[/quote]
,
be the set of non-negative integers and 
be the set of positive real numbers. Let 
be a function such that 
and 
for all integers 
,
for all integers 
.
.
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 
.
at 
respectively. Let 
be the symmetric points of 
intersects circles 
respectively which are distinct from 
and 
are equal.
,
.
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
and 
.
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 
Y by tiendung2006, ImSh95, ehuseyinyigit
In an acute triangle 
, its inscribed circle touches the sides 
at the points 
respectively. Let 
be the midpoint of the side 
, 
be the midpoint of the arc on the circumcircle of triangle , and 
be the projection of on the segment 
.
Prove that the points
and the incenter 
of the triangle lie on the same line.
, its inscribed circle touches the sides 
at the points 
respectively. Let 
be the midpoint of the side 
, 
be the midpoint of the arc on the circumcircle of triangle , and 
be the projection of on the segment 
.Prove that the points
and the incenter 
of the triangle lie on the same line.
is symmedian in 
.
;
;
.
is cyclic with center 
.
.
is the midpoint of 
.
is perpendicular to 
.
be the circumcircle 
Intersects 
radius of 
,
and 
![\[\color{magenta} \boxed{\mathcal{E}}\]](http://latex.artofproblemsolving.com/1/e/1/1e168c66dfa08d393a47f4aa29b82ab30bfc8af5.png)
.
.
and check that 
with strong EFFT which establishes 
,
and 
which is equivalent to solving the system 
after some manipulation which helps a lot.
be the antipode of 
the midpoint of 
.
.
.
implies 
which implies our desired result.
and 
be the 
and 
.
and 
,
and 
are collinear, we are done by homothety in 
https://artofproblemsolving.com/community/u998255h3355708p31160241
