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Difference between revisions of "1991 AHSME Problems/Problem 30"

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For any set <math>S</math>, let <math>|S|</math> denote the number of elements in <math>S</math>, and let <math>n(S)</math> be the number of subsets of <math>S</math>, including the empty set and the set <math>S</math> itself. If <math>A</math>, <math>B</math>, and <math>C</math> are sets for which <math>n(A)+n(B)+n(C)=n(A\cup B\cup C)</math> and <math>|A|=|B|=100</math>, then what is the minimum possible value of <math>|A\cap B\cap C|</math>?
 
For any set <math>S</math>, let <math>|S|</math> denote the number of elements in <math>S</math>, and let <math>n(S)</math> be the number of subsets of <math>S</math>, including the empty set and the set <math>S</math> itself. If <math>A</math>, <math>B</math>, and <math>C</math> are sets for which <math>n(A)+n(B)+n(C)=n(A\cup B\cup C)</math> and <math>|A|=|B|=100</math>, then what is the minimum possible value of <math>|A\cap B\cap C|</math>?
  
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<math>(A) 96 \space (B) 97 \space (C) 98 \space (D) 99 \space (E) 100</math>
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<math>(A) 96 \ (B) 97 \ (C) 98 \ (D) 99 \ (E) 100</math>
  
 
== Solution ==
 
== Solution ==

Revision as of 14:16, 13 February 2018

Problem

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

$(A) 96 \ (B) 97 \ (C) 98 \ (D) 99 \ (E) 100$

Solution

$n(A)=n(B)=2^{100}$, so as $n(C)$ and $n(A \cup B \cup C)$ are integral powers of $2$, $n(C)=2^{101}$ 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 $\fbox{B=97}$

See also

1991 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 29 		Followed by
Problem 30
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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