  In this paper, we study a transient spatially inhomogeneous random walk with
asymptotically zero drifts on the lattice of the positive half line. We give
criteria for the finiteness of the number of points having exactly the same
local time and/or upcrossing time and weak cutpoints (a point $x$ is called a
weak cutpoint if the walk never returns to $x-1$ after its first upcrossing
from $x$ to $x+1$). In addition, for the walk with some special local drifts,
we also give the order of the expected number of these points in $[1,n].$
Finally, we show that, when properly scaled, the number of these points in
$[1,n]$ converges in distribution to a random variable with the standard
exponential distribution. Our results answer three conjectures related to the
local time, the upcrossing time, and the weak cutpoints proposed by E. Cs\'aki,
A. F\"oldes, P. R\'ev\'esz [J. Theoret. Probab. 23 (2) (2010) 624-638].
