[/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 
be the circumferences of diameters 
and 
,
at 
,
at 
and 
.
.
,
and 
.
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 
.
is a perfect square.
Y by Adventure10, Mango247
In a triangle 
, 
. 
is the altitude of the triangle. 
is on such that 
( is in the triangle). and 
cut the circumcircle of in 
and 
respectively. if tangent to circumcircle of at cuts 
at 
, prove that ,
and 
are collinear.(25 points)
the exam time was 4 hours and 30 minutes.
, 
. 
is the altitude of the triangle. 
is on such that 
( is in the triangle). and 
cut the circumcircle of in 
and 
respectively. if tangent to circumcircle of at cuts 
at 
, prove that ,
and 
are collinear.(25 points)the exam time was 4 hours and 30 minutes.
This post has been edited 1 time. Last edited by goodar2006, Aug 8, 2010, 11:12 AM
be the reflection of 
the intersection point of 
and 
.
be the point where 
be the point where the tangent of the circumcircle of 
.
and 
we have![\[tan \angle AP_1O= \frac{OA}{OP_1}\]](http://latex.artofproblemsolving.com/7/1/d/71db19d42bba03763660c33ee11249a729cabf12.png)
![\[sin\angle OP_2M= \frac{OM}{OP_2}\]](http://latex.artofproblemsolving.com/e/a/7/ea780a1ca8febc310d392b98c7f329f7f6b7694b.png)
,
we are done. But it´s straightforward to check that 
and 
.
we are done....
and 
are equal, we conclude 
,
