[/quote]
,
with 
.![\[
\sqrt{a + (b - c)^2} + \sqrt{b + (c - a)^2} + \sqrt{c + (a - b)^2} \geq \sqrt{3},
\]](http://latex.artofproblemsolving.com/c/b/5/cb551e6d6ce94902b7bcb78f6da210d6c27b96d5.png)
with equality if and only if 
.
be a triangle with orthocenter 
and let 
be the midpoint of 
Suppose that 
and 
are distinct points on the circle with diameter 
different from 
such that 
Prove that the orthocenter of 
lies on the circumcircle of 
.
.
Where 
,
,
be real numbers, such that 
and 
.
.
,
,
around a circle so that the sum of every 
.
Let 
.
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 
.
.
is tangent to 
and 
at 
,
is tangent to 
at 
and both circles pass through the inside of triangle 
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)
Y by kiyoras_2001
Kuba has two finite families 
of convex polygons (in the plane). It turns out that every point of the plane lies in the same number of elements of 
as elements of 
. Prove that 
.
\textit{Note:} We treat segments and points as degenerate convex polygons, and they can be elements of or .
of convex polygons (in the plane). It turns out that every point of the plane lies in the same number of elements of 
as elements of 
. Prove that 
.\textit{Note:} We treat segments and points as degenerate convex polygons, and they can be elements of
or .
