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|x_1c_1 + x_2c_2 +... + x_nc_n| >= |b-a|/2(|c_1|+|c_2|+\ldots+|c_n|)

Source: Polish MO Recond Round 1977 p1

September 9, 2024
algebrainequalities

Problem Statement

Let a a and b b be different real numbers. Prove that for any real numbers c1,c2,…,cn c_1, c_2, \ldots,c_n there exists a sequence of n n -elements (xi) (x_i) , each term of which is equal to one of the numbers a a or b b such that ∣x1c1+x2c2+…+xncn∣≥∣b−a∣2(∣c1∣+∣c2∣+…+∣cn∣). |x_1c_1 + x_2c_2 + \ldots + x_nc_n| \geq \frac{|b-a|}{2}(|c_1|+|c_2|+\ldots+|c_n|).
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