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Least value of n [Mediterranean Mathematics Olympiad 2017, P2]

Source: Mediterranean Mathematics Olympiad 2017, Problem 2

June 15, 2017

algebra

Problem Statement

Determine the smallest integer

n

for which there exist integers

x_1,\ldots,x_n

and positive integers

a_1,\ldots,a_n

so that \begin{align*} x_1+\cdots+x_n &=0,\\ a_1x_1+\cdots+a_nx_n&>0, \text{ and }\\ a_1^2x_1+\cdots+a_n^2x_n &<0. \end{align*}