[/quote]
,
such that 
![\[
C(p-1, k-1) \equiv (-1)^{k-1} \pmod{p}
\]](http://latex.artofproblemsolving.com/4/0/7/4072235afbaa1f2d8d4c9ca044615a113a4306ce.png)
for 
,
is the binomial coefficient 
choose 
.
satisfy that: 
and 
.
be a sequence defined by 
.
if 
.
be a convex quadrilateral. Suppose that the circles with diameters 
and 
intersect at points 
and 
.
and 
.
,
,
be positive integers and 
be an odd prime such that:![\[
a^p \equiv 1 \pmod{p^n}.
\]](http://latex.artofproblemsolving.com/3/4/8/34815b8211ded2f37b0653525dc4b436356e6eee.png)
Prove that:![\[
a \equiv 1 \pmod{p^{n-1}}.
\]](http://latex.artofproblemsolving.com/0/7/a/07a4ed2ab3c94f41028080f7789cfb6e2973955c.png)
be a function such that for every regular 
-
we have 
.
for all reals 
.
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
be positive integers, where 
.
such that
.
.
is the sum of digits of 
taken in base 
)
with 
.
lie on 
respectively, such that 
.
to meet the circumcircle of 
respectively. Let the circumcircles of 
meet at points 
.
to 
at 
,
,![$[ABC]$](http://latex.artofproblemsolving.com/d/3/3/d33cc80fa8f093e155c5be46d2e5d9da3d7e1ef5.png)
.
Y by
Given an acute, non-isosceles triangle 
.
Let
and 
be points on sides 
and 
, respectively, such that 
.
Denote
and 
as the centers of the circumcircles of 
and 
, respectively.
The line
intersects and at two distinct points 
and 
.
Let
be the circumcenter of 
.
Given that all these points are distinct, prove that the lines
, 
, and 
are concurrent at a single point.
.Let
and 
be points on sides 
and 
, respectively, such that 
.Denote
and 
as the centers of the circumcircles of 
and 
, respectively.The line
intersects and at two distinct points 
and 
.Let
be the circumcenter of 
.Given that all these points are distinct, prove that the lines
, 
, and 
are concurrent at a single point.
