AoPS Academy
. Let 
be the circumcenter of this triangle . Let 
, 
, 
be the feet of the altitudes of triangle from the vertices 
, 
, 
, respectively. Denote by 
, 
, 
the midpoints of these altitudes 
, 
, 
, respectively. The incircle of triangle has center 
and touches the sides 
, 
, 
at the points 
, 
, 
, respectively. Prove that the four lines 
, 
, 
and 
are concurrent. (When the point concides with , we consider the line as an arbitrary line passing through .)
be an acute triangle and let 
be an interior point for it such that 
. Prove that
When does equality hold?
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 . Consider a 
grid, a set 
of squares is called connected if for any points 
in , there exists an integer 
and 
squares 
in such that and shares a common side (
). satisfying that however the squares are colored black or white, there always exists a connected set for which the absolute value of the difference between the number of black and white squares is at least .
we can appeal to the result above.
and 
we have:
are similar (
)
, such that 
(since 
is inside the segment 
)
is the midpoint of 
and 
intersects 
at its midpoint . we find that the line 
is also median for 
.
and 
be , and the intersection of 
and be . Let 
, use area ratios:![\[\frac{AM}{BM}=\frac{\triangle{AOE}}{\triangle{AOB}}\cdot \frac{\triangle{AOB}}{\triangle{EOB}}=\frac{EC}{CB} \cdot \frac{ED}{DC} = 1\]](http://latex.artofproblemsolving.com/6/f/c/6fc7dbde616a493fec59f9010feb4785db208a3e.png)
be a quadrilateral with 
,
.
and 
passes through the midpoints of 
.
be three collinear points and a point 
.
intersects the lines 
at the points 
respectively.
intersects the segment 
on its midpoint 
passes through 
be the midpoints of 
respectively.
passes through 
passes through