CN-TST 2023 Problem 3

(1) Let $a,b$ be coprime positive integers. Prove that there exists constants $\lambda$ and $\beta$ such that for all integers $m$ ,

$$\left| \sum\limits_{k=1}^{m-1} \left\{ \frac{ak}{m} \right\}\left\{ \frac{bk}{m} \right\} - \lambda m \right| \le \beta$$ (2) Prove that there exists $N$ such that for all $p>N$ (where $p$ is a prime number), and any positive integers $a,b,c$ positive integers satisfying $p\nmid (a+b)(b+c)(c+a)$ , there are at least $\lfloor \frac{p}{12} \rfloor$ solutions $k\in \{1,\cdots,p-1\}$ such that $$\left\{\frac{ak}{p}\right\} + \left\{\frac{bk}{p}\right\} + \left\{\frac{ck}{p}\right\} \le 1$$