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 
.
that satisfy
for all integers 
and rational numbers 
.
such that 
are one of such 
be a scalene oblique triangle with circumcenter 
and orthocenter 
,
,
)
,
be the circumcevian triangles of 
be the pedal triangle of 
be the reflection in 
of 
.
,
cyclically.
be the reflection in 
.
,
cyclically.
,
be the circumcenters of 
,
,
)
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 
