AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
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 
.
. Prajit then tries to place n points such that no four points are concyclic and no 
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 again at 
and 
, respectively. Let 
and 
be the circumferences of diameters 
and 
, respectively. The circle intersects 
at 
and 
, and the circle intersects 
at 
and 
. Show that , , and are concyclic.
be a point inside 
. It is known that there exists a line tangent to the incircles of 
, 
and 
. Prove that if 
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 
.
be positive real numbers such that : . Prove that : 
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 finite set of cells in a square grid. On each cell of we place a grasshopper. Each grasshopper can face up, down, left or right. A grasshopper arrangement is Asturian if, when each grasshopper moves one cell forward in the direction in which it faces, each cell of still contains one grasshopper., the number of Asturian arrangements is a perfect square. is the following set: an acute triangle inscribed in a circle with its center . Let the lines 
and 
the perpendiculars in to 
respectively, where 
, 
and 
. Let 
be the circumcircles of triangles 
.
and have the same centroid.
such that 
and 
.
is a perfect square.
and it's orthocenter . Point is arbitrary chosen on the side 
. Let and be reflections of point over sides 
. It is given that points 
are collinear. Prove that is right angled.
?
