$A$
, $B$
, $C$
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$A \cup B=A+BA\cap B$
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.