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IMAR Test 2018 P3

Source: IMAR Test 2018 P3

April 21, 2021

combinatoricsSetsromania

Problem Statement

Let

S

be a finite set and let

\mathcal{P}(S)

be its power set, i.e., the set of all subsets of

S

, the empty set and

S

, inclusive. If

\mathcal{A}

and

\mathcal{B}

are non-empty subsets of

\mathcal{P}(S),

let

\mathcal{A}\vee \mathcal{B}=\{X:X\subseteq A\cup B,A\in\mathcal{A},B\in\mathcal{B}\}.

Given a non-negative integer

n\leqslant |S|,

determine the minimal size

\mathcal{A}\vee \mathcal{B}

may have, where

\mathcal{A}

and

\mathcal{B}

are non-empty subsets of

\mathcal{P}(S)

such that

|\mathcal{A}|+|\mathcal{B}|>2^n

.
Amer. Math. Monthly

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