such that for all 
,![\[xf(x+f(y))=(y-x)f(f(x)).\]](http://latex.artofproblemsolving.com/e/4/a/e4a3bbb8b91d2aa62d699c24df342fa59be71915.png)
be the number of all positive divisors of 
.
of real numbers that satisfy the system of equations![\[ \begin{cases}x^3 = 3x-12y+50, \\ y^3 = 12y+3z-2, \\ z^3 = 27z + 27x. \end{cases}\]](http://latex.artofproblemsolving.com/d/4/d/d4d6c485b4653577e732ff235448bb779914a576.png)
be an acute triangle with orthocenter 
,
be a point on the side 
,
and 
and 
are the feet of the altitudes from 
is the circumcircle of 
,
be the point on 
is a diameter of 
the circumcircle of triangle 
,
be the point such that 
is a diameter of 
and 
Y by Adventure10, Mango247
In the plane a point 
is and a sequence of points 
are given. The distances 
are 
Let 
satisfies 
Suppose that for every 
the distance from the point 
to any other point of the sequence is 
Determine the exponent 
, as large as possible such that for some 
independent of
![\[r_n \geq Cn^{\beta}, n = 1,2, \ldots\]](//latex.artofproblemsolving.com/f/7/f/f7f319a7b43f4cec591a9d210f64de2aad799114.png)
is and a sequence of points 
are given. The distances 
are 
Let 
satisfies 
Suppose that for every 
the distance from the point 
to any other point of the sequence is 
Determine the exponent 
, as large as possible such that for some 
independent of ![\[r_n \geq Cn^{\beta}, n = 1,2, \ldots\]](http://latex.artofproblemsolving.com/f/7/f/f7f319a7b43f4cec591a9d210f64de2aad799114.png)
for idiotic reasons. However, I compared the problem to the statement in the IMO Compendium, and there are two differences:
.
"
meant instead?)
