and 
be two circles with respective centres 
and 
intersecting in two distinct points 
and 
such that 
is an obtuse angle. Let the circumcircle of 
intersect 
and 
.
intersect 
; let the line 
intersect 
.
are concyclic.
from the set of positive integers to the set of positive integers such that, for all positive integers 
and 
,![\[ a, f(b) \text{ and } f(b + f(a) - 1).\]](http://latex.artofproblemsolving.com/4/a/b/4ab97da27e05d0f9e7762f74d0d0bd2dc8b9842f.png)
and 
.![$S=\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9}$](http://latex.artofproblemsolving.com/9/d/9/9d93ff9d9f7355e452f1007aeb87423e3491353a.png)
Y by
Let 
be two nonzero real numbers such that 
Consider the functions 
and 
satisfying the condition ![$$g(f(x+by))=a[f(x)+2yg(x)]+2xg(y)+bg(y)[y+g(y)],\quad\forall x,y\in\mathbb{R}.$$](//latex.artofproblemsolving.com/3/c/9/3c974cb76aae7f529dcd85202b0c7d441a07d002.png)
(a) Prove that there exists a real number 
such that 
(b) Find all functions 
and 
that satisfy the given condition.
be two nonzero real numbers such that 
Consider the functions 
and 
satisfying the condition ![$$g(f(x+by))=a[f(x)+2yg(x)]+2xg(y)+bg(y)[y+g(y)],\quad\forall x,y\in\mathbb{R}.$$](http://latex.artofproblemsolving.com/3/c/9/3c974cb76aae7f529dcd85202b0c7d441a07d002.png)
(a) Prove that there exists a real number 
such that 
(b) Find all functions 
and 
that satisfy the given condition.
