21th Eötvös Competition Problems 1914

1.  Circles C and C' meet at A and B. The arc AB of C' divides the area inside C into two equal parts. Show that its length is greater than the diameter of C. 2.  a, b, c are reals such that |ax2 + bx + c| ≤ 1 for all x ≤ |1|. Show that |2ax + b| ≤ 4 for all |x| ≤ 1.