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 
Y by centslordm
On the sides 
and 
of the triangle 
were chosen respectively points 
and 
such that each of them divides the corresponding side in the ratio 
(counting from the vertex 
). In what ratio does the intersection point of the segments 
and 
divide each of these segments?
and 
of the triangle 
were chosen respectively points 
and 
such that each of them divides the corresponding side in the ratio 
(counting from the vertex 
). In what ratio does the intersection point of the segments 
and 
divide each of these segments?This post has been edited 1 time. Last edited by parmenides51, Jul 26, 2021, 3:07 PM
