AoPS Academy
. The point 
is in the interior of . The following ratio equalities hold:![\[\angle PAD:\angle PBA:\angle DPA=1:2:3=\angle CBP:\angle BAP:\angle BPC\]](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 
such that for all positive real numbers 
and 
![\[f(x+f(xy))+y=f(x)f(y)+1\]](http://latex.artofproblemsolving.com/c/6/a/c6a968a17f6fd5fd637c1ee5d855bdff11e5423a.png)
be inscribed in a circle with diameter 
. Points 
and 
are symmetric to 
and 
with respect to and 
respectively. If the lines 
, intersect at and , 
intersect at 
, show that 
is perpendicular to .
be inscribed in a circle with diameter . Points and are symmetric to and with respect to and respectively. If the lines , intersect at and , intersect at , show that is perpendicular to .
are the incenter and excenter opposite to 
respectievly of 
. Now draw, 
, now we'll prove 
and 
are similar, but how? 
, and we denote 
where 
are colinear which also implies 
and 
Prove that
