10.5. Let's consider the cut lines formed in the original 100 x 100 square. It is easy to see that one cut of the folded square corresponds to cutting the original square along several parallel lines, with the distances between these lines being no less than 2 (since at least one fold line must pass between any two cut lines). Therefore, there can be no more than 50 such lines, and they divide the square into no more than 51 rectangles. With two such cuts, the maximum number of parts formed is $51^{2}=2601$, since each rectangle from the first division is split into no more than 51 parts by the second cut. An example with 2601 parts can be obtained by folding the square so that the fold lines form a 2 x 2 grid, and then cutting the resulting 2 x 2 square after the folds along its two middle lines.
Answer: 2601

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