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10th ibmo - chile 1995/q2.

Source: Spanish Communities

May 7, 2006
inequalitiesquadraticsalgebra unsolvedalgebra

Problem Statement

Let nn be a positive integer greater than 1. Determine all the collections of real numbers x_1,\ x_2,\dots,\ x_n\geq1\mbox{ and }x_{n+1}\leq0 such that the next two conditions hold:
(i) x112+x232+⋯+xnn−12=nxn+112x_1^{\frac12}+x_2^{\frac32}+\cdots+x_n^{n-\frac12}= nx_{n+1}^\frac12
(ii) x1+x2+⋯+xnn=xn+1\frac{x_1+x_2+\cdots+x_n}{n}=x_{n+1}
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