Question 153
Professor Piraldo takes part in soccer matches with a lot of goals and judges a match in his own peculiar way. A match with score of $m$ goals to $n$ goals, $m \ge n$, is *tough* when $m \le f(n)$, where $f(n)$ is defined by $f(0) = 0$ and, for $n \ge 1$, $f(n) = 2n - f(r) + r$, where $r$ is the largest integer such that $r \lt n$ and $f(r) \le n$.
Let $\phi = \frac{1+\sqrt{5}}{2}$. Prove that a match with score of $m$ goals to $n$, $m \ge n$, is tough if $m \le \phi n$ and is not tough if $m \ge \phi n + 1$.
Let $\phi = \frac{1+\sqrt{5}}{2}$. Prove that a match with score of $m$ goals to $n$, $m \ge n$, is tough if $m \le \phi n$ and is not tough if $m \ge \phi n + 1$.