MathDB

4

Part of 2019 China Western Mathematical Olympiad

Problems(1)

Sum of distances to nearest integer

Source: 2019 CWMI P4

8/13/2019
Let nn be a given integer such that n≥2n\ge 2. Find the smallest real number λ\lambda with the following property: for any real numbers x1,x2,…,xn∈[0,1]x_1,x_2,\ldots ,x_n\in [0,1] , there exists integers ε1,ε2,…,εn∈{0,1}\varepsilon_1,\varepsilon_2,\ldots ,\varepsilon_n\in\{0,1\} such that the inequality ∣∑k=ij(εk−xk)∣≤λ\left\vert \sum^j_{k=i} (\varepsilon_k-x_k)\right\vert\le \lambdaholds for all pairs of integers (i,j)(i,j) where 1≤i≤j≤n1\le i\le j\le n.
inequalities