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Sum of Decimal Parts

Source: 2023 Taiwan Round 1 Mock Exam P6

March 18, 2023
Taiwanalgebra

Problem Statement

For every positive integer M≥2M \geq 2, find the smallest real number CMC_M such that for any integers a1,a2,…,a2023a_1, a_2,\ldots , a_{2023}, there always exist some integer 1≤k<M1 \leq k < M such that  {ka1M}+{ka2M}+⋯+{ka2023M}≤CM.\left\{\frac{ka_1}{M}\right\}+\left\{\frac{ka_2}{M}\right\}+\cdots+\left\{\frac{ka_{2023}}{M}\right\}\leq C_M. Here, {x}\{x\} is the unique number in the interval [0,1)[0, 1) such that x−{x}x - \{x\} is an integer.
Proposed by usjl
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