RU-SPB 2012 Problem 114

$x_1,...,x_n$ are reals and $x_1^2+...+x_n^2=1$ Prove, that exists such $y_1,...,y_n$ and $z_1,...,z_n$ such that $|y_1|+...+|y_n| \leq 1$ ; $max(|z_1|,...,|z_n|) \leq 1$ and $2x_i=y_i+z_i$ for every $i$