[/quote]
,
Prove that
be integers, let 
be a set with 
elements, and let 
be pairwise distinct non-empty, not necessary disjoint subset of 
is called nice if there exists an index 
such that ![\[\sum_{x \in X_k} f(x)>\sum_{x \in X_i} f(x) \quad \text{for all } i \ne k.\]](http://latex.artofproblemsolving.com/3/d/a/3da5ad1bcade98ee08caa35763de226027bcb2c6.png)
Prove that the number of nice functions is at least 
.
.![\[\sqrt{a+\frac{9}{b+2c}}+\sqrt{b+\frac{9}{c+2a}}+\sqrt{c+\frac{9}{a+2b}}+\frac{2(ab+bc+ca)}{9}\ge\frac{20}{3}\]](http://latex.artofproblemsolving.com/b/3/2/b32e0bcba967ca847e878ac186c3168cd6c5fc66.png)
be positive real numbers such that 
Find the minimum value of 
Prove that
Prove that
,
,
are fixed on a line in this order. Let 
be a circle passing through 
.
the intersection of the tangents to 
at 
.
and the line 
Prove that
Where 
be a triangle with circumcentre 
.
and 
are interior points of the sides 
and 
respectively. Let 
and 
be the midpoints of the segments 
and 
.
be the circle passing through 
satisfying
Prove
.
has at most 
and 
Prove that
such that:
for all 
be an acute-angled scalene triangle with circumcenter 
.
,
,
the midpoints of sides 
,
,
,
be the circle passing through 
and tangent to 
at 
at points 
and 
,
be the midpoint of segment 
,
be the intersection of lines 
.
and that triangle 
is isosceles.
Y by
In 
of the vertices of the regular polygon 
we write the number 
and in the remaining ones we write the number 
Let 
be the number written on the vertex 
A vertex is good if ![\[x_i+x_{i+1}+\cdots+x_j>0\quad\text{and}\quad x_i+x_{i-1}+\cdots+x_k>0,\]](//latex.artofproblemsolving.com/b/f/4/bf4cf984af7f310c1eb90b6b37126bfca9181305.png)
for any integers 
and 
such that 
Note that the indices are taken modulo 
Determine the greatest possible value of such that, regardless of numbering, there always exists a good vertex.
of the vertices of the regular polygon 
we write the number 
and in the remaining ones we write the number 
Let 
be the number written on the vertex 
A vertex is good if ![\[x_i+x_{i+1}+\cdots+x_j>0\quad\text{and}\quad x_i+x_{i-1}+\cdots+x_k>0,\]](http://latex.artofproblemsolving.com/b/f/4/bf4cf984af7f310c1eb90b6b37126bfca9181305.png)
for any integers 
and 
such that 
Note that the indices are taken modulo 
Determine the greatest possible value of such that, regardless of numbering, there always exists a good vertex.
?
.
,
is a counterexample. Now we show 
values of 
such that the first condition is satisfied for all 
.
denote 
(in particular 
)
.
,
.
for all 
.
be the largest 
satisfies the first condition. Note that![\[x_{i_k+1}+x_{i_k+2}+\cdots+x_j=s_j-s_{i_k}=s_j-k\]](http://latex.artofproblemsolving.com/b/d/0/bd0430e61dce446a7095aa85eca4d73bc0a790d9.png)
however, if there exists 
such that 
,
such that 
,
as 
.
written on it, there must exist a vertex that satisfies both conditions as there are only 
