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
, angle 
is right. On the leg , the point 
is chosen so that 
, and on point 
, point 
is chosen so that 
. Find the acute angle between the lines 
and 
.
, angle is right. On the leg , the point is chosen so that , and on point , point is chosen so that . Find the acute angle between the lines and .
. Prove that
