be positive real numbers such that : 
. Prove that : 
with![\[2x^3zf(z)+yf(y) \ge 3yz^2f(x)\]](http://latex.artofproblemsolving.com/a/b/f/abf10d7e679f6f04db556424a0a0eda26f371fd4.png)
for all 
.
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 
,
and 
.
meets the side 
be the second intersection points of the circumcircles of 
and 
with the circumcircle of 
,
and 
Y by
Line 
intersects circle 
at points 
and 
. The perpendicular bisector of segment 
intersects at points 
and 
. Let 
be a point of which lies outside the circle . The parallels to 
and 
passing through intersect 
and at points 
and 
, respectively. Prove that 
and 
are perpendicular.
intersects circle 
at points 
and 
. The perpendicular bisector of segment 
intersects at points 
and 
. Let 
be a point of which lies outside the circle . The parallels to 
and 
passing through intersect 
and at points 
and 
, respectively. Prove that 
and 
are perpendicular.
