On your turn, you
must engage the target you challenged or challenge a different target. To engage the target, you must either attack it or end your turn adjacent to it. If none of these events occur by the end of your turn, the marked condition ends and you can’t use divine challenge on your next turn.
Possibilities:
1) Challenge target, attack or move adjacent to target, target is still challenged after the turn.
2) Attack target, challenge target, target not adjacent, the challenge ends at the end of the turn. The reason is the phrase "must engage" is not a past tense phrase, but "challenged" is past tense. The phrase is not "must have engaged this turn", it's "must engage".
3) Attack target, challenge target, target adjacent, the challenge continues into the next turn. Unlike the previous scenario (#2), the non-past tense "must engage" phrase is satisfied since the Paladin is engaging after challenging.
The challenge must occur before the engaging or the power fails at the end of the turn.
Now, the phrase "or challenge a different target" could supercede this (as people here have claimed) except for a different sentence in the power description:
The target remains marked until you use this power against another target, or if you fail to engage the target.
4) At end of turn, challenge a different target. The challenge ends at the end of the turn since "you fail to engage the target" in a non-adjacent case. The target is no longer marked, hence, it is no longer challenged. It works in an adjacent case because the target is engaged.