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Yes. They bounced millions of queries off of ChatGPT to teach/form/train their DeepSeek model. This bot-like querying was the "distillation."

They definitely didn't. They demonstrated their stuff long before OAI and the models were nothing like each other.

Why would OpenAI allow someone to do that?

They didn't, but how do you stop it? Presuming the scale that OpenAI is running at?

They don't anymore. They introduced ID verification shortly after, but it's hard to stop completely while also scaling fast.