Hidden Roots Part 3 (maybe)

Calculus Level 4

F ( x ) = 1 x − 1 + 1 x − 2 + 1 x − 3 + ⋯ + 1 x − ( 2 k − 1 ) + 1 x − 2 k \mathfrak{F}(x)=\dfrac{1}{x-1}+\dfrac{1}{x-2}+\dfrac{1}{x-3} + \cdots +\dfrac{1}{x-(2k-1)}+\dfrac{1}{x-2k}

Given the above function F ( x ) \mathfrak{F}(x) ,

x 1 , x 2 , … , x n x_{1},x_{2}, \ldots ,x_{n} are its roots in some order and k ∈ N k \in \mathbb{N} .

Find the maximum possible value of value [ x 1 ] − [ x 2 ] + [ x 3 ] − [ x 4 ] + ⋯ ± [ x n ] n + 1 . \large\dfrac{ [x_{1} ]-[ x_{2} ]+ [ x_{3} ]-[ x_{4} ]+ \cdots \pm[ x_{n} ]} {n+1}.

Details and Assumptions:

  • [ m ] [m] represents greatest integer function of m m

  • [ x m ] [ x_{m} ] represents the greatest integer of the smallest root of F ( x ) \mathfrak{F}(x)

Inspired from Shubhendra Singh and his problem Hidden Roots
k + 1 2 \frac{k+1}{2} k 2 \frac{k}{2} k + 0.5 2 \frac{k+0.5}{2} n − [ x m ] 2 \frac{ n - [ x_{m} ] }{2} n + [ x m ] 2 \frac{ n + [ x_{m} ] }{2} 2 k − n + 1 \ 2k-n+1 2 k + n 2k+n

Omar Sehlouli by Omar Sehlouli , 21, Morocco

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