1. ## Help with proof?

Let x1, x2, . . . , xk be real numbers such that the set A = {cos(nπx1)+cos(nπx2)+
· · ·+cos(nπxk) | n ≥ 1} is finite. Prove that all the xi are rational numbers.

Apparently, it requires use of the pigeonhole principle, but I can't seem to figure out where or how to apply it

2. apparently , its a question fromt he book 'putnam ad beyond' by titu anreescuu

http://aam.org.in/st_material/1.pdf

the solution is in page number 340