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
Two circles 
and 
are tangent at 
. A common tangent, not through , touches at 
and at 
. Points 
and 
, on and respectively, are outside the triangle 
and are such that is on the line 
. Prove that 
is perpendicular to 
.
and 
are tangent at 
. A common tangent, not through , touches at 
and at 
. Points 
and 
, on and respectively, are outside the triangle 
and are such that is on the line 
. Prove that 
is perpendicular to 
.This post has been edited 1 time. Last edited by parmenides51, Feb 23, 2020, 11:23 PM
at 
.
,
.
as desired.
