Can't believe I actually tried to answer this...

Anyway, bear with me...

Circumference of earth = 2*pi*r metres where r is the radius at the equator

But, the circumference of belt = 2*pi*r + 1 metre

The tightend belt will from a 'pinch' at h metres above the earth

Fig.1

The straight distance to the horizon will be the distance from the 'pinch' to the horizon = B

The curved distance to the horizon is from a point directly below the pinch to the horizon following the curvature of the earth. = d

So, the length of the belt = 2*pi*r +1 = distance round the earth - 2*curved distance to horizon + 2*straight distance to horizon

Equation A: 2*pi*r + 1 = 2*pi*r - 2d + 2B

The distance of the pinch to the horizon (straight and curved) can be calculated using the formulas stated

here.

Expressing d and B (as shown in Fig.1 above) in these terms

d = r cos^-1(r/(r+h)

B = root((h(2r+h))

Put this into equation A above

2*pi*r + 1 = 2*pi*r - 2[r cos^-1(r/(r+h)] + 2[root((h(2r+h))]

or

1 = - 2[r cos^-1(r/(r+h)] + 2[root((h(2r+h))]

1/2 = - [r cos^-1(r/(r+h)] + [root((h(2r+h))]

...erm?

.....erm.....?

...and now I havn't a clue how to calculate h.