(x+y)^z+(y+z)^x+(z+x)^y > 2.

Note: Link to posted solution is not working.

Without loss of generality, let x>y>z

(x+y)^z+(y+z)^x+(z+x)^y > (x+y)^x+(y+z)^x+(z+x)^x

> (x+x)^x+(x+x)^x+(x+x)^x = 3*(2x)^x

Prove (2x)^x>2/3

Differentiating and setting = 0 yields x = (e^-1) / 2

and (2x)^x > 0.832 > 2/3

QED