6th Eötvös Competition Problems 1899

1.  ABCDE is a regular pentagon (with vertices in that order) inscribed in a circle of radius 1. Show that AB·AC = √5. 2.  The roots of the quadratic x2 - (a + d) x + ad - bc = 0 are α and β. Show that α3 and β3 are the roots of x2 - (a3 + d3 + 3abc + 3bcd) x + (ad - bc)3 = 0.