[/quote]
,
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 
is an acute angle triangle such that 
and 
.
the center of the circumscribed circle of the triangle and 
the intersection of altitudes of this triangle. Line 
intersects 
in point 
and 
in point 
.
.
.
be a point on side 
,
be a point on the extension of 
such that 
.
,
satisfying 
.
of positive integers with 
prime and ![\[ a^p=b!+p. \]](http://latex.artofproblemsolving.com/9/f/6/9f674dabcb364872cef4153198e2e362ae40a2fd.png)
satsfying 
and:![\[ f(2n)f(2n+1)=9f(n)^2+3f(n)\]](http://latex.artofproblemsolving.com/b/c/1/bc18d6ad72589f012dcab7836228b3ed7ee061c2.png)
be positive reals such that 
.![\[18\sum_{\text{cyc}}\frac{1}{(3-c)(4-c)}+2(ab+bc+ca)\ge 15. \]](http://latex.artofproblemsolving.com/5/d/5/5d5c47a52d2d6ea7d13ce1c2fa14a08425bbfa5d.png)
Proposed by David Stoner
such that 
.
consisting of a prime number 
and 
such that 
and 
are both powers of 
and let 
be reals such that 
satisfying
Prove
Y by Adventure10, Mango247
Point 
is chosen at the diagonal 
of parallelogram 
The straight line 
intersects the side 
and the straight line 
at points 
and 
respectively. Let 
be the circle with centre and radius 
and 
be the circumcircle of triangle 
Denote by 
and 
the intersection points of circles and 
Prove that the circle is inscribed into the angle 
is chosen at the diagonal 
of parallelogram 
The straight line 
intersects the side 
and the straight line 
at points 
and 
respectively. Let 
be the circle with centre and radius 
and 
be the circumcircle of triangle 
Denote by 
and 
the intersection points of circles and 
Prove that the circle is inscribed into the angle 
be the center of the parallelogram and 
the reflection of 
about 
Since 
Let 
be the infinity point of this direction. Since 
it follows that the pencil 
is harmonic 
By Newton's theorem, we have then 
