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All possible values of the sum

Source: Romanian District Olympiad 2014, Grade 8, P4

June 15, 2014
floor functionalgebra proposedalgebra

Problem Statement

Let n≥2n\geq2 be a positive integer. Determine all possible values of the sum S=⌊x2−x1⌋+⌊x3−x2⌋+...+⌊xn−xn−1⌋ S=\left\lfloor x_{2}-x_{1}\right\rfloor +\left\lfloor x_{3}-x_{2}\right\rfloor+...+\left\lfloor x_{n}-x_{n-1}\right\rfloor where xi∈Rx_i\in \mathbb{R} satisfying ⌊xi⌋=i\lfloor{x_i}\rfloor=i for i=1,2,…ni=1,2,\ldots n.
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