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There doesnt exist integers such that...

Source: Greek national M.O. 2004, Final Round,problem 2

November 15, 2011
inductionalgebra unsolvedalgebra

Problem Statement

If m≥2m\geq 2 show that there does not exist positive integers x1,x2,...,xm,x_1, x_2, ..., x_m, such that x1<x2<...<xm  and  1x13+1x23+...+1xm3=1.x_1< x_2<...< x_m \ \ \text{and} \ \ \frac{1}{x_1^3}+\frac{1}{x_2^3}+...+\frac{1}{x_m^3}=1.
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