be distinct natural numbers each of which has exactly 
positive integer factors. For instance, 
has exactly 
.
divides any of the 
'
be an integer. Prove that 
can be written as a polynomial with integer coefficients in the variables 
,
and 
.
,
are real numbers such that 
is an integral multiple of 
.
,
for all positive integers 
.
be the orthocenter of an acute-angled triangle 
.
centered at the midpoint of 
and passing through 
and 
.
,
,
and 
.
with real coefficient. It is known that 
and 
.
can't have 
distinct real roots.
Y by Adventure10, Mango247
Let 
be a point on circumcircle of triangle 
on arc 
which does not contain point 
. Let lines 
and 
intersect at point 
, and lines 
and 
intersect at 
. If perpendicular bisector of side intersects in point 
, and perpendicular bisector of side intersects side in point 
, prove that:
be a point on circumcircle of triangle 
on arc 
which does not contain point 
. Let lines 
and 
intersect at point 
, and lines 
and 
intersect at 
. If perpendicular bisector of side intersects in point 
, and perpendicular bisector of side intersects side in point 
, prove that:
be the angles at 
in the cyclic quadrilateral 
(so that 
)
.
and 
.
and 
as well as 
and 
,
from our equation we are then left to prove that![\[\frac{\sin (\alpha+\beta)^2}{\sin (\alpha+\gamma)^2}=\frac{\frac{1}{2\cos \alpha}-\frac{\sin \gamma}{\sin \alpha+\gamma}}{\frac{1}{2\cos \alpha}-\frac{\sin \beta}{\sin \alpha+\beta}}\]](http://latex.artofproblemsolving.com/6/b/5/6b50a3bc29a0b54380f25586fa6fbe1ec702e251.png)
Multiplying both sides with 
we need to prove that![\[\frac{\sin \alpha+\beta}{\sin \alpha+\gamma}=\frac{\sin (\alpha+\gamma)-2\cos \alpha\sin \gamma}{\sin(\alpha+\beta)-2\cos \alpha\sin\beta}\]](http://latex.artofproblemsolving.com/c/6/f/c6fe5675dab3f886cd778e534c057dd37b62a058.png)
But 
and similarly 
and we are done.
