26th IMO 1985 shortlist

26th IMO 1985 shortlisted problems


1. Show that if n is a positive integer and a and b are integers, then n! divides a(a + b)(a + 2b) ... (a + (n-1)b) b^n-1.
2. A convex quadrilateral ABCD is inscribed in a circle radius 1. Show that 0 < \|AB + BC + CD + DA - AC - BD\| < 2.
3. Given n > 1, find the maximum value of sin²x₁ + sin²x₂ + ... + sin²x_n, where x_i are non-negative and have sum π.
4. Show that x₁²/(x₁² + x₂x₃) + x₂²/(x₂² + x₃x₄) + ... + x_n-1²/(x_n-1² + x_nx₁) + x_n²/(x_n² + x₁x₂) ≤ n-1 for all positive reals x_i.
5. T is the set of all lattice points in space. Two lattice points are neighbors if they have two coordinates the same and the third differs by 1. Show that there is a subset S of T such that if a lattice point x belongs to S then none of its neighbors belong to S, and if x does not belong to S, then exactly one of its neighbors belongs to S.
6. Let A be a set of positive integers such that \|m - n\| ≥ mn/25 for any m, n in A. Show that A cannot have more than 9 elements. Give an example of such a set with 9 elements.
7. Do there exist 100 distinct lines in the plane having just 1985 distinct points of intersection?
8. Find 8 positive integers n₁, n₂, ... , n₈ such that we can express every integer n with \|n\| < 1986 as a₁n₁ + ... + a₈n₈ with each a_i = 0, ±1.
9. The points A, B, C are not collinear. There are three ellipses, each pair of which intersects. One has foci A and B, the second has foci B and C and the third has foci C and A. Show that the common chords of each pair intersect.
10. The polynomials p₀(x, y, z), p₁(x, y, z), p₂(x, y, z), ... are defined by p₀(x, y, z) = 1 and p_n+1(x, y, z) = (x + z)(y + z) p_n(x, y, z+1) - z²p_n(x, y, z). Show that each polynomial is symmetric in x, y, z.
11. Show that if there are a_i = ±1 such that a₁a₂a₃a₄ + a₂a₃a₄a₅ + ... + a_na₁a₂a₃ = 0, then n is divisible by 4.
12. Given 1985 points inside a unit cube, show that we can always choose 32 such that any polygon with these points as vertices has perimeter less than 8√3.
13. A die is tossed repeatedly. A wins if it is 1 or 2 on two consecutive tosses. B wins if it is 3 - 6 on two consecutive tosses. Find the probability of each player winning if the die is tossed at most 5 times. Find the probability of each player winning if the die is tossed until a player wins.
14. At time t = 0 a point starts to move clockwise around a regular n-gon from each vertex. Each of the n points moves at constant speed. At time T all the points reach vertex A simultaneously. Show that they will never all be simultaneously at any other vertex. Can they be together again at vertex A?
15. On each edge of a regular tetrahedron of side 1 there is a sphere with that edge as diameter. S be the intersection of the spheres (so it is all points whose distance from the midpoint of every edge is at most 1/2). Show that the distance between any two points of S is at most 1/√6.
16. Let x₂ = 2^1/2, x₃ = (2 + 3^1/3)^1/2, x₄ = (2 + (3 + 4^1/4)^1/3)^1/2, ... , x_n = (2 + (3 + ... + n^1/n ... )^1/3)^1/2 (where the positive root is taken in every case). Show that x_n+1 - x_n < 1/n! .
17. p is a prime. For which k can the set {1, 2, ... , k} be partitioned into p subsets such that each subset has the same sum?
18. a, b, c, ... , k are positive integers such that a divides 2^b - 1, b divides 2^c - 1, ... , k divides 2^a - 1. Show that a = b = c = ... = k = 1.
19. Show that the sequence [n √2] for n = 1, 2, 3, ... contains infinitely many powers of 2.
20. Two equilateral triangles are inscribed in a circle radius r. Show that the area common to both triangles is at least r²(√3)/2.
21. Show that if the real numbers x, y, z satisfy 1/(yz - x²) + 1/(zx - y²) + 1/(xy - z²) = 0, then x/(yz - x²)² + y/(zx - y²)² + z/(xy - z²)² = 0.
22. Show how to construct the triangle ABC given the distance between the circumcenter O and the orthocenter H, the fact that OH is parallel to the side AB, and the length of the side AB.
23. Find all positive integers a, b, c such that 1/a + 1/b + 1/c = 4/5.
24. Factorise 5¹⁹⁸⁵ - 1 as a product of three integers, each greater than 5¹⁰⁰.
25. 34 countries each sent a leader and a deputy leader to a meeting. Some of the participants shook hands before the meeting, but no leader shook hands with his deputy. Let S be the set of all 68 participants except the leader of country X. Every member of S shook hands with a different number of people (possibly zero). How many people shook hands with the leader or deputy leader of X?
26. Find the smallest positive integer n such that n has exactly 144 positive divisors including 10 consecutive integers.
27. Find the largest and smallest values of w(w + x)(w + y)(w + z) for reals w, x, y, z such that w + x + y + z = 0 and w⁷ + x⁷ + y⁷ + z⁷ = 0.
28. X is the set {1, 2, ... , n}. P₁, P₂, ... , P_n are distinct pairs of elements of X. P_i and P_j have an element in common iff {i, j} is one of the pairs. Show that every element of X belongs to exactly two of the pairs.
29. Show that for any point P on the surface of a regular tetrahedron we can find another point Q such that there are at least three different paths of minimal length from P to Q.
30. C is a circle and L a line not meeting it. M and N are variable points on L such that the circle diameter MN touches C but does not contain it. Show that there is a fixed point P such that the ∠MPN is constant.
Note: This list does not incude the problems used in the Olympiad.