[/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?
?
be a triangle and 
its incenter. The lines 
and 
intersect the circumcircle of 
and 
,
and 
be the circumferences of diameters 
and 
,
,
at 
and 
.
.
,
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 
.
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.
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 
.
Y by
Different lines 
and 
intersect at point 
. Consider all possible segments 
of length 
, the ends of 
and 
of which lie on and , respectively, and denote by 
the point the intersection of the perpendiculars on the straight lines and , drawn from and , respectively. Find the locus of points .
and 
intersect at point 
. Consider all possible segments 
of length 
, the ends of 
and 
of which lie on and , respectively, and denote by 
the point the intersection of the perpendiculars on the straight lines and , drawn from and , respectively. Find the locus of points .This post has been edited 1 time. Last edited by parmenides51, Jul 5, 2020, 8:14 AM
.
which is constant thus all the points 
