KöMaL 2012 A563

Let 1$\le p$<2 be a real number. Prove that ($x$+$y$)$^p$+($z$+$v$)$^p$+($x$+$z$)$^p$+($y$+$v$)$^p \le x^p$+$y^p$+$z^p$+$v^p$+($x$+$y$+$z$+$v$)$^p$ for all nonnegative real numbers $x$, $y$, $z$ and $v$.