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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)
. Find the radius of the circle.
. Find the radius of the circle.
be the triangle, with right angle at 
. Then the power of 
wrt this circle is 
. However, the power of is also equivalent to 
.
and 
.
