[/quote]
,
be a trapezoid. By 
. Let 
and 
.
is cyclic.
be a triangle such that 
and 
.
be the foot of the interior bissector of 
(its intersection with 
)
calculate the sidelength 
is the remainder obtained by dividing it by 
and 
and it's digits are all different and nonzero. How many piolas are there between 
and 
?
be the midpoint of the arc 
of its circumcircle which does not contain 
or 
and 
meet at 
and the lines 
and 
meet at 
.
and 
and orthocenter 
.
denote the circumcircle of triangle 
the midpoint of 
.
and 
,
.
and 
similarly.
.
such that 
be the midpoint of 
and let 
and the circumcircle of 
.
passes through a fixed point.
and 
intersect each other at 
and 
.
and 
respectively. Let 
and 
be the reflection of 
intersect circles 
and 
(both distinct from 
is the second tangent to 
is the midpoint of the arc 
,
,
,
be the 
intersects 
.
is perpendicular to 
,
is the incenter of 
and the circle 
of radius 
Let 
be the midpoint of 
The line 
intersects 
Let 
Let 
and 
and excentres 
and 
,
and 
and 
and 
respectively. Prove that the circles 
and 
have a common point other than 
and 
.
and 
,
intersect 
.
be an inscribed quadrilateral. Circles of diameters 
and 
intersect at points 
and 
,
and 
intersect at points 
and 
.
and 
intersect in two points 
and 
.
and 
are concurrent.
Y by
I was practicing problems and came across one as such:
"Find all integers
such that 
is a positive integral power of a prime positive integer."
I mean after factoring I don't really know where to go...
A hint would be appreciated, and if you want to solve it, please hide your solutions!
Thanks
"Find all integers
such that 
is a positive integral power of a prime positive integer."I mean after factoring I don't really know where to go...
A hint would be appreciated, and if you want to solve it, please hide your solutions!
Thanks
,
is prime and 
is a positive integer.
.
o
or 
or 
.
,
.
.
or 
or 
