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) bn-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 + ... + sin2xn, where xi are non-negative and have sum π.
4.  Show that x12/(x12 + x2x3) + x22/(x22 + x3x4) + ... + xn-12/(xn-12 + xnx1) + xn2/(xn2 + x1x2) ≤ n-1 for all positive reals xi.
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 ai = 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 pn+1(x, y, z) = (x + z)(y + z) pn(x, y, z+1) - z2pn(x, y, z). Show that each polynomial is symmetric in x, y, z.
11.  Show that if there are ai = ±1 such that a1a2a3a4 + a2a3a4a5 + ... + ana1a2a3 = 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 = 21/2, x3 = (2 + 31/3)1/2, x4 = (2 + (3 + 41/4)1/3)1/2, ... , xn = (2 + (3 + ... + n1/n ... )1/3)1/2 (where the positive root is taken in every case). Show that xn+1 - xn < 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 2b - 1, b divides 2c - 1, ... , k divides 2a - 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, ... , Pn are distinct pairs of elements of X. Pi and Pj 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.

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© John Scholes
6 Aug 2002