d) + (c3)/(a + b + d) + (d3)/(a + b + c) ³ 1/3

Problem 25. (TUR 4)
Let Q+ be the set of positive rational numbers. Construct a function f: Q+ ® Q+ such that

f(xf(y)) = (f(x))/y,

for all x, y in Q+.

Problem 26. (USA 2)
Let P be a cubic polynomial with rational coefficients, and let q1, q2, q3, ... be a sequence of rational numbers such that qn = P (qn+1) for all n ³ 1. Prove that there exist k ³ 1 such that for all n ³ 1, qn+k = qn.

Problem 27. (USS 1)
Find all natural numbers n for which every natural number whose decimal representation has n-1 digits 1 and one digit 7 is prime.

Problem 28. (USS 3)
Prove that on a coordinate plane it is impossible to draw a closed broken line such that