11th Swedish Mathematical Society Problems 1971

1.  Show that (1 + a + a2)2 < 3(1 + a2 + a4) for real a ≠ 1. 2.  An arbitrary number of lines divide the plane into regions. Show that the regions can be colored red and blue so that neighboring regions have different colors. 3.  A table is covered by 15 pieces of paper. Show that we can remove 7 pieces so that the remaining 8 cover at least 8/15 of