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min n such that x_i \in [1/2 , 2], sum x_i >= 7n/6, sum 1/x_i >= 4n/4

Source: 2000 Moldova JBMO TST p4

February 20, 2021
algebrainequalitiesSystem

Problem Statement

Find the smallest natural number nonzero n so that it exists in real numbers x1,x2,...,xnx_1, x_2,..., x_n which simultaneously check the conditions: 1) xi∈[1/2,2]x_i \in [1/2 , 2] , i=1,2,...,ni = 1, 2,... , n 2) x1+x2+...+xn≥7n6x_1+x_2+...+x_n \ge \frac{7n}{6} 3) 1x1+1x2+...+1xn≥4n3\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}\ge \frac{4n}{3}
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