Three planar regions $A$, $B$, $C$ partially overlap each other, with $|A| = 90,$ $|B| = 90,$ $|C| = 60$ and $|A \cup B \cup C| = 100,$ where $|\cdot|$ denotes the area. Find the minimum possible $|A \cap B \cap C|$.

Try to find the minimum $|A\cap B|.$

… by considering the maximum $|A\cup B|.$

What does that minimal case imply about $C$ in relation to $A$ and $B,$ given the question conditions?

… more specifically, in relation to $A\cup B,$ given $|A \cup B \cup C| = 100?$

Given $C$ must be contained within $A\cup B,$ what does that imply about $C$ in relation to $A\cap B?$

… more specifically, how must $C$ be distributed outside of $A\cap B$ in order to minimize the full intersection?

First consider the minimum size of $A \cap B$. We know that $|A \cup B|=|A|+|B|-|A\cap B|$ and since $|A \cup B\cup C|=100$ then $|A \cup B|\le100$ so $|A \cap B|\ge80$. In this minimal case, $C$ must be contained within $A \cup B,$ because otherwise $|A \cup B \cup C| > 100$. We want to have as much of $C$ as possible outside of $A\cap B,$ hence the minimal size of the intersection is $|C| - (|A \cup B| - |A \cap B|)=40$.

Note: Drawing standard (circular) Venn diagrams for $A,B,C$ to meet the above conditions is not possible, but it is possible using other shapes. Give it a try.