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Finally an inequality

Source: China TST 2023 Problem 14

March 27, 2023
inequalitiesSets

Problem Statement

For any nonempty, finite set BB and real xx, define
dB(x)=min⁡b∈B∣x−b∣d_B(x) = \min_{b\in B} |x-b|
(1) Given positive integer mm. Find the smallest real number λ\lambda (possibly depending on mm) such that for any positive integer nn and any reals x1,⋯ ,xn∈[0,1]x_1,\cdots,x_n \in [0,1], there exists an mm-element set BB of real numbers satisfying dB(x1)+⋯+dB(xn)≤λnd_B(x_1)+\cdots+d_B(x_n) \le \lambda n
(2) Given positive integer mm and positive real ϵ\epsilon. Prove that there exists a positive integer nn and nonnegative reals x1,⋯ ,xnx_1,\cdots,x_n, satisfying for any mm-element set BB of real numbers, we have
dB(x1)+⋯+dB(xn)>(1−ϵ)(x1+⋯+xn)d_B(x_1)+\cdots+d_B(x_n) > (1-\epsilon)(x_1+\cdots+x_n)
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