Problem

Source: 2023 Taiwan Round 1 Mock Exam P6

Tags: Taiwan, algebra



For every positive integer $M \geq 2$, find the smallest real number $C_M$ such that for any integers $a_1, a_2,\ldots , a_{2023}$, there always exist some integer $1 \leq k < M$ such that \[\left\{\frac{ka_1}{M}\right\}+\left\{\frac{ka_2}{M}\right\}+\cdots+\left\{\frac{ka_{2023}}{M}\right\}\leq C_M.\]Here, $\{x\}$ is the unique number in the interval $[0, 1)$ such that $x - \{x\}$ is an integer. Proposed by usjl