Problem - 2068
Let positive real number $x$, $y$, and $z$ satisfy $x+y+z=1$. Find the minimal value of $u=\sqrt{x^2 + y^2 + xy} + \sqrt{y^2 +z^2 +yz} +\sqrt{z^2 +x^2 + xz}$
Let $z_1=(x+\frac{1}{2}y)+\frac{3}{2}yi$, $z_2=(y+\frac{1}{2}z)+\frac{3}{2}zi$, and $z_3=(z+\frac{1}{2}x)+\frac{3}{2}xi$.
Then $u=|z_1|+|z_2|+|z_3| \ge |z_1+z_2+z_3| = \boxed{\sqrt{3}}$