More Parity Puzzles

...because I'm odd that way. Both are in the same 'can-do' spirit of the last one, yet neither use quite the same techniques; still simple.
Guaranteed correct, 'cause they're not mine, but lifted from the problem sets I'll credit.

1. Two delegations A and B, with the same number of delegates, arrived at a conference. Some of the delegates knew each other already. Prove that there is a non-empty subset A' of A such that either each member in B knew an odd number of members from A', or each member of B knew an even number of members from A'.

2. Given a finite set X of positive integers and a subset A of X, there exists a subset B of X, such that A are precisely the elements of X that divide by an odd number of elements of B.

(Hint: not a number theory puzzle.)


First is from the 1996 Kurschak Competition (Hungarian), via the Kalva site (see sidebar). Second is from the book '102 Combinatorial Problems' by Andreescu and Feng. Enjoy!