Why does my car turn off in the mornings?

Okay so I just bought an 88 Honda Accord 5 speed. I love it! The guy before me maintained it really well and it drives great. The engine looks good and everything on it is awesome. One problem though that he said he never figured out and neither can I. He was an older guy so and didn't have much time for the car anymore so I'm sure he didn't look that far into it. In the morning, and after the car has been cooled down for a few hours, when I start it, it won't stay on. I have to start in first gear and drive it instantly for it to stay on until it warms up and then it gives me no problem. What could this be. I though maybe a spark plug but the guys said he had it tuned recently and I also get regular oil changes. It's not a big deal for now but I worry about the winter when it's time to warm it up how will I keep it on.

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