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, Mango247
Let 
be a triangle and 
a point inside the triangle such that the centers 
and 
of the circumcircles 
and 
of triangles 
and 
, respectively, lie outside the triangle . In addition, we assume that the three points 
and are collinear as well as the three points 
and . The line through parallel to side 
intersects circles and in points 
and 
, respectively, where 
. Show that 
.
(Proposed by Walther Janous)
be a triangle and 
a point inside the triangle such that the centers 
and 
of the circumcircles 
and 
of triangles 
and 
, respectively, lie outside the triangle . In addition, we assume that the three points 
and are collinear as well as the three points 
and . The line through parallel to side 
intersects circles and in points 
and 
, respectively, where 
. Show that 
.(Proposed by Walther Janous)
.
and 
.
and 
are collinear, 
,
.
,
and thus 
where 
is the incenter of 
.
and similarly 
.
.
,
.
and 
collinear. Angle chasing, we see
,
,
