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Part of 2015 China Western Mathematical Olympiad

Problems(1)

China Western Mathematical Olympiad 2015 ,Problem 1

Source: China Yinchuan Aug 16, 2015

8/17/2015
Let the integer n≥2n \ge 2 , and x1,x2,⋯ ,xnx_1,x_2,\cdots,x_n be real numbers such that ∑k=1nxk\sum_{k=1}^nx_k be integer . dk=min⁡m∈Z∣xk−m∣d_k=\underset{m\in {Z}}{\min}\left|x_k-m\right| , 1≤k≤n1\leq k\leq n .Find the maximum value of ∑k=1ndk\sum_{k=1}^nd_k.
inequalitiesalgebra