For 2) solution by Mikhail
. Applying Cauchy-Schwarz inequality
$\sum_{cyc}\sqrt{\frac{x}{x+y}}\le\sqrt{(\sum_{cyc}x+z)(\sum_{cyc}\frac{x}{(x+y)(x+z)})}$
It suffices to show that
$\sum_{cyc}\frac{x}{(x+y)(x+z)}\le\frac{9}{4(x+y+z)}$
You can expand to prove the last inequality.
