5. How many diagonals in a regular 32-sided polygon are not parallel to any of its sides
Answer: 240.

Solution: In a 32-sided polygon, there are $32*(32-3)/2=464$ diagonals in total. We can divide the sides into 16 pairs of parallel sides. It is not hard to notice that if we fix a pair, i.e., 4 vertices, the remaining vertices can be connected in pairs by diagonals parallel to this pair. There will be a total of (32-4)/2 = 14 such diagonals. Therefore, the number of diagonals parallel to any side is $14 * 16=224$. The number of diagonals not parallel to any side is $464-224=240$.