such that![\[ x f(x + xy) = x f(x) + f \left( x^2 \right) f(y) \quad \forall x,y \in \mathbb{R}.\]](http://latex.artofproblemsolving.com/b/0/f/b0f1a1d3b9de5d23f1b5a2e9be3b88d8bd575883.png)
be points on the sides 
respectively of a triangle 
.
and 
meet at 
and 
,
and 
,
,
.
to the area of 
.
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
be positive integers. Prove that it is impossible to have all of the three numbers 
to be perfect squares.
be a metric space and 
be a function such that 
.
Prove that for all 
,
exists, where 
is 
.
Prove that the amount of the limit does not depend on choosing 
.
![\[\iint_{R} \sin(xy) \,dx\,dy, \quad R = \left[0, \frac{\pi}{2}\right] \times \left[0, \frac{\pi}{2}\right]\]](http://latex.artofproblemsolving.com/a/2/0/a205868dd99f4794de53ad1db50f34b17c70a923.png)
![$\{x_n\} \subseteq [0,1]$](http://latex.artofproblemsolving.com/8/e/4/8e4206325387485d73765c9d7070ecb1ce18ff60.png)
,
,
,
and 
could be chosen to satisfy 
and 
.
be external direct sum of vector spaces 
and 
over a field 
.
and 
and 
is subspaces.
.
be the external direct sum of 
Then 
is
and whose roots are all real.
.
with a finitely generated k-algebra with one generator such that make 
Y by kiyoras_2001
Kuba has two finite families 
of convex polygons (in the plane). It turns out that every point of the plane lies in the same number of elements of 
as elements of 
. Prove that 
.
\textit{Note:} We treat segments and points as degenerate convex polygons, and they can be elements of or .
of convex polygons (in the plane). It turns out that every point of the plane lies in the same number of elements of 
as elements of 
. Prove that 
.\textit{Note:} We treat segments and points as degenerate convex polygons, and they can be elements of
or .
