I've always been inspired by examples of this through history including:
- Longitude Rewards (17th century): https://en.wikipedia.org/wiki/Longitude_rewards
- Orteig Prize for Trans-Atlantic Flight (1920s): https://en.wikipedia.org/wiki/Orteig_Prize
- Fermat's Last Theorem (1800s/1908): https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem#Monetary_prizes
- Ansari X Prize (1996): https://en.wikipedia.org/wiki/Ansari_X_Prize
- DARPA Grand Challenge (2000s): https://en.wikipedia.org/wiki/DARPA_Grand_Challenge
What specific problems would you want to donate a few dollars to see solved and that will work with prize amounts in the range $1k-$500k? What referees would you recommend to determine if the problem is "solved"?