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{x_i-a}+{x_{i+1}-b} at most 1/2024

Source: Malaysian SST 2023 P5

August 27, 2023
algebra

Problem Statement

Find the maximal value of c>0c>0 such that for any n≥1n\ge 1, and for any nn real numbers x1,⋯ ,xnx_1, \cdots, x_n there exists real numbers a,ba ,b such that {xi−a}+{xi+1−b}≤12024\{x_i-a\}+\{x_{i+1}-b\}\le \frac{1}{2024} for at least cncn indices ii. Here, xn+1=x1x_{n+1}=x_1 and {x}\{x\} denotes the fractional part of xx.
Proposed by Wong Jer Ren
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