Question 70
Let $f: N \times Z \to N$ be a given function satisfying simultaneously the following conditions:
i/ $f(0, 0) = 5^{2003}$, $f(0, n) = 0$ for every integer $n \neq 0$;
ii/ $f(m, n) = f(m - 1, n) - 2\left[\frac{f(m - 1, n)}{2}\right] + \left[\frac{f(m - 1, n - 1)}{2}\right] + \left[\frac{f(m - 1, n + 1)}{2}\right]$ for every natural number $m \gt 0$ and for every integer $n$.
Prove that there exists natural number $M$ such that
$f(M, n) = 1$ for all integers $n$ such that $|n| \le (5^{2003} - 1)/2$
and $f(M, n) = 0$ for all integers $n$ such that $|n| \gt (5^{2003} - 1)/2$.
i/ $f(0, 0) = 5^{2003}$, $f(0, n) = 0$ for every integer $n \neq 0$;
ii/ $f(m, n) = f(m - 1, n) - 2\left[\frac{f(m - 1, n)}{2}\right] + \left[\frac{f(m - 1, n - 1)}{2}\right] + \left[\frac{f(m - 1, n + 1)}{2}\right]$ for every natural number $m \gt 0$ and for every integer $n$.
Prove that there exists natural number $M$ such that
$f(M, n) = 1$ for all integers $n$ such that $|n| \le (5^{2003} - 1)/2$
and $f(M, n) = 0$ for all integers $n$ such that $|n| \gt (5^{2003} - 1)/2$.