  In this paper, we study a class of fully nonlinear contracting curvature
flows of closed, uniformly convex hypersurfaces in the Euclidean space $\mathbb
R^{n+1}$ with the normal speed $\Phi$ given by $r^\alpha F^\beta$ or $u^\alpha
F^\beta$, where $F$ is a monotone, symmetric, inverse-concave, homogeneous of
degree one function of the principal curvatures, $r$ is the distance from the
hypersurface to the origin and $u$ is the support function of hypersurface. If
$\alpha\geq \beta+1$ when $\Phi=r^\alpha F^\beta$ or $\alpha> \beta+1$ when
$\Phi=u^\alpha F^\beta$, we prove that the flow exists for all times and
converges to the origin. After proper rescaling, we prove that the normalized
flow converges exponentially in the $C^\infty$ topology to a sphere centered at
the origin. Furthermore, for special inverse concave curvature function
$F=K^{\frac{s}{n}}F_1^{1-s}(s\in(0, 1])$, where $K$ is Gauss curvature and
$F_1$ is inverse-concave, we obtain the asymptotic convergence for the flow
with $\Phi=u^\alpha F^\beta$ when $\alpha=\beta+1$. If $\alpha<\beta+1$, a
counterexample is given for the above convergence when speed equals to
$r^\alpha F^\beta$.
