Baba Yaga's Hut says instead of rolling for your movement you automatically move five spaces a turn. The Path of the Determined says instead of rolling for your movement you automatically move one space per turn. Which prevails over the other when you have both?!?!

"A character with a Path must always follow the Travel effect on his Path Card."

Therefore Path of the Determined overrides Baba Yaga's Hut!

Surely the rule from the Woodland rulebooks matters here, but the reason why Path of the Determined overrides Baba Yaga's Hut is subtler.

Baba Yaga's Hut says "instead of rolling for your movement, you must always move 5 spaces."; Path of the Determined says "Instead of your normal move, you only move 1 space".

If you have Path of the Determined and then take Baba Yaga's Hut as a Follower, you're not rolling for your movement so the Hut effect is not applicable as long as you have Path of the Determined.

Vice versa, if you have Baba Yaga's Hut and take Path of the Determined, the instruction "instead of your normal move" overrides the special movement instructions from Baba Yaga's Hut.

This just expands an answer already given by adjogi on Board Game Geek.

http://boardgamegeek.com/thread/1243715/playing-now-baba-yagas-hut-vs-path-determined