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
Let 
be two points on a give circle, and 
be the midpoint of one of the arcs 
. Point 
is the orthogonal projection of 
onto the tangent 
to the circle at 
. The tangent at to the circle meets 
at 
respectively. Prove that if 
then 
.
be two points on a give circle, and 
be the midpoint of one of the arcs 
. Point 
is the orthogonal projection of 
onto the tangent 
to the circle at 
. The tangent at to the circle meets 
at 
respectively. Prove that if 
then 
.
?
and setting 
we have: 
and 
where 
is the radius of the circle. Then ![$f(\theta)=\frac{[ABC]}{[A'B'C]}=\frac{AC^2}{A'C^2}=\frac{\sin^2(2\theta)}{\left(\sin(2\theta)-\tan\frac{\theta}{2}\right)^2}.$](http://latex.artofproblemsolving.com/5/a/8/5a8033b3165c5371c22afdfc99d90bdd8f8a9315.png)
To simplify set 
and rewrite 
Since 
it is easy to show that as long as 
(or 
)
in the denominator is positive, and therefore we can write 
We are asked to show that 
which is obvious by solving the resulting quadratic in 
notation mean in this context.
.
