Two real problems

I pick two reals x and y. I show one of them to you randomly by tossing a fair coin. You have to tell me if the number you see is the larger or the smaller of the two. How do you do this so that for any choice of x and y, the probability that you are correct is strictly larger than .5?

On a sold out flight, 100 people line up to board the plane. The first passenger in the line has lost his boarding pass, but was allowed in, regardless. He takes a random seat. Each subsequent passenger takes his or her assigned seat if available, or a random unoccupied seat, otherwise. What is the probability that the last passenger to board the plane finds his seat unoccupied?

Stringer Verlag

What is the longest string you can make using only C's and G's within the following constraints:

1. No GG can occur within the string,
2. No sub-string can repeat itself 3 times consecutively eg: CGCGCG is not allowed.

Why?

Suspension bridges

You must have seen the Golden Gate bridge or one of the other suspension bridges during your travels. What curve do the cables follow?

Admittedly, this is a bit mathematical. Hence, I shall abstain from giving points if someone complains. Otherwise, 1 point!

It gets somewhat interesting after you solve the problem. Shown below is a stolen picture of the chelsea bridge. In red diamond symbols are second order polynomial fits to the main cable. I simply digitized the middle cable and fitted it in matlab. While the good fit makes it very tempting to conclude that its a proof for a parabolic curve, the catenoid can also be series expanded to a parabola accurate to 4th order! Any small error in curve fit could push it this way or that. So, I guess the fitting does not say anything useful other than the fact that the cable is roughly a parabola.

ps: Check out Manoj's angle trisection comments at http://xyfactor.blogspot.com/2006/05/trisect-angle.html