Is there a way to create a "captcha"-like problem that computers can verify, but only humans can solve?
Yes.  Computer A can generate a CAPTCHA encoding a 24-digit code, and also compute a cryptographic hash of the correct response.  Computer A can give the CAPTCHA and the hash to B and C.  Now if B claims to have solved the CAPTCHA, computer C can verify the claimed answer using the cryptographic hash.  The cryptographic hash doesn't help computer C solve the challenge, because the hash is one-way and cannot be inverted (except by trying all possible codes, but that would take far too long).

In practice, this is not very effective, because no one knows how to build a CAPTCHA that humans can solve but that is expensive for computers to solve.  For one thing, there are black-market services that will solve CAPTCHAs for you at the price of about $1-2 per thousand CAPTCHAs solved, i.e., about 0.1-0.2 cents per CAPTCHA.  For another, there are algorithms that break most CAPTCHAs with probability higher than we'd like.  So ultimately, CAPTCHAs kinda suck.  But the reasons they suck have nothing to do with verifiability.  If we could make a CAPTCHA that didn't suck, we could probably build one that was verifiable in the sense you mention.

Also, if you were implementing this in real life, you'd make sure to use a slow hash (like PBKDF2, scrypt, or bcrypt) when hashing the code.  But again, because CAPTCHAs have so many limitations, I'm not sure you'd want to implement this scheme in real life anyway.