and 
Prove that
.![\[\angle PAD:\angle PBA:\angle DPA=1:2:3=\angle CBP:\angle BAP:\angle BPC\]](http://latex.artofproblemsolving.com/9/3/8/9388a91325842f1ec67a4693f0f10af46ed66c6d.png)
Prove that the following three lines meet in a point: the internal bisectors of angles 
and 
and the perpendicular bisector of segment 
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)
be the set of positive real numbers. Determine all functions 
and 
![\[f(x+f(xy))+y=f(x)f(y)+1\]](http://latex.artofproblemsolving.com/c/6/a/c6a968a17f6fd5fd637c1ee5d855bdff11e5423a.png)
Y by Adventure10
Let 
a triangle, and let 
and 
the excircles relatives to the sides 
and 
, respectively. touches at 
and the extensions of the sides 
at the points 
, respectively. touches at 
and the extensions of the sides 
at the points 
, respectively. The line 
intersects line 
at the point 
, 
intersects 
at 
. Prove that 
are collinear.
a triangle, and let 
and 
the excircles relatives to the sides 
and 
, respectively. touches at 
and the extensions of the sides 
at the points 
, respectively. touches at 
and the extensions of the sides 
at the points 
, respectively. The line 
intersects line 
at the point 
, 
intersects 
at 
. Prove that 
are collinear.
meets 
at 
and 
meets 
at 
.
and 
are both tangent to 
.If we define 
and 
, they share a ray 
and also both equal to 
, so their 
other intersections namely 
are co-linear .
as intersection of 
with 
then 
)
