RSA challenge is well-known and it has a wiki page.
Is there a discrete log for $\mathbb F_p$ where $p$ is Sophie-Germain prime?
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AFAIK the RSA challenge has ended as well. Very basically: it's not needed anymore; if you can publish a paper where you break RSA in any significant way then you should be set for an interesting career. I'd guess it is the same thing for the DLP.
These kind of challenges are the odd one(s) out, not the norm at all. And that kind of makes sense; any significant break is probably coming from the academic field, where getting the academic recognition is a reward in itself.
With regards to RSA factoring; the latest breaks are usually a more efficient way of performing an existing attack, together with a significant amount of distributed computing power. Parallel computing has its own challenges, and the attack may be more interesting from that point of view.
As such, I don't think that a challenge would make much sense. RSA for high key sizes (say 2048 bit keys or higher) is still very much out of reach. Furthermore, the cost of the distributed system would likely dwarf any price money.
As a final remark: these kind of challenges are generally also a marketing tool. "Look: if anybody would be able to break our system they could apply for the price money, right?" As such they are mostly important for a company that makes money out of the system or algorithm used.
Nowadays these algorithms are wide-spread and the RSA labs from the old days is long gone. I don't think that there is a company that would directly benefit from issuing a challenge.