Problem 2

Let

f(x)

and

g(x)

be given by

f(x)=\dfrac1x+\dfrac1{x-2}+\dfrac1{x-4}+\cdots+\dfrac1{x-2018}

and

g(x)=\dfrac1{x-1}+\dfrac1{x-3}+\dfrac1{x-5}+\cdots+\dfrac1{x-2017}

. Prove that

|f(x)-g(x)|>2

for any non-integer real number

x

satisfying

0<x<2018

Step 2 of 6: Use the half-turn symmetry

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f(2018-x)=f(x),\qquad g(2018-x)=g(x)

Detailed analysis

Replacing $x$ by $2018-x$ permutes the shifts $2k\mapsto 2018-2k$ and $2k+1\mapsto 2017-2k$ among themselves, so both sums are invariant. Consequently the interval type $2n<x<2n+1$ maps to the interval type $2m-1<x<2m$ with $m=1009-n$ , so it suffices to treat only intervals of the form $2n-1<x<2n$ for $1\le n\le 1009$ .