We prove some estimates for convex ancient solutions (the existence time forthe solution starts from $-\infty$) to the power-of-mean curvature flow, whenthe power is strictly greater than 1/2. As an application, we prove that in twodimension, the blow-down of the entire convex translating solution, namely$u_{h}=\frac{1}{h}u(h^{\frac{1}{1+\alpha}}x),$ locally uniformly converges to$\frac{1}{1+\alpha}|x|^{1+\alpha}$ as $h\rightarrow\infty$. Another applicationis that for generalized curve shortening flow (convex curve evolving in itsnormal direction with speed equal to a power of its curvature), if the convexcompact ancient solution sweeps $\textbf{R}^{2}$, it it has to be a shrinkingcircle. Otherwise the solution is defined in a strip region.
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