Infocus 250 projector won't power up

Teach it a lesson. Leave it plugged in until it decides to come back on.
If you have a remote control, you can try turning it on with that.
Another option is carefully remove the lamp and put it back in again.
It might be a false connection problem.
One thing I always try is while plugged in and "supposedly" on, try dropping it about one half of an inch on the table top. If it has a loose connection, that sometimes fixes things.

Could simply be a defective power cord I suggest that you find a different power cord and see if the problem goes away