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. 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?
be a positive integer. Paul has a 
rectangular strip consisting of 
unit squares, where the 
square is labelled with 
for all 
. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then translate (without rotating or flipping) the pieces to obtain an 
square satisfying the following property: if the unit square in the row and 
column is labelled with 
, then 
is divisible by 
.
+2 w 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 three rays, and 
a point inside the trihedron 
. Consider all planes passing through and cutting at points 
, respectively. How is the plane to be placed in order to yield a tetrahedron 
with minimal perimeter ?
