# Practice Paper 2 Question 4

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 \(|A \cap B \cap C|\).

## Related topics

## Hints

- Hint 1Try to find the minimum \(|A\cap B|.\)
- Hint 2... by considering the maximum \(|A\cup B|.\)
- Hint 3What does that minimal case imply about \(C\) in relation to \(A\) and \(B,\) given the question conditions?
- Hint 4... more specifically, in relation to \(A\cup B,\) given \(|A \cup B \cup C| = 100?\)
- Hint 5Given \(C\) must be contained within \(A\cup B,\) what does that imply about \(C\) in relation to \(A\cap B?\)
- Hint 6... more specifically, how must \(C\) be distributed outside of \(A\cap B\) in order to minimize the full intersection?

## Solution

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.

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