IR-MO 2011 Problem 7

Suppose that $f:P(\mathbb N)\longrightarrow \mathbb N$ and $A$ is a subset of $\mathbb N$ . We call $f$ $A$ -predicting if the set $\{x\in \mathbb N|x\notin A, f(A\cup x)\neq x \}$ is finite. Prove that there exists a function that for every subset $A$ of natural numbers, it's $A$ -predicting.

proposed by Sepehr Ghazi-Nezami