Problem

Source: 2010 Romania JBMO TST 1.3

Tags: system of equations, algebra



Let $n \ne 0$ be a natural number and integers $x_1, x_2, ...., x_n, y_1, y_2, ...., y_n$ with the properties: a) $x_1 + x_2 + .... + x_n = y_1 + y_2 + .... + y_n = 0,$ b) $x_1 ^ 2 + y_1 ^ 2 = x_2 ^ 2 + y_2 ^ 2 = .... = x_n ^ 2 + y_n ^ 2$. Show that $n$ is even.