7th Eötvös Competition Problems 1990

1.  d is not divisible by 5. For some integer n, a n3 + b n2 + c n + d is divisible by 5. Show that for some integer m, a + b m + c m2 + d m3 is divisible by 5. 2.  Construct the triangle ABC given c, r and r', where c = |AB|, r is the radius of the inscribed circle, and r' is the radius of the other circle tangent to the segment AB and the lines BC and CA