Question 32
Let $p$ be an odd prime number and $\mathbb{Z}_{\gt 0}$ be the set of positive integers. Suppose that a function $f: \mathbb{Z}_{\gt 0} \times \mathbb{Z}_{\gt 0} \rightarrow\{0,1\}$ satisfies the following properties:
- $f(1,1)=0$;
- $f(a, b)+f(b, a)=1$ for any pair of relatively prime positive integers $(a, b)$ not both equal to 1 ;
- $f(a+b, b)=f(a, b)$ for any pair of relatively prime positive integers $(a, b)$.
Prove that
$$
\sum_{n=1}^{p-1} f\left(n^{2}, p\right) \geqslant \sqrt{2 p}-2
$$
- $f(1,1)=0$;
- $f(a, b)+f(b, a)=1$ for any pair of relatively prime positive integers $(a, b)$ not both equal to 1 ;
- $f(a+b, b)=f(a, b)$ for any pair of relatively prime positive integers $(a, b)$.
Prove that
$$
\sum_{n=1}^{p-1} f\left(n^{2}, p\right) \geqslant \sqrt{2 p}-2
$$