$A$, $B$, $C$ 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 $|A \cap B \cap C|$.
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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?
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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.