number 
.
and 
need not be different.
and 
Find the minimum 
satisfying![\[f(x+f(y))=x+f(f(y))\]](http://latex.artofproblemsolving.com/8/d/a/8daa6bcb3155768ee1ba4bd22beb53a85a6e3923.png)
and 
,
.
be positive odd integers. Prove that 
doesn't divide 
.
Y by Adventure10, Mango247
Let 
be a convex quadrilateral. 
and 
meet at 
, with 
. Let 
, and 
be the midpoints of 
and 
respectively. Find the greatest positive real number 
for which
![\[EG+3HF\ge kd+(1-k)s \]](//latex.artofproblemsolving.com/b/1/b/b1b8edd269a3ae32f89f4db32def60c057167005.png)
where
is the semi-perimeter of the quadrilateral and 
is the sum of the lengths of its diagonals. When does the equality hold?
be a convex quadrilateral. 
and 
meet at 
, with 
. Let 
, and 
be the midpoints of 
and 
respectively. Find the greatest positive real number 
for which![\[EG+3HF\ge kd+(1-k)s \]](http://latex.artofproblemsolving.com/b/1/b/b1b8edd269a3ae32f89f4db32def60c057167005.png)
where
is the semi-perimeter of the quadrilateral and 
is the sum of the lengths of its diagonals. When does the equality hold?
