An algebra problem by أحمد الحلاق

Algebra Level 3

\[ \large \dfrac{ \log_{10} 20^2 \cdot \log_{20} 30^2 \cdot \log_{30} 40^2 \cdots \log_{990} 1000^2 }{ \log_{10} 11^2 \cdot \log_{11} 12^2 \cdot \log_{12} 13^2 \cdots \log_{99} 100^2} =\, ? \]

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