2021 AIME II Problems/Problem 6

Problem

For any finite set $S$ , let $|S|$ denote the number of elements in $S$ . FInd the number of ordered pairs $(A,B)$ such that $A$ and $B$ are (not necessarily distinct) subsets of $\{1,2,3,4,5\}$ that satisfy $|A| \cdot |B| = |A \cap B| \cdot |A \cup B|$

Solution 1

By PIE, $|A|+|B|-|A \cap B| = |A \cup B|$ , and after some algebra you see that we need $A \subseteq B$ or $B \subseteq A$ . WLOG $A\subseteq B$ , then for each element there are $3$ possibilities, either it is in both $A$ and $B$ , it is in $B$ but not $A$ , or it is in neither $A$ nor $B$ . This gives us $3^{5}$ possibilities, and we multiply by $2$ since it could of also been the other way around. Now we need to subtract the overlaps where $A=B$ , and this case has $2^{5}=32$ ways that could happen. It is $32$ because each number could be in the subset or it could not be in the subset. So the final answer is $2\cdot 3^5 - 2^5 = \boxed{454}$ .

~ math31415926535

Solution 2

We denote $\Omega = \left\{ 1 , 2 , 3 , 4 , 5 \right\}$ . We denote $X = A \intersect B$ (Error compiling LaTeX. Unknown error_msg), $Y = A \backslash \left( A \intersect B \right)$ (Error compiling LaTeX. Unknown error_msg), $Z = B \backslash \left( A \intersect B \right)$ (Error compiling LaTeX. Unknown error_msg), $W = \Omega \backslash \left( A \union B \right)$ (Error compiling LaTeX. Unknown error_msg).

Therefore, $X \cup Y \cup Z \cup W = \Omega$ and the intersection of any two out of sets $X$ , $Y$ , $Z$ , $W$ is an empty set. Therefore, $\left( X , Y , Z , W \right)$ is a partition of $\Omega$ .

Following from our definition of $X$ , $Y$ , $Z$ , we have $A \cup B = X \cup Y \cup Z$ .

Therefore, the equation

$|A| \cdot |B| = |A \cap B| \cdot |A \cup B|$

can be equivalently written as

$\left( | X | + | Y | \right) \left( | X | + | Z | \right) = | X | \left( | X | + | Y | + | Z | \right) .$

This equality can be simplified as

$| Y | \cdot | Z | = 0 .$

~ Steven Chen (www.professorchenedu.com)

2021 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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2021 AIME II Problems/Problem 6

Problem

Solution 1

Solution 2