[/quote]
,
and 
Prove that
Let 
and 
Prove that
which satisfy 
![$ \frac{a+b+c}{3} \ge \sqrt[10]{\frac{a^3+b^3+c^3}{3}}$](http://latex.artofproblemsolving.com/c/d/a/cdae05ad34239366b0dd3921243522c711b6ae58.png)
method, I can assum 
![$ \frac{2x+\frac{1}{x^2}}{3} \ge \sqrt[10]{ \frac{2x^3+ \frac{1}{x^6}}{3}} $](http://latex.artofproblemsolving.com/c/d/4/cd481672bee1e87d8c97ac623436e0ec75cfa73a.png)
are given such that 
.
in natural numbers m and n.
,
denote the smallest possible value of 
where 
are sets such that 
and 
whenever 
.
and 
intersect at 
.
of the two circles touches 
at 
respectively. Let 
be the symmetric points of 
intersects circles 
respectively which are distinct from 
and 
are equal.
,
.
intersects 
at 
.
cuts segment 
at 
respectively. If 
and 
,
equals 
.
are points on a circle 
,
,
.
at 
.
is on 
such that 
.
intersects 
.
is the reflection of 
wrt 
.
lies on the same circle.
in its exterio
so that 
,
,
.
and
such that 
and 
.
be the points on 
and 
,
and 
.
and 
is on the circumference of triangle 
has a right angle at 
.
such that 
and 
lies between 
.
is chosen so that 
and 
.
is chosen so that 
and 
is the bisector of 
.
be the point such that 
is a parallelogram. Prove that 
and 
are concurrent.
such that the number of 
which satisfy the following condition is minimal:
has periodic 
and 
are tangent to the circumcircle of 
be the reflection of 
be the reflection of 
.
is the circumcenter of 
,
is perpendicular to 
consisting of some 
grid so that each column contains a string from 
such that if 
Y by Adventure10, Mango247
In a country with 
cities there is exactly one two-way flight between each city. The three flights made between three cities belong to at most two different airline companies. No matter how the flights are shared between some number of companies, if there is always a city in which 
flights belong to the same airline, what is the maximum value of ?
cities there is exactly one two-way flight between each city. The three flights made between three cities belong to at most two different airline companies. No matter how the flights are shared between some number of companies, if there is always a city in which 
flights belong to the same airline, what is the maximum value of ?
.
groups numbered from 
to 
with color 
.
.
.
whose subgraph is disconnected. Denote the connected components of 
.
,
and 
are of same color.
to all 
are of same color. Considering that for all 
analogously, will imply the desired result.
and 
on this spanning tree, edges 
and 
must be of same color (otherwise consider triangle 
)
, 
and 
are of different colors, the path from 
as single vertices in order to simplify our graph. Now we get rid of color 
with least vertices could have at most 
vertices. In this case 
edges from 
and 
