IMO 2019 Shortlist A7
Let Z be the set of integers. We consider functions f : Z Ñ Z satisfying f fpx yq y ˘ “ f fpxq y ˘ for all integers x an...
Category: Algebra
Problem
Let Z be the set of integers. We consider functions f : Z Ñ Z satisfying
f
fpx yq y ˘ “ f
fpxq ` y
˘
for all integers x and y. For such a function, we say that an integer v is f-rare if the set
Xv “ tx P Z: fpxq “ vu
is finite and nonempty.
(a) Prove that there exists such a function f for which there is an f-rare integer.
(b) Prove that no such function f can have more than one f-rare integer.
(Netherlands) 6 Bath — UK, 11th–22nd July 2019