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.
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 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.
. Adamu wins if the polynomial obtained has an integer root; otherwise, Afaafa wins. Afaafa plays first if is odd; otherwise Adamu plays first. Prove that: is odd. is even.
that satisfies![\[ f(x+yf(x))f(y)=f(x)(1+yf(y+1)) \]](http://latex.artofproblemsolving.com/f/7/a/f7a138fd2dc76f034525491a10f62f1e83de294d.png)
for all 
., 
be positive integers such that 
is a multiple of 
for all positive integers . Prove that 
.
with 
is called historic if 
. Show that the set of all nonnegative integers can be written as the union of pairwise disjoint historic sets.
satisfying 
for all pairs of positive reals 
and 
. Here, 
denotes the set of all positive reals. be positive real numbers such that : . Prove that : 
be a convex hexagon inscribed in a circle . Prove that the diagonals 
and 
intersect at one point if and only if 
says that 
which is equivalent by sine rule to
QED
)
,
)
.
as the second intersection of line 
with circumscirbed circle. Because 
lies on arc 
that doesn't contain point 
.
Hence 
We will prove that function 
is strictly increasing when we move point 
to point 
.
In other words we need to prove the property for 
where 
.
or 
Hence 
QED.