Question 30
Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial $f(x)$ with the following properties:
(a) for any integer $n$, $f(n)$ is an integer;
(b) the degree of $f(x)$ is less than 187.
Alice knows that $f(x)$ satisfies (a) and (b), but she does not know what $f(x)$ is. In every turn, Alice picks a number $k$ from the set $\{1, 2, \dots, 187\}$, and Bob will tell Alice the value of $f(k)$. Determine the smallest positive integer $N$ so that Alice always knows for sure the parity of $f(0)$ within $N$ turns.
(a) for any integer $n$, $f(n)$ is an integer;
(b) the degree of $f(x)$ is less than 187.
Alice knows that $f(x)$ satisfies (a) and (b), but she does not know what $f(x)$ is. In every turn, Alice picks a number $k$ from the set $\{1, 2, \dots, 187\}$, and Bob will tell Alice the value of $f(k)$. Determine the smallest positive integer $N$ so that Alice always knows for sure the parity of $f(0)$ within $N$ turns.