A. 412. Let t be an irrational number and let
for all integers x, y. The function f(x,y) has many decompositions of the form f(x,y)=u(x)+v(y)+w(x-y), where the functions u, v and w map the set of integers to the set of reals. Prove that (a) There exists a case when u, v and w are bounded; (b) There exists a case when u, v and w attain only integers; (c) There exists no case when u, v and w are bounded and attain only integers.
(Proposed by Tamás Keleti, Budapest)
Deadline expired on 15 December 2006.