[/quote]
,
with![\[2x^3zf(z)+yf(y) \ge 3yz^2f(x)\]](http://latex.artofproblemsolving.com/a/b/f/abf10d7e679f6f04db556424a0a0eda26f371fd4.png)
for all 
.
be positive real numbers such that : 
. Prove that : 
such that ![\[f(f(x)) + xf(xy) = x + f(y)\]](http://latex.artofproblemsolving.com/a/6/f/a6f84dd53eec985b90bad3df6580eeaf29d73195.png)
for all positive real numbers 
and 
.
.
points are collinear in Euclidean plane, such that Shining cannot find a group of three points such that their circumcircle contains none of the other remaining points. Is he always able to do so?
?
its incenter. The lines 
and 
intersect the circumcircle of 
and 
,
and 
be the circumferences of diameters 
and 
,
at 
at 
.
,
and 
.
is the intersection point of the common external tangents of a random pair of these incircles and Y is the intersection point of common external tangents of some other pair of these three incircles, then 
passes through either 
or 
.
grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning strategy.
and 
be the circumcenter and orthocenter of a scalene triangle 
be the intersection point of the lines 
and 
.
meets the side 
and 
with the circumcircle of 
,
and 
and 
the perpendiculars in 
respectively, where 
,
and 
.
be the circumcircles of triangles 
.
and 
such that 
and 
.
is a perfect square.
Y by Mango247
In triangle 
, let 
and 
be the feet of the perpendiculars from 
onto the external and internal bisectors of 
, respectively; and let 
and 
be the feet of the perpendiculars from onto the internal and external bisectors of 
, respectively. If 
and 
, what is the area of triangle ?
, let 
and 
be the feet of the perpendiculars from 
onto the external and internal bisectors of 
, respectively; and let 
and 
be the feet of the perpendiculars from onto the internal and external bisectors of 
, respectively. If 
and 
, what is the area of triangle ?
and 
are rectangles yields 
,
.
and 
are the reflections of 
Hence area 
.
