be a sequence of positive real numbers, and 
be a positive integer, such that![\[a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.\]](http://latex.artofproblemsolving.com/f/1/2/f12c9c5e0f154549118a33a2f359329333f432a7.png)
and 
,![\[a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.\]](http://latex.artofproblemsolving.com/b/0/b/b0b2bd4507621302eeee27ae50ed9c0afd72c6ef.png)
and 
on a given circle meet at point 
be a point on segment 
and let 
meet the circle again at 
.
parallel to 
intersects 
at 
.
is parallel to 
.
,
be positive real numbers, such that 
.![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
of positive integers such that ![\[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \]](http://latex.artofproblemsolving.com/3/5/3/353bbb15bac205053c26208af8cac7c32a296f55.png)
Proposed by Gabriel Chicas Reyes, El Salvador
Y by Adventure10, Mango247
Let 
be a triangle such that it's circumcircle radius is equal to the radius of outer inscribed circle with respect to 
.
Suppose that the outer inscribed circle with respect to touches 
at 
.
Prove that
(Center of circumcircle) is the orthocenter of 
.
be a triangle such that it's circumcircle radius is equal to the radius of outer inscribed circle with respect to 
.Suppose that the outer inscribed circle with respect to
touches 
at 
.Prove that
(Center of circumcircle) is the orthocenter of 
.This post has been edited 1 time. Last edited by Nima Ahmadi Pour, Apr 19, 2006, 9:57 AM
at 
.
meet the circumcircle of 
at 
.
, thus 
.
and 
we conclude that 
,
.
...
of the vertex ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
.
be the 
meet 
again at points 
,
.
are parallel to each other and the two triangles have opposite orientations.
such that reflection in 
)
.
are corresponding cirucmenters of 
,
.
