
1. Show that if n is a positive integer and a and b are integers, then n! divides a(a + b)(a + 2b) ... (a + (n1)b) b^{n1}.


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 sin^{2}x_{1} + sin^{2}x_{2} + ... + sin^{2}x_{n}, where x_{i} are nonnegative and have sum π.


4. Show that x_{1}^{2}/(x_{1}^{2} + x_{2}x_{3}) + x_{2}^{2}/(x_{2}^{2} + x_{3}x_{4}) + ... + x_{n1}^{2}/(x_{n1}^{2} + x_{n}x_{1}) + x_{n}^{2}/(x_{n}^{2} + x_{1}x_{2}) ≤ n1 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 n_{1}, n_{2}, ... , n_{8} such that we can express every integer n with n < 1986 as a_{1}n_{1} + ... + a_{8}n_{8} 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 p_{0}(x, y, z), p_{1}(x, y, z), p_{2}(x, y, z), ... are defined by p_{0}(x, y, z) = 1 and p_{n+1}(x, y, z) = (x + z)(y + z) p_{n}(x, y, z+1)  z^{2}p_{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 a_{1}a_{2}a_{3}a_{4} + a_{2}a_{3}a_{4}a_{5} + ... + a_{n}a_{1}a_{2}a_{3} = 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 ngon 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 x_{2} = 2^{1/2}, x_{3} = (2 + 3^{1/3})^{1/2}, x_{4} = (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 r^{2}(√3)/2.


21. Show that if the real numbers x, y, z satisfy 1/(yz  x^{2}) + 1/(zx  y^{2}) + 1/(xy  z^{2}) = 0, then x/(yz  x^{2})^{2} + y/(zx  y^{2})^{2} + z/(xy  z^{2})^{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 5^{1985}  1 as a product of three integers, each greater than 5^{100}.


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 w^{7} + x^{7} + y^{7} + z^{7} = 0.


28. X is the set {1, 2, ... , n}. P_{1}, P_{2}, ... , 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.

