Problem 1.

There are several boys and girls in the class, and it is given that any boy is friends with at least one girl. Prove that it is possible to select at least half of the class so that each boy is friends with odd number of girls.

Problem 2.

Prove that for any finite set of positive integers, it is possible to colour at least a third of its elements in such a way that there were no three coloured numbers $a,b,c$ such that $a+b=c$.

Problem 3.

There is a problem about graphs in the previous homework, it is in the box that goes after the five problems. Now that you know more about mathematical expectation, you can actually solve it by yourself :)