AoPS Academy
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)
, 
, 
, ... be an infinite sequence of real numbers satisfying the equation 
for all 
, where and are two different positive reals., , , ... be bounded?
and 
in a non-isosceles triangle 
with incenter 
intersect its circumcircle 
at 
and 
, respectively. The line through , parallel to 
, intersects 
at 
. Prove that 
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 
. be an isosceles triangle such that 
. Let be a dot on the side.
at point intersects the circumcircle of 
at 
. Prove that CD
AB
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 . Let 
intersect 
at , 
intersect 
at 
, and 
intersect 
at 
. If the area of 
can be written as 
for relatively prime integers 
and 
and squarefree 
, find the sum of all prime factors of , counting multiplicities (so the sum of prime factors of 
is 
), given that has 
divisors.
.
such that 
is a positive integral power of a prime positive integer."
touches the sides 
and 
in points 
and 
respectively. 
is a diameter of the incircle. Finally, let 
be the intersection of the lines 
and 
. Prove that the segments 
and 
have equal length.
touches the sides and in points and respectively. is a diameter of the incircle. Finally, let be the intersection of the lines and . Prove that the segments and have equal length.
, let 
, by brokard's on 
we get that the polar of passes through but by La Hire's, the polar of passes through , hence 
is the polar of so 
and since 
we are done
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 
