**A. 480.** Let *p*(*z*) be a complex polynomial of degree *n*, and suppose that all (complex) roots of *p*(*z*) are of unit modulus. For every real number *c*0, show that the roots of the polynomial

2*z*(*z*-1)*p*'(*z*)+((*c*-*n*)*z*+(*c*+*n*))*p*(*z*)

are of unit modulus as well.

(5 points)

**A. 481.** Prove that there are infinitely many *n*, for which there exist simple graphs *S*_{1},...,*S*_{n} with the following properties:

(*a*) each *S*_{i} is a complete bipartite graph;

(*b*) the union of the graphs *S*_{1},...,*S*_{n} is a complete graph on 2*n* vertices;

(*c*) each edge of this complete graph is contained in an odd number of the graphs *S*_{i}.

(5 points)