A Contest Venn Puzzle

A contest has three problems $A, B, C$. It is known that:

1. $30$ students solved at least one problem; 2. among students who solved exactly one problem, half solved $C$; 3. the number of students who solved $A$ equals the number who solved only $B$; 4. the number who solved exactly $A$ and $B$ equals the number who solved exactly $B$ and $C$; 5. exactly $4$ students solved only $A$; 6. the number who solved exactly $A$ and $C$ is half the number who solved exactly $A$ and $B$.

How many students solved all three problems?