4th Eötvös Competition Problems 1897

1.  ABC is a right-angled triangle. Show that sin A sin B sin(A - B) + sin B sin C sin(B - C) + sin C sin A sin(C - A) + sin(A - B) sin(B - C) sin(C - A) = 0. 2.  ABC is an arbitrary triangle. Show that sin(A/2) sin(B/2) sin(C/2) < 1/4. 3.  The line L contains the distinct points A, B, C, D in that order. Construct a rectangle whose sides (or their