잘 지내셨어요? 잘 지냈어요? Both correct and different in honorifics. All verbs and adjectives can be converted into an honorific form by adding the infix -시- (-si-) or -으시- (-eusi-) after the stem and before the ending. 지내셨어요 is an honorific expression of 지냈어요 지내셨어요 is composed of 지내(다)+시+었+어요 지냈어요 is composed of 지내(다)+었+어요 지내셨어요 used when talking with someone superior in status, for example older relative, a stranger of roughly equal or greater age, an employer, a teacher, a customer, or the like. 지냈어요 used when talking with someone not superior in status, for example younger stranger, a student, an employee or the like. Refer to http://en.wikipedia.org/wiki/Korean_honorifics