  In this paper, we construct a new class of solutions for five dimensional
third order quasi-topological black holes coupled to a power-law Maxwell
nonlinear electrodynamics. To have real solutions, we should establish
condition $\mu<-\frac{\lambda^2}{3}$ and to have finite solutions at infinity,
the parameter of power-law Maxwell theory "s" should obey $\frac{1}{2}<s\leq
2$. Power-law Maxwell lagrangian is successful to set conformal invariance in
higher dimensions. Also, this theory can reduce the divergence of the
electrical field at the origin that is caused in linear Maxwell theory. As the
value of parameter "s" increases, this divergence reduces more. In
asymptotically anti-de sitter spacetimes, these obtained solutions lead to a
black hole with two horizons for small values of $s$ and $q$. Also, solutions
for $s=2$ have different behaviors with respect to the ones for other values of
$s$. For negative and small values of parameter $\mu$, these solutions can
describe a black hole with two horizons. An other important tip is that this
black hole has thermal stability just for anti-de sitter solutions if its
temperature is positive.
