be a positive integer . Prove that there exists infinitely many pairs of positive integers 
such that 
and :
such that 
and 
are coprime if and only if 
?
Y by Adventure10, Mango247, and 1 other user
Let 
be an scalene triangle and 
and 
its incenter, ortocenter respectively.
The incircle touchs
, 
and 
at 
an 
. 
and 
intersects at 
while 
and intersets at 
.
Shows that
cannot be paralel to 
.
PS1: The original problem without the adaptation apeared at the Brazilian Olympic Revenge 2011 but it was incorrect.
PS2:The Brazilian Olympic Revenge is a competition for teachers, and the problems are created by the students.
Sorry if I had some English mistakes here.
be an scalene triangle and 
and 
its incenter, ortocenter respectively.The incircle touchs
, 
and 
at 
an 
. 
and 
intersects at 
while 
and intersets at 
.Shows that
cannot be paralel to 
.PS1: The original problem without the adaptation apeared at the Brazilian Olympic Revenge 2011 but it was incorrect.
PS2:The Brazilian Olympic Revenge is a competition for teachers, and the problems are created by the students.
Sorry if I had some English mistakes here.
This post has been edited 1 time. Last edited by andrembcosta, Mar 27, 2012, 2:02 AM
and 
.
are the orthocenters of 
and 
,
are collinear on the radical axis of these two circles. Suppose, 
are the circumcenters of 
and 
.
and 
are anti-parallel wrt 
.
.
.
.
passes through the circumcenter of 
and 
.
is the Simson's line of 
wrt 
.
lies on 
.
.
,
.
can't be perpendicular to 
and 
meet the opposite sides at 
and 
,
be the midpoint 
Lines 
and 
meet at 
and lines 
meet at 
.
cannot be parallel to 
.
and
if you sum the equities: 
,
,
