Is deciding if there is a set that is intersected with some given sets and that has at most one common element with other given sets NP-complete?
You can reduce 3SAT to this problem by converting each variable $v_i$ to two elements $e_{i1}$ and $e_{i2}$ (one for its positive literal and one for its negative literal) and a set $B_i$ containing $e_{i1}$ and $e_{i2}$, and converting each clause $c_j=l_{j1}\vee l_{j2}\vee l_{j3}$ to a set $A_j$ containing elements representing $l_{j1},l_{j2},l_{j2}$. 

Now it's not hard to prove this problem is NP-complete using the reduction above.