30th IMO 1989 shortlist

30th IMO 1989 shortlisted problems


1. Ali Baba has a rectangular piece of carpet. He finds that if he lays if flat on the floor of either of his storerooms then each corner of the carpet touches a different wall of the room. He knows that the storerooms have widths 38 and 50 feet and the same (unknown but integral) length. What are the dimensions of the carpet?
2. Prove that for n > 1 the polynomial xⁿ/n! + x^n-1/(n-1)! + ... + x/1! + 1 has no rational roots.
3. The polynomial xⁿ + n x^n-1 + a₂x^n-2 + ... + a₀ has n roots whose 16th powers have sum n. Find the roots.
4. The angle bisectors of the triangle ABC meet the circumcircle again at A'B'C'. Show that 16 (area A'B'C')³ ≥ 27 area ABC R⁴, where R is the circumradius of ABC.
5. Show that any two points P and Q inside a regular n-gon can be joined by two circular arcs PQ which lie inside the n-gon and meet at an angle at least (1 - 2/n)π.
6. The rectangle R is covered by a finite number of rectangles R₁, ... , R_n such that (1) each R_i is a subset of R, (2) the sides of each R_i are parallel to the sides of R, (3) the rectangles R_i have disjoint interiors, and (4) each R_i has a side of integral length. Show that R has a side of integral length.
7. For each n > 0 we write (1 + 2^1/34 - 4^1/34)ⁿ as a_n + b_n 2^1/3 + c_n 4^1/3, where a_n, b_n, c_n are integers. Show that c_n is non-zero.
8. Let C represent the complex numbers. Let g: C → C be an arbitrary function. Let w be a cube root of 1 other than 1 and let v be any complex number. Find a function f: C → C such that f(z) + f(wz + v) = g(z) for all z and show that it is unique.
9. Define the sequence a₁, a₂, a₃, ... by 2ⁿ = the sum of a_d such that d divides n. Show that a_n is divisible by n. [For example, a₁ = 2, a₂ = 2, a₃ = 6.]
10. There are n cars waiting at distinct points of a circular race track. At the starting signal each car starts. Each car may choose arbitrarily which of the two possible directions to go. Each car has the same constant speed. Whenever two cars meet they both change direction (but not speed). Show that at some time each car is back at its starting point.
11. A quadrilateral has both a circumcircle and an incircle. Show that intersection point of the diagonals lies on the line joining the centers of the two circles.
12. a, b, c, d, m, n are positive integers such that a² + b² + c² + d² = 1989, a + b + c + d = m² and the largest of a, b, c, d is n². Find m and n.
13. The real numbers a₀, a₁, ... , a_n satisfy a₀ = a_n = 0, a_k = c + ∑_i=k^n-1 a_i-k(a_i + a_i+1). Show that c ≤ 1/(4n).
14. Given 7 points in the plane, how many segments (each joining two points) are needed so that given any three points at least two have a segment joining them?
15. Given a convex n-gon A₁A₂ ... A_n with area A and a point P, we rotate P through an angle x about A_i to get the point P_i. Find the area of the polygon P₁P₂ ... P_n.
16. A positive integer is written in each square of an m x n board. The allowed move is to add an integer k to each of two adjacent numbers (whose squares have a common side). Find a necessary and sufficient condition for it to be possible to get all numbers zero by a finite sequence of moves.
17. Show that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle, but that the obtuse angle cannot exceed 120^o.
18. Five points are placed on a sphere of radius 1. That is the largest possible value for the shortest distance between two of the points? Find all configurations for which the maximum is attained.
19. a and b are non-square integers. Show that x² - ay² - bz² + abw² = 0 has a solution in integers not all zero iff x² - ay² - bz² = 0 has a solution in integers not all zero.
20. b > 0 and a are fixed real numbers and n is a fixed positive integer. The real numbers x₀, x₁, ... , x_n satisfy x₀ + x₁ + ... + x_n = a and x₀² + ... + x_n² = b. Find the range of x₀.
21. Let m > 1 be a positive odd integer. Find the smallest positive integer n such that 2¹⁹⁸⁹ divides mⁿ - 1.
22. The points O, A, B, C, D in the plane are such that the six triangles OAB, OAC, OAD, OBC, OBD, OCD all have area at least 1. Show that one of the triangles must have area at least √2.
23. 155 birds sit on a circle center O. Birds at A and B are mutually visible iff ∠AOB ≤ 10^o. More than one bird may sit at the same point. What is the smallest possible number of mutually visible pairs?
24. Given positive integers a ≥ b ≥ c, let N(a, b, c) be the number of solutions in positive integers x, y, z to a/x + b/y + c/z = 1. Show that N(a, b, c) ≤ 6ab(3 + ln(2a) ).
25. ABC is an acute-angled triangle with circumcenter O and orthocenter H. AO = AH. Find all possible values for the ∠A.
There were 109 problems proposed in total, although this included one problems which was merely a variant. 32 were shortlisted (from which 6 were chosen and are not shown above, nor is the variant shown).