AoPS Academy
with 
, the foot of altitudes from 
to the sides 
are 
, respectively. 
is the orthocenter. 
is the midpoint of segment 
. Lines 
and 
intersect at 
. Let the tangents drawn to circumcircle 
from 
and 
intersect at 
. Prove that 
are colinear
and 
with real coefficients satisfying
be three positive reals such that 
. Prove that![\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]](http://latex.artofproblemsolving.com/a/1/4/a14adf0f1e35df57c734c7df701d5a67da22dc3c.png)
s.t. 
and 
be positiv integer numbers.
be a positive integer.Prove that there is a polynomial 
with integer coefficients so that 
,then![$$a^{2n+1}+b^{2n+1}+c^{2n+1}=abc[P(a,b)+P(b,c)+P(a,c)]$$](http://latex.artofproblemsolving.com/5/2/0/520813bebb2f297928208f7b7b2eea76d24043ac.png)
.
are of the form 
, where 
, 2, 3, . For each , let 
be the greatest common divisor of 
and 
. Find the maximum value of as ranges through the positive integers.
are positive real number such that 
. Prove that
, where 
always occurs at 
is the set of positive real solutions to the system![\[
x^3 + y^3 + z^3 = x + y + z
\]](http://latex.artofproblemsolving.com/7/7/a/77a748a8b55a6b44761bced9bb1031193b6127dc.png)
![\[
x^2 + y^2 + z^2 = xyz
\]](http://latex.artofproblemsolving.com/9/3/a/93a4dffc6fd17b9f19f738d3b3865f56aa0151a9.png)
then find the number of elements in .
, where integers and 
is a positive integer. Suppose that the integers
satisfy: 
are divisible by k. Prove that:
is divisible by k for which 
is divisible by 
?
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle with circumcenter 
, centroid 
and orthocenter . Prove that 
is constant.
, and let 
. Let 
be any points on the plane
, such that the centroid of 
, 
are 
respectively [*]. Suppose 
and 
intersects at and yeilds
thus 
with circumcenter , centroid and orthocenter .
is midpoint of 
be circumcenter, centroid and orthocenter of 
.
analogously
and some calculation
be reflection of 
wrt point 
notice that 
and 
Prove that
