[/quote]
,
,
lie on a line in this order and satisfy 
.
,
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
,
and 
and 
and 
be a positive integer. Given a sequence of nonnegative real numbers 
we define the transformed sequence 
as follows: the number 
is the greatest possible value of the average of consecutive terms of the sequence that contain 
.
is 
.
,
is less than or equal to 
.
holds.
be positive real numbers. Prove that:![$$\sqrt[3]{a^3+b^3}+\sqrt[3]{b^3+c^3}+\sqrt[3]{c^3+a^3}\geq \sqrt[3]{2}(a+b+c)$$](http://latex.artofproblemsolving.com/2/6/e/26ebcdae35153bee73abbcfaf4a767d3a32b8912.png)
be all the positive integers smaller than 
such that
for all 
.
is written on the board, with 
linear factors on each side. What is the least possible value of 
linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?
of positive integers is called central if for every positive integer 
terms of the sequence is equal to 
,
,
,
of positive integers such that for every central sequence 
,
,
,
.
are two odd positive integers such that 
.
.
.
be the center of the 
and 
intersectlines 
and 
at 
and 
,
and 
intersect at 
is perpendicular to line 
be a real number. Find all functions 
such that 
for all 
is given . The positive integer 
satisfing condition :
;
are all perfect squares .
but not 
consecutive positive integers, each one is a divisor of 
such that 
be the incenter of triangle 
be the perpendicular bisector of 
.
and 
is on 
.
and the midsegment of 
is defined recursively by ![\[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]](http://latex.artofproblemsolving.com/f/4/b/f4b8f95e49de7274cccd8ee498e0b496f31ddfea.png)
Show that 
for all 
.
Y by Adventure10, Mango247
Let the incircle 
of the triangle 
touch its side 
at 
. Let the line 
intersect at 
and denote the excentre of opposite to 
by 
. Let 
and 
be the midpoints of and 
respectively.
Prove that the points
and 
are concyclic.
of the triangle 
touch its side 
at 
. Let the line 
intersect at 
and denote the excentre of opposite to 
by 
. Let 
and 
be the midpoints of and 
respectively.Prove that the points
and 
are concyclic.
T
e
.
T
at 
, 
.
.
.
, 
.
, 
.
The facts that 
are concurrent and 
and 
a
is cyclic.
.
(
be the reflection of 
with respect to 
.
be the intersection of the polar of 
with respect to 
and 
.
be the reflection of 
with respect to the perpendicular bisector of 
are collinear.
,
.
is the polar of 
is tangent to 
.
and 
,
.
are concyclic
,
are concyclic.
,
be points of tangency of the incircle with 
and 
.
.
.
and 
,
.
is a symmedian of the triangle 
so 
,
is a midpoint of 
.
.
,
are concyclic.
makes it clear.
and 
as the images of 
be a point on line 
and note that 
is tangent to 
.
are concyclic, so 
are also concyclic. By power of point we get 
so 
are concyclic. Thus, 
so 
are collinear.
,
are concyclic. By power of point, 
so 
are concyclic. The result follows as 
is an isosceles trapezoid. 
triangle 
similar triangle 
.
Similarity:
concyclic.
and 
.
thus by MPT in 
we get 
be reflection of 
be other touch point of incircle with 
then 
.
.
and 
is 
cyclic.
by using power of point we get 
Which give us 
.
hence 
.
which yields 
.
cyclic.
