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with 
, the foot of altitudes from 
to the sides 
are 
, respectively. 
is the orthocenter. 
is the midpoint of segment 
. Lines 
and 
intersect at 
. Let the tangents drawn to circumcircle 
from 
and 
intersect at 
. Prove that 
are colinear
be an integer and let 
be positive integers. In a group of 
people, some games of chess are played. Two people can play each other at most once. Prove that it is possible for the following two conditions to hold at the same time:
.
with 
, there is someone who has played exactly 
games of chess.
be an acute triangle with orthocenter and circumcircle 
. Let be the midpoint of side . Point 
is chosen from the minor arc on 
such that 
. Let 
be a point on such that 
is perpendicular to 
, and 
be a point on line such that 
is perpendicular to . Lines 
and intersect at point 
, and point 
is the reflection point of with respect to .
.
inside a square 
we raise a segment 
perpendicular to the plane of the square. Show that the projections of on the planes 
, 
, 
and 
are coplanar if and only if ![$O\in [AC]\cup [BD]$](http://latex.artofproblemsolving.com/6/6/6/666df8f2afa87f15c79f9c96d9d1cea17a2a7af0.png)
.
