Postby **conrado.math** » Tue Sep 29, 2009 1:48 pm

There's a much simpler solution to this problem. It's rather informal, but with a little work (and time, which we don't have during the test) it can be made rigorous. Here it goes:

Let b the inner angle to the right and a the one above. A simple application of the sine law shows that b goes to zero when r and s go to infinity. In particular, a converges to 70 degrees. So, the picture we have after taking limits is two parallel half-lines joined together by a segment whose angle with the bottom half-line is 110 degrees and to the top one is 70 degrees.

If we lift a perpendicular segment at the point of intersection between the bottom half-line and the segment described previously, we form a right triangle whose inner angles are 70 degrees on the top left and 20 degrees on the bottom. The quantity we seek is precisely the length of the base of this triangle, which can be easily seen to be cos(70 degrees), a real number greater than zero but less than one.

It took me about 20 minutes to rigorously prove all these claims, and if anyone wants a detailed proof just ask.