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JBMO 2018 P3, the shortlist version, min k for the nxn system

Source: JBMO Shortlist 2018 A4

July 22, 2019
algebraminimumsystem of equationsProduct

Problem Statement

Let k>1,n>2018k > 1, n > 2018 be positive integers, and let nn be odd. The nonzero rational numbers x1,x2,…,xnx_1,x_2,\ldots,x_n are not all equal and satisfy x1+kx2=x2+kx3=x3+kx4=…=xn−1+kxn=xn+kx1x_1+\frac{k}{x_2}=x_2+\frac{k}{x_3}=x_3+\frac{k}{x_4}=\ldots=x_{n-1}+\frac{k}{x_n}=x_n+\frac{k}{x_1} Find: a) the product x1x2…xnx_1 x_2 \ldots x_n as a function of kk and nn b) the least value of kk, such that there exist n,x1,x2,…,xnn,x_1,x_2,\ldots,x_n satisfying the given conditions.
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