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# Inequalities

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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

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

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

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

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Jul 28, 2019