Even though the battery is a year old, it can still be a problem. Manufacturing defects, severe shock (bumps) can damage a battery. Also, if the car just sat at the mechanic's with the battery installed, this can pull the battery charge down.
Also, you could have a failing alternator, one which cannot maintain the battery charge sufficiently, or one which is throwing too much current and maybe damaging the battery.
Best to have the battery and alternator tested. Any AutoZone will do this for you. Probably a good idea to charge the battery up - low current battery maintainer is best - takes longer, but better for the battery.
Doubtful it's the starter because once the car starts, it's job is finished and wouldn't effect the way the car runs after it's running.
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