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be an acute scalene triangle with circumcenter 
. Let 
be the foot of the altitude from 
to the side 
. The lines and 
intersect at 
. Let 
be the line through perpendicular to . The line intersects 
and 
at 
and 
, respectively. Denote by 
the circumcircle of triangle 
. Line 
intersects again at 
. and the circumcircles of triangles and 
have a common point.
cities in a country, where 
is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities and 
we define:
A 
between and as a sequence of distinct cities 
, 
, such that there are direct flights between 
and 
for every 
;
A 
between and as a path between and such that no other path between and has more cities;
A 
between and as a path between and such that no other path between and has fewer cities. and in the country, there exist a long path and a short path between them that have no cities in common (except and ). Let 
be the total number of pairs of cities in the country that are connected by direct flights. In terms of , find all possible values 
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle with circumcenter , centroid 
and orthocenter 
. Prove that 
is constant.
that satisfies:![\[
\sum_{i=1}^{n} a_{\left\lfloor \frac{n}{i} \right\rfloor} = n^k
\]](http://latex.artofproblemsolving.com/b/b/e/bbe7fe81e366fe390be5d17193d1581cc40dbece.png)
be a positive integer. Prove that for all integers 
, we have:![\[
\frac{c^{a_n} - c^{a_{n-1}}}{n} \in \mathbb{Z}
\]](http://latex.artofproblemsolving.com/0/4/4/04464c1a350e3f9a929ab51534018498680068fc.png)
and let the given incircle and excercle touch at 
interset in wehave to prove
that 
which is trivial to prove by the sine law in 
, the area of triangle BPQ is 
and the length of the segment PQ is 
.Prove that the distance between the centroid of triangle ABC and vertex B is 
and let 
and the excircle of 
corresponding to 