be a cyclic quadrilateral with 
and let 
and 
be the feet of altitudes from 
to the lines 
and 
,
bisects 
.
is strictly increasing : 
be a real numbers such that 
P
Where 
,
be the set of positive integers that do not exceed 
as the smallest positive integer that allows 
satisfying 
,
,
.
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 
