MathDB

Let

n

be an even positive integer. Alice and Bob play the following game. Before the start of the game, Alice chooses a set

S

containing

m

integers and announces it to Bob. The players then alternate turns, with Bob going first, choosing

i\in\{1,2,\dots, n\}

that has not been chosen and setting the value of

v_i

to either

0

1

. At the end of the game, when all of

v_1,v_2,\dots,v_n

have been set, the expression

E=v_1\cdot 2^0 + v_2 \cdot 2^1 + \dots + v_n \cdot 2^{n-1}

is calculated. Determine the minimum

m

such that Alice can always ensure that

E\in S

regardless of how Bob plays.

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