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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 
, respectively. Let 
be the intersection point of the lines 
and 
. Suppose the line 
meets the side at 
. Let 
and 
be the second intersection points of the circumcircles of 
and 
with the circumcircle of 
, respectively. Show that the four points 
and lie on a circle.
be a parallelogram with 
. Inside the parallelogram there is a point 
such that the triangle 
is equilateral and 
. Let K be the midpoint of side 
. Calculate the measure of the angle 
.
be a parallelogram with . Inside the parallelogram there is a point such that the triangle is equilateral and . Let K be the midpoint of side . Calculate the measure of the angle .
to be the symmetric point to wrt to 
Then triangle 
is equilateral, triangle 
is isosceles with base angle 
and triangle 
is isosceles and right. Therefore, 
Let 
intersect at 
But then 
and therefore triangle 
is isosceles and right; thus 
But triangles 
and 
are congruent because 
and 
Therefore 
From the above we get 
and therefore 
is the midpoint of and thus 
Finally, 
because 
is a rhombus.
we get 
where 
Similarly from triangle 
we get 
where 
From 
we get 
and substituting it in 
we get 
and therefore 
