with real coefficients is the average of two monic polynomials of degree 
of positive integers, for which![$$(a, b) + 3[a, b] = a^3 - b^3$$](http://latex.artofproblemsolving.com/b/8/3/b83f52a0152dc39b6d966880a1d6ca824a0f3786.png)
denotes the greatest common divisor of ![$[a, b]$](http://latex.artofproblemsolving.com/d/a/2/da2e551d2ca2155b8d8f4935d2e9757722c9bab6.png)
denotes the least common multiple of 
and 
.
the center of the circumscribed circle of the triangle and 
the intersection of altitudes of this triangle. Line 
intersects 
and 
.
.
Y by
Triangle 
is given. The line perpendicular to 
and going and through 
intersects the circle with diameter at the points 
and 
where lies closer to than . Analogously the line perpendicular to 
and going through 
intersects the circle with diameter at the points 
and 
where lies closer to than . Prove that the intersection point of 
and 
lies on the line 
.
is given. The line perpendicular to 
and going and through 
intersects the circle with diameter at the points 
and 
where lies closer to than . Analogously the line perpendicular to 
and going through 
intersects the circle with diameter at the points 
and 
where lies closer to than . Prove that the intersection point of 
and 
lies on the line 
.
,
.
be the foot of altitude from 
to 
is cyclic.
and 
.
lies on 
and 
lies on 
.
is the radical axis of 
and 
and 
is the radical axis of 
and 
and 
are 
;
are collinear.
is cyclic by easy angle chase,
Now we are able to show the collinearity,
We are done.
