[/quote]
,
.
Where 
,
,
be real numbers, such that 
and 
.
.
,
,
around a circle so that the sum of every 
.
Let 
.
be a positive integer such that ![\[
1 + \sum_{i=1}^{n-1} i^{n-1}.
\]](http://latex.artofproblemsolving.com/e/2/d/e2daf7800da3f984cd761ed1fecf10083444db25.png)
Prove that 
amount of people stand on coordinates 
where 
.
.
got 
people's water cup is empty. Every minute, two neighbouring people are chosen that does not have the same amount of water in their water cups, and they equalize the amount of water in their water cups.
litres of water.
,
,
.![\[\sum_{i=1}^{n}\frac{1}{1+x_{i}}\cdot \sum_{i=1}^{n}x_{i}\leq \sum_{i=1}^{n}\frac{x_{i}^{k+1}}{1+x_{i}}\cdot \sum_{i=1}^{n}\frac{1}{x_{i}^{k}}\]](http://latex.artofproblemsolving.com/2/6/4/264d110e2a40c7f787c650aa8b8dc30af5d28263.png)
be an integer. You have two 
boards. Each board contains the numbers 
and 
,
row and 
column of the first board is different from the number that is in the 
be a convex pentagon such that![\[ \angle BAC = \angle CAD = \angle DAE\qquad \text{and}\qquad \angle ABC = \angle ACD = \angle ADE.
\]](http://latex.artofproblemsolving.com/d/3/d/d3d9a82f4318190298a8f008d417a364e03f1fca.png)
The diagonals 
and 
meet at 
.
bisects the side 
.
,
,
is the incenter, 
is the intersection of 
-
.
lies on the external angle bisector of 
such that 
and 
.
lies on 
.
is tangent to 
and 
at 
,
is tangent to 
at 
and both circles pass through the inside of triangle 
is the Midpoint od the arc 
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)
meet the circle again at 
parallel to 
.
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
Two circles 
and 
with equal radii intersect at P and Q. Points B and C are located on the circles and so that they are inside the circles and , respectively. Also, points X and Y distinct from P are located on and , respectively, so that:
The intersection point of the circumcircles of triangles XPC and YPB is called S. Prove that BC, XY and QS are concurrent.
Thanks.
and 
with equal radii intersect at P and Q. Points B and C are located on the circles and so that they are inside the circles and , respectively. Also, points X and Y distinct from P are located on and , respectively, so that:
The intersection point of the circumcircles of triangles XPC and YPB is called S. Prove that BC, XY and QS are concurrent.Thanks.
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