23rd Austrian-Polish 2000


1. Find all polynomials p(x) with real coefficients such that for some n > 0, p(x+2) - p(x+3) + 2p(x+4) - 2p(x+5) + ... - n p(x+2n) + n p(x+2n+1) = 0 holds for infinitely many real x.
2. O is a vertex of a cube side 1. OABC and OADE are faces of the cube. Find the shortest distance between a point of the circle inscribed in OABC and a point of the circumcircle of OAD.
3. For n > 2 find all real solutions to: x₁³ = x₂ + x₃, x₂³ = x₃ + x₄, ... , x_n-2³ = x_n-1 + x_n, x_n-1³ = x_n + x₁, x_n³ = x₁ + x₂.
4. Find all positive integers n, not divisible by any primes except (possibly) 2 and 5, such that n + 25 is a square.
5. For which n > 4 can we color the vertices of a regular n-gon with 6 colors so that every 5 adjacent vertices have different colors?
6. A unit cube is glued onto each face of a central unit cube (so that the glued faces coincide). Can copies of the resulting solid fill space?
7. ABC is a triangle. The points A', B', C' lie on the lines BC, CA, AB respectively and A'B'C' is similar to ABC. Find all possible positions for the circumcenter of A'B'C'.
8. Given 27 points in the plane. Four of the points form a unit square. The other points are all inside the square. No three points are collinear. Show that we can find three points forming a triangle with area at most 1/48.
9. Show that 2 ≤ (1 - x²)² + (1 - y²)² + (1 - z²)² ≤ (1 + x)(1 + y)(1 + z) for non-negative reals x, y, z with sum 1.
10. The diagram shows the plan of the castle. There are 16 nodes. Eight pairs are connected by two links each (at the four corners). How many closed paths pass through each node just once (only count a path once irrespective of whether it is traversed clockwise or counter-clockwise)? How many closed paths pass through each link once? In this case, treat paths as different if a link is traversed in opposite directions.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.