76th Kürschák 1976


1. ABCD is a parallelogram. P is a point outside the parallelogram such that angles PAB and PCB have the same value but opposite orientation. Show that angle APB = angle DPC.
2. A lottery ticket is a choice of 5 distinct numbers from 1, 2, 3, ... , 90. Suppose that 5⁵ distinct lottery tickets are such that any two of them have a common number. Prove that one can find four numbers such that every ticket contains at least one of the four.
3. Prove that if the quadratic x² + ax + b is always positive (for all real x) then it can be written as the quotient of two polynomials whose coefficients are all positive.

The original problems are in Hungarian. Many thanks to Péter Dombi for the translation.