RO-NMO 2009 Problem 101

a) Show that two real numbers $x,y>1$ chosen so that $x^y=y^x,$ are equal or there exists a positive real number $m\neq 1$ such that $x=m^{\frac{1}{m-1}}$ and $y=m^{\frac{m}{m-1}} .$ b) Solve in $\left( 1,\infty \right)^2$ the equation: $x^y+x^{x^{y-1}}=y^x+y^{y^{x-1}} .$