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.
satisfy the equality, 
prove that the triangle must have a right angle.
be a cyclic quadrilateral and let the intersection point of lines 
and 
be 
. Let the points 
and 
be arbitrary points on sides and respectively, which satisfy the conditions
Prove that 
.
square is filled with numbers 
.The numbers inside four 
squares is summed,and arranged in an increasing order. Is it possible to obtain the following sequences as a result of this operation?
satisfying 
, find the biggest value the following equation could acquire:
![\[\frac{x^3+1}{(y-1)(z+1)}+\frac{y^3+1}{(z-1)(x+1)}+\frac{z^3+1}{(x-1)(y+1)}\]](http://latex.artofproblemsolving.com/5/c/6/5c6894b617b990ec69c62ed1cf7d062d4660af53.png)
where 
are reals.
+1 w
so that for any 
, we have ![\[ f(xf(x^2)+yf(x))=xf(y+xf(x)). \]](http://latex.artofproblemsolving.com/1/d/4/1d4fbf3f80306aaae5cfc11cc0e4be7ea41491d7.png)
+1 w
such that a positive integer power of 
equals to a positive integer power of 
.
positive real numbers satisfying 
,show that:![$\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9} \geq \frac{4}{3}$](http://latex.artofproblemsolving.com/c/b/3/cb39e7f838c99ed8a209c6844f3d2810838d1dfd.png)
.
, where 
and 
. Prove that 
couldn't be expressed as product of two non-constant polynomials with integer coefficients.
be positive real numbers .Prove that
the rows of the grid, from top to bottom, respectively 
the columns of the grid, from left to right. The row 
and the column 
split the initial grid into 
smaller 
grids (excluding and ).
grid contains totally 
cells: 
black cells and 
white cells. grid with only black cells.
is a white cell, any grid of dimension 
contains this white cell. is 
.
. Prove that
square with all unit square painted black.
