and 
Prove 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 
.
satisfying![\[\begin{aligned}
\begin{cases}
a+b+c+d=0,\\
a^2+b^2+c^2+d^2=12,\\
abcd=-3.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/8/b/1/8b1923e0303c0aa79332ee4540ee1885a9cf3f08.png)
Y by sami1618
In triangle 
, let 
be the 
-excenter. Points 
and 
are placed on line 
such that 
is between and 
, and is between and . Moreover, and are the contact points of with the -excircle of triangles 
and 
, respectively. Let 
be the -excenter of triangle 
, and let 
be the reflection of the orthocenter of triangle with respect to its circumcenter. Prove that , , and are collinear.
Proposed by Ali Nazarboland
, let 
be the 
-excenter. Points 
and 
are placed on line 
such that 
is between and 
, and is between and . Moreover, and are the contact points of with the -excircle of triangles 
and 
, respectively. Let 
be the -excenter of triangle 
, and let 
be the reflection of the orthocenter of triangle with respect to its circumcenter. Prove that , , and are collinear.Proposed by Ali Nazarboland
This post has been edited 2 times. Last edited by jannatiar, Mar 4, 2025, 7:58 AM
be the ellipse passing through 
with foci 
and 
and let 
be the ellipse passing through 
and 
be a point the lies on both 
and 
.
be the circle centered at 
and define 
and 
similarly. Then, 
Assume that 
.
centered at 
is externally tangent to 
that is internally tangent to 
lies on 
,
and 
similarly. Then notice 
,
,
.
.
lies on 
and 
are tangent to 
gives that 
lies on 
lie on 
and 
are tangent to 
,
,
and hence the conjugate 
.
intersects circle 
at 
at 
which implies the existence of a circle tangent at 
using the Bogdanov's theorem as well. The fact that 
was just a fact I heard somewhere and it also appeared in Sharygin 2023 first round. I didn't prove it myself but there exist a lot of different methods to prove that like coordinate bash or even taking the configuration into the 3D space. The fact that 
could be easily proven by 2 point-DIT (having 
)
