Buy applecare more than 1 year after purchase

Hi everyone

I've seen that you can't buy AppleCare more than one year after purchase and I understand that, after the warranty ends, people would just wait until they encounter problems and then purchase AppleCare. I bought my MacBook Pro just over a year ago. When I was about ten months in I had a motherboard failure and the warranty covered it. The service took almost a month and the technicians failed to mention that the hard drive was fixed to the motherboard and would also be replaced. Lucky me that I had everything on the cloud. So even though it took a long time, I was happy with the result. Now I find out that I'm two weeks late for buying AppleCare.

If I subtract the nearly one month of waiting for the repair then I would still be within the one year warranty right now 🙂 ! Is it possible to take my MacBook to an apple center to prove that everything is working as it should and then still buy AppleCare to cover until three years after purchase?

Thanks for any replies