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Let \(x_1, x_2, \ldots, x_n\) be real numbers which satisfy \(|x_i| < 1\) for \(i = 1, 2, \dots, n,\) and \(|x_1| + |x_2| + \dots + |x_n| = 19 + |x_1 + x_2 + \dots + x_n|.\)What is the smallest possible value of \(n\)?

 Jul 28, 2019

\(\text{First off by the triangle inequality $\sum \limits_{n=1}^N|x_n|\geq \left|\sum \limits_{n=1}^N x_n\right|$}\)


\(\text{So if we want to minimize $n$ the best we can do is have $\sum \limits_{n=1}^N x_n = 0$}\\ \text{This would make the smallest possible value of $N=20$}\\ x_n = (-1)^n \dfrac{19}{20}\)


\(\text{It should be pretty clear as $|x_n|< 1,~\forall n$ that we can't do any better than this}\)

 Jul 28, 2019

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