The function F is defined and continuous on the whole real line.  

F(1000) = 999.

For every real number x, the product of F(x) and F(F(x)) is 1.

What is F(101)?

(This problem is Macalester College Math Problem of the Week #891, taken
from a Polish mathematical olympiad.)

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All midshipmen submitting correct solutions to problem #101 by 3:30 pm on
Tuesday, November 2, win a cookie. The best solution will be posted on the
problem bulletin board.
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A solution is "correct" if it gives a correct solution and explains why it
is correct; a solution is "best" if it includes the clearest correct
explanation.  Submit solutions to Prof. Hanna at mathprob@usna.edu, or via
the mailbox in Chauvenet 301.

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Optimal solutions to problem #100 (15 wires) came from Midn. Milev and
LCDR Skufca. Two crossings are sufficient: wires connected in groups of 1,
2, 3, 4, and 5 at each end of the conduit will do the trick if you pick
the second groups carefully.