19th All Soviet Union Mathematical Olympiad Problems 1985

1.  ABC is an acute angled triangle. The midpoints of BC, CA and AB are D, E, F respectively. Perpendiculars are drawn from D to AB and CA, from E to BC and AB, and from F to CA and BC. The perpendiculars form a hexagon. Show that its area is half the area of the triangle. 2.  Is there an integer n such that the sum of the (decimal) digits of n is 1000 and the sum of