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,
,
,
is __
satisfying 
be a triangle and 
denote its inradius and ex-radii opposite to the vertices 
respectively. If 
and 
,
is obtuse
is acute
,
is semi perimeter
,![$ a,b\in[\frac{1}{2},1] $](http://latex.artofproblemsolving.com/0/d/2/0d24f742e1f7db553b3a8795c061c796f07a897f.png)
.
Let ![$ a,b\in[\frac{1}{2},2] $](http://latex.artofproblemsolving.com/a/8/d/a8d4c8205bac0e34129f582a8f255355eda12e0b.png)
.
satisfy that: 
and 
.
, such that 
Prove that :
equality cases : ?
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)
and 
such that![\[
1^n + 2^n + 3^n + \cdots + n^n = x!
\]](http://latex.artofproblemsolving.com/6/4/b/64b5f5919fd3710cc15e96724918eae216c8e4f0.png)
.
meet at point 
.
on line 
which is not on the segment 
be 
.
meet 
respectively. Let 
lie on a circle.
red cards, 
white cards, and 
and 
the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
Y by Adventure10, Mango247
Two different points 
and 
and a circle 
that passes through and are given. 
is a variable point on (different from and ). 
is a point such that 
is the bisector of the angle 
( lies outside of ) and 
. Find the geometrical locus of .
and 
and a circle 
that passes through and are given. 
is a variable point on (different from and ). 
is a point such that 
is the bisector of the angle 
( lies outside of ) and 
. Find the geometrical locus of .
on half of arc 
.
ok.
and 
be on 
and 
beyond 
such that 
and 
on circle 
without 
of the circles 
and 
and point 
which is the intersection of 
and 
.
,
be the midpoint of 
.
such that 
.
are constants, 
,
.
.
.
,
.
and 
.
not containing 
of 
not containing 
?
and 
and 
.
