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such that![\[(a^2 + b^2 + c^2)^2 = 3(a^3b + b^3c + c^3a) + 1. \]](http://latex.artofproblemsolving.com/9/e/4/9e400fd807a9535ede983b75dcc307d8b75a35b6.png)
so that 
is an integer.
, 
and 
be midpoints of sides 
and 
, respectively, and let 
be circumcenter of acute-angled triangle 
. Circumcircles of triangles 
and 
intersect at two different points 
and 
inside of triangle . Prove that ![\[\angle BAX=\angle CAY.\]](http://latex.artofproblemsolving.com/3/9/9/39910fdd000cbed820b33ac08a8200ca06ed8193.png)
in the interior of a fixed triangle lines 
are drawn parallel to the medians through 
respectively to meet 
at 
respectively. Prove that 
is constant (independent of ).
in the interior of a fixed triangle lines are drawn parallel to the medians through respectively to meet at respectively. Prove that is constant (independent of ).
is 
constant.
; choose 
., point 
and 
.
, point 
and 
.
, point 
and 
.
.
and 
.
