Presumably if you made the prize large enough you wold get to the point where it would be economical and profitable to start fabricating your own hardware like the EFF did for DES.
Ship yourself custom ASICs in the few months allowed by the contest guidelines and you'd earn it. FPGAs are another possibility, but even that would be a bit of an undertaking in a short timeframe and it might not actually get you much.
Also the DES hardware is slightly different than prime number factorization in terms of workload. Supercomputers are fairly well optimized for some of the types of matrix operations you'd need to do for a GNFS, which is generally the method of choice for factoring large numbers on a classical computer. Custom hardware isn't going to give you the huge boost like you'd see for brute forcing DES.
Custom hardware isn't a silver bullet and it is a large engineering problem that takes usually more time than the telegraph contest allows.
This is true, but you could easily respond by increasing the RSA key size. This would make the protocol Telegram-secure without meaningfully improving its actual security profile in any way.
http://en.wikipedia.org/wiki/EFF_DES_cracker