[/quote]
,
be a triangle and the points 
and 
on 
,
and 
on 
,
and 
on 
be such that 
and 
.
with 
and 
are 
and 
respectively, and the intersections of 
with 
and 
are 
and 
,
and 
are concurrent.
Prove that![$$\frac{a}{b}+ \frac{kb^2}{c^2} + \frac{c}{a}\geq 5\sqrt[5]{\frac{k}{16}}$$](http://latex.artofproblemsolving.com/8/b/e/8bed6605f3988f59b8395b091b92a8b9e8b40419.png)
Where 
.
to the side 
intersect at 
be the line through 
the circumcircle of triangle 
.
intersects 
.
have a common point.
cities in a country, where 
is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities 
we define:
A 
between 
,
,
and 
for every 
;
A 
between 
A 
between 
be the total number of pairs of cities in the country that are connected by direct flights. In terms of 
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
and orthocenter 
.
is constant.
that satisfies:![\[
\sum_{i=1}^{n} a_{\left\lfloor \frac{n}{i} \right\rfloor} = n^k
\]](http://latex.artofproblemsolving.com/b/b/e/bbe7fe81e366fe390be5d17193d1581cc40dbece.png)
be a positive integer. Prove that for all integers 
,![\[
\frac{c^{a_n} - c^{a_{n-1}}}{n} \in \mathbb{Z}
\]](http://latex.artofproblemsolving.com/0/4/4/04464c1a350e3f9a929ab51534018498680068fc.png)
,
for all positive reals 
.
in the 
-
,
such that the minimum value of 
is positive.
such that![\[(a^2 + b^2 + c^2)^2 = 3(a^3b + b^3c + c^3a) + 1. \]](http://latex.artofproblemsolving.com/9/e/4/9e400fd807a9535ede983b75dcc307d8b75a35b6.png)
Y by Adventure10, Mango247
Two circles with centers 
and 
are externally tangent at point 
. These circles are also internally tangent to circle 
at points 
and 
, respectively. Denote by 
one of the intersection points of and common tangent to and at .Line 
intersects at 
while 
intersects at 
.
Prove that
is common tangent of circles and .
and 
are externally tangent at point 
. These circles are also internally tangent to circle 
at points 
and 
, respectively. Denote by 
one of the intersection points of and common tangent to and at .Line 
intersects at 
while 
intersects at 
.Prove that
is common tangent of circles and .
to 
meet 
at 
.
so that 
.
,
is cyclic. Thus,![\[ \angle B_2A_2Q = \angle A_2A_1P = \angle B_1B_2P = \angle B_2A_2Q
\]](http://latex.artofproblemsolving.com/c/f/0/cf0453e9692a350ca824000410af7148d2ce38fc.png)
.
and 
are common tangents from the same circle. We have that 
,
.
since 
and 
is tangent to 
,
,
be the second intersections of 
with 
and 
are parallel. It implies 
,
.
