$\large \dfrac{x^{100}}{1+x+x^{2}+x^{3}+ \cdots +x^{200}}$

For $x>0$ , the maximum value of the expression above is $\dfrac{1}{N}$ for some integer $N$ . Find $N$ .

This is a part of 10-seconds challenge .

The answer is 201.

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Expression can be written as : $\dfrac{1}{\color{#0C6AC7}{\frac{1}{x^{100}}}+\color{#D61F06}{\frac{1}{x^{99}}}+\cdots \color{#D61F06}{x^{99}}+\color{#0C6AC7}{x^{100}}}$ Applying $AM\geq GM$ to denominator , we can see minimum value of denominator is $201$ . Hence maximum value of expression is $\boxed{\large {\color{#20A900}{\dfrac{1}{201}}}}$ , $\therefore N=201$ . Equality occurs when $x=1$ .