HMMT 2009 A7

Simplify the product

$\prod_{m=1}^{100} \prod_{n=1}^{100} \frac{x^{n+m}+x^{n+m+2}+x^{2 n+1}+x^{2 m+1}}{x^{2 n}+2 x^{n+m}+x^{2 m}}$

Express your answer in terms of $x$ .

Dataset text NuminaMath olympiads. Quoted with source.

Show the answer

x^{9900}\left(\frac{1+x^{100}}{2}\right)^{2}

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