21st Austrian-Polish 1998


1. Show that (wx + yz - 1)² ≥ (w² + y² - 1)(x² + z² - 1) for reals w, x, y, z such that w² + y² ≤ 1.
2. n points lie in a line. How many ways are there of coloring the points with 5 colors, one of them red, so that every two adjacent points are either the same color or one or more of them is red?
3. Find all real solutions to x³ = 2 - y, y³ = 2 - x.
4. Show that [1^m/1] + [2^m/4] + [3^m/9] + ... + [n^m/N] ≤ n + m(2^m/4 - 1), where N = n², and m, n are any positive integers.
5. Find all positive integers m, n such that the roots of x³ - 17x² + mx - n² are all integral.
6. A, B, C, D, E, F lie on a circle in that order. The tangents at A and D meet at P and the lines BF and CE pass through P. Show that the lines AD, BC, EF are parallel or concurrent.
7. Find positive integers m, n with the smallest possible product mn such that the number m^mnⁿ ends in exactly 98 zeros.
8. Given an infinite sheet of squared paper. A positive integer is written in each small square. Each small square has area 1. For some n > 2, every two congruent polygons (even if mirror images) with area n and sides along the rulings on the paper have the same sum for the numbers inside. Show that all the numbers in the squares must be equal.
9. ABC is a triangle. K, L, M are the midpoints of the sides BC, CA, AB respectively, and D, E, F are the midpoints of the arcs BC (not containing A), CA (not containing B), AB (not containing C) respectively. Show that r + KD + LE + MF = R, where r is the inradius and R the circumradius.

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