9th Austrian-Polish 1986


1. ABC is a triangle which does not contain a right-angle. A' is the center of a circle through B and C. Similarly B' is the center of a circle through C and A, and C' is the center of a circle through A and B. Each pair of circles touches. The triangles ABC and A'B'C' are similar. Find their angles.
2. The monic polynomial p(x) has degree n > 1 and all its roots distinct negative reals. The coefficient of x is A and the constant term is B. Show that Ap(1) > 2n²B. (A monic polynomial has leading coefficient 1).
3. Every point in space is colored red or blue. Show that we can either find a unit square with red vertices, or a unit square with blue vertices, or a unit square with just one blue vertex.
4. Find all positive integer solutions (m, n, N) to m^N - n^N = 2¹⁰⁰ with N > 1.
5. Find all real solutions to: x₁² + x₂² + x₃² + x₄² = 4; x₁x₃ + x₂x₄ + x₃x₂ + x₄x₁ = 0; x₁x₂x₃ + x₁x₂x₄ + x₁x₃x₄ + x₂x₃x₄ = -2; x₁x₂x₃x₄ = -1.
6. The inscribed and circumscribed spheres of a tetrahedron have radii r and R and are concentric. Find the range of possible values for R/r.
7. k < n²/4 is a positive integer with no prime divisor greater than n. Show that k divides n! .
8. mn distinct reals are arranged in an m x n array so that the entries in each row increase from left to right. Each column is then rearranged so that the entries increase from bottom to top. Show that the elements in each row still increase from left to right.
9. Find all continuous real-valued functions f on the reals such that (1) f(1) = 1, (2) f(f(x)) = f(x)² for all x, (3) either f(x) ≥ f(y) for all x ≥ y, or f(x) ≤ f(y) for all x ≤ y.

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