has an inscribed circle with center 
.
and 
be the incenters of the triangles 
and 
,
and 
meet at 
,
and 
meet at 
.
.
such that for all integers 
,
with
be the incircle of an arbitrary triangle 
with tangency points 
on 
and 
respectively. 
intersects 
.
and 
meet 
and 
.
and 
are concurrent.
chessboard into 
non-overlapping rectangles subject to the following conditions:
is the number of white squares in the 
-
.
.
Y by
I'm reading Pang Cheng Wu Functional Equation handout. I stopped at one problem, which is:
Find all functions
such that for all reals 
, we have
![\[f(x)^2+f(y)^2+f((x+y)^2)=2x^2+2y^2+f(xf(y)+y(f(x))\]](//latex.artofproblemsolving.com/6/b/9/6b93f24d3061853a70965c4b57d3565d6646702f.png)
Shortly, if we define 
, the statement becomes:
![\[g(x+y)+g(x-y)=2g(x)+2g(y)\]](//latex.artofproblemsolving.com/3/b/4/3b4840c15e0cb0c3f61cf14b166fcd1d73a5369e.png)
Where 
(which makes 
bounded on some interval). I have proved that 
. From here I have to prove that 
, and I want to use the fact that is bounded on some interval, but the form of 
is not the form of regular Cauchy Equation. Does anyone know how to transform the form become Cauchy? So we can use the information of is bounded?
Find all functions
such that for all reals 
, we have![\[f(x)^2+f(y)^2+f((x+y)^2)=2x^2+2y^2+f(xf(y)+y(f(x))\]](http://latex.artofproblemsolving.com/6/b/9/6b93f24d3061853a70965c4b57d3565d6646702f.png)
Shortly, if we define 
, the statement becomes:![\[g(x+y)+g(x-y)=2g(x)+2g(y)\]](http://latex.artofproblemsolving.com/3/b/4/3b4840c15e0cb0c3f61cf14b166fcd1d73a5369e.png)
Where 
(which makes 
bounded on some interval). I have proved that 
. From here I have to prove that 
, and I want to use the fact that is bounded on some interval, but the form of 
is not the form of regular Cauchy Equation. Does anyone know how to transform the form become Cauchy? So we can use the information of is bounded?
