Track / Stage 3 / 6 of 260 #6 of 1964

Problem 6

AMC 10/12, early questions

Combinatorics Difficulty 3.0 Multiple choice

For any set $S$ , let $|S|$ denote the number of elements in $S$ , and let $n(S)$ be the number of subsets of $S$ , including the empty set and the set $S$ itself. If $A$ , $B$ , and $C$ are sets for which $n(A)+n(B)+n(C)=n(A\cup B\cup C)$ and $|A|=|B|=100$ , then what is the minimum possible value of $|A\cap B\cap C|$ ?

Pick one

Official solution

$n(A)=n(B)=2^{100}$ , so $n(C)$ and $n(A \cup B \cup C)$ are integral powers of $2$ $\Longrightarrow$ $n(C)=2^{101}$ and $n(A \cup B \cup C)=2^{102}$ . Let $A=\{s_1,s_2,s_3,...,s_{100}\}$ , $B=\{s_3,s_4,s_5,...,s_{102}\}$ , and $C=\{s_1,s_2,s_3,...,s_{k-2},s_{k-1},s_{k+1},s_{k+2},...,s_{100},s_{101},s_{102}\}$ where $s_k \in A \cap B$
Thus, the minimum value of $|A\cap B \cap C|$ is $\fbox{B=97}$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.