and 
.
![$$a+b+\frac{3}{2}c\geq \frac{9}{8}\sqrt [3]{\frac{3}{2}}-\frac{3}{16}$$](http://latex.artofproblemsolving.com/2/2/e/22e6ec2f98777db80e39333542664e903663bf81.png)
![$$a+b+\frac{8}{5}c\geq \frac{9\sqrt [3]{25}-4}{20}$$](http://latex.artofproblemsolving.com/f/c/a/fca343ee701eeff4c309b31beb3426ec96b49b55.png)
Let 
.
![$$a+b+\frac{27}{25}c\geq\frac{35\sqrt [3]5-9}{50}$$](http://latex.artofproblemsolving.com/6/9/5/695cc35986a391fdc15e4863d361ce373fbbf64c.png)
.
Equality holds when 
Equality holds when 
is a regular polygon with 
diagonals. Randomly select a triangle whose three vertices are the three vertices of polygon 
be a sequence of positive integers with 
,
and![\[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\]](http://latex.artofproblemsolving.com/1/b/f/1bf1a88979e7d2b33530de979d66a4e81032953d.png)
for all 
.
is a prime less than 
for some positive integer 
,
such that 
.
Y by kiyoras_2001
Let 
be triangle with an arbitrary point 
. Line 
is the line pass through and perpendicular to side 
respectively . Point 
is reflection of 
across line 
respectively . Define 
analogously . Point 
is intersection of 
and 
. Define point 
in the same way . Point 
is inversion image of point wrt 
. Prove that quadrilaterals 
and 
are similar .
be triangle with an arbitrary point 
. Line 
is the line pass through and perpendicular to side 
respectively . Point 
is reflection of 
across line 
respectively . Define 
analogously . Point 
is intersection of 
and 
. Define point 
in the same way . Point 
is inversion image of point wrt 
. Prove that quadrilaterals 
and 
are similar .
,
and 
as the radii. Then 
is cyclic with center 
and 
.
is inversely similar to 
be the inverse of 
means that 
and 
are similar. So 
.
.
and so on.
,
,
with the ratios being 
etc., then the two (possibly degenerate) common intersection points of these circles are 
.
,
,
