Question 85
A flea moves by jumping back and forth along a line. The flea's burrow is a point on the line. The jump rules are as follows:
- if the flea is at a distance less than or equal to one metre from the burrow, after the next jump it will be at twice the previous distance, moving even farther from the burrow.
- if the flea is at a distance $d$ greater than one metre from the burrow, after the next jump it will be at distance $\frac{1}{d}$ from the burrow, but on the opposite side from the one where it currently is.
If after 5 jumps the flea is at $80~\mathrm{cm}$ from the burrow in a certain direction, with how many distinct sequences of jumps can it have reached that position?
- if the flea is at a distance less than or equal to one metre from the burrow, after the next jump it will be at twice the previous distance, moving even farther from the burrow.
- if the flea is at a distance $d$ greater than one metre from the burrow, after the next jump it will be at distance $\frac{1}{d}$ from the burrow, but on the opposite side from the one where it currently is.
If after 5 jumps the flea is at $80~\mathrm{cm}$ from the burrow in a certain direction, with how many distinct sequences of jumps can it have reached that position?