Split two peas in a pod

Discrete Mathematics Level 5

What is the smallest integer \(N\), such that no matter how we split the set \( S_N = \{ 1, 2, \ldots, N \} \) into two sets \( A \) and \(B\), there exists one set such that we can find 20 (not necessarily distinct) elements \( x_1, x_2, \ldots x_{20} \) satisfying

\[ x_1 + x_2 + \ldots + x_{19} = x_{20}? \]

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