be a positive integer. Find all real solutions 
to the system:![\[a_1^2 + a_1 - 1 = a_2\]](http://latex.artofproblemsolving.com/a/4/e/a4e26b83c929b82800576bac11e12de0c3aa1cf6.png)
![\[ a_2^2 + a_2 - 1 = a_3\]](http://latex.artofproblemsolving.com/9/9/5/995e8eec93828b82901b0639463f907f5ff95d08.png)
![\[\hspace*{3.3em} \vdots \]](http://latex.artofproblemsolving.com/e/e/1/ee15e1fefe2ad550d3c9b1c51207eaf66fbfa41f.png)
![\[a_{n}^2 + a_n - 1 = a_1\]](http://latex.artofproblemsolving.com/7/c/7/7c727d1f2044c1bf7c80981e1074bb3cde807b8f.png)
,
,
,
for all 
,
and 
with incenter 
intersect its circumcircle 
at 
and 
,
,
at 
is tangent to 
,
such that ![\[\left(a_1+\frac1{a_1}\right)\left(a_2+\frac1{a_2}\right)\cdots\left(a_n+\frac1{a_n}\right)=2020.\]](http://latex.artofproblemsolving.com/3/6/2/362fccfa6353ffcbc4d29f24ccbfce73269080bf.png)
Compute 
.
.
at point 
at 
.
A
have angles 
and 
such that 
and 
. Moreover, suppose that the product of the side lengths of the triangle is equal to its area. Let 
denote the circumcircle of 
intersect 
at 
intersect 
at 
,
intersect 
at 
.
can be written as 
for relatively prime integers 
and 
and squarefree 
,
is 
)
divisors.
.
such that 
is a positive integral power of a prime positive integer."
Prove that![$$\frac{a}{b}+ \frac{kb^3}{c^3} + \frac{c}{a}\geq 7\sqrt[7]{\frac{k}{729}}$$](http://latex.artofproblemsolving.com/5/d/c/5dcb84c12e880c66ecf09af425d216a37510bbe9.png)
Where 
![$$\frac{a}{b}+ \frac{kb^4}{c^4} + \frac{c}{a}\geq \frac{9}{2}\sqrt[9]{\frac{k}{128}}$$](http://latex.artofproblemsolving.com/b/8/e/b8ef8dec5b29f4d6ec3f834f644ce30d6db8bb3c.png)
Where 
Prove that
such that 
.
Where 
Y by Adventure10 and 1 other user
The in-circle of 
, where 
, touches 
at 
and 
is a diameter of the in-circle. 
produced cuts at 
.
(i) Prove
.
(ii) A circle
of variable radius touches at 
. The tangents (other than ) from 
and 
to intersect at 
. moves as the radius of varies. Find the locus of .
, where 
, touches 
at 
and 
is a diameter of the in-circle. 
produced cuts at 
.(i) Prove
.(ii) A circle
of variable radius touches at 
. The tangents (other than ) from 
and 
to intersect at 
. moves as the radius of varies. Find the locus of .
.
which is one of the two similar parts of a hyperbola with focuses 
