CN-TST 2023 Problem 14

For any nonempty, finite set $B$ and real $x$ , define $$d_B(x) = \min_{b\in B} |x-b|$$ (1) Given positive integer $m$ . Find the smallest real number $\lambda$ (possibly depending on $m$ ) such that for any positive integer $n$ and any reals $x_1,\cdots,x_n \in [0,1]$ , there exists an $m$ -element set $B$ of real numbers satisfying $$d_B(x_1)+\cdots+d_B(x_n) \le \lambda n$$ (2) Given positive integer $m$ and positive real $\epsilon$ . Prove that there exists a positive integer $n$ and nonnegative reals $x_1,\cdots,x_n$ , satisfying for any $m$ -element set $B$ of real numbers, we have $$d_B(x_1)+\cdots+d_B(x_n) > (1-\epsilon)(x_1+\cdots+x_n)$$