BALKAN 1996 Problem 4

Suppse that $X=\{1,2, \ldots, 2^{1996}-1\}$ , prove that there exist a subset $A$ that satisfies these conditions:

a) $1\in A$ and $2^{1996}-1\in A$ ;

b) Every element of $A$ except $1$ is equal to the sum of two (possibly equal) elements from $A$ ;

c) The maximum number of elements of $A$ is $2012$ .

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