I do not think there was any specific project that prompted this, but in 2021, OpenAI consistently used the phrase “Solve Impossible Problems” in their job postings and promotional materials, from twitter job postings (Example 1, Example 2) to the source code of their website (as spotted here)
The slogan was/is a great way to highlight their mission to push the boundaries of AI research and achieve unprecedented levels of performance - so I think it makes sense they would trademark this phrase to solidify their brand identity and vision.