There is a natural number $\displaystyle k$. Prove that from any set consisting of integer numbers that has more than $\displaystyle 3^k$ elements we can take a subset S that has $\displaystyle (k+1)$ elements and:

For any two different subsets $\displaystyle A, B \subseteq S$ the sum of all elements of set $\displaystyle A$ is different from the sum of all elements of set $\displaystyle B$. (We assume that the sum of all elements of an empty set equals 0).

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