AoPS Academy
be an acute triangle and 
its orthocenter. Let 
be a point on the parallel through 
to 
such that 
. Define 
and 
as points on the parallels through 
to 
and through 
to 
similarly. If 
are positioned around the sides of as in the given configuration, prove that are collinear.
are collinear, and 
are arbitrary points
, let 
be the sum of the squares of the digits of . For example, 
. Determine all integers 
such that 
.
with centroid 
and circumcircle 
centred at 
, the extension of 
meets at 
; lines and 
intersect at ; and lines and 
intersect at . Suppose the circumcentre 
of the triangle 
lies on and 
are collinear. Prove that 
.
, let 
and 
be points on sides and , respectively, such that 
is cyclic. Let lines 
and 
intersect at point , and and 
be the midpoints of 
and 
, respectively. If 
is the foot of the perpendicular from to , and the circumcircles of triangles 
and 
intersect at second point 
different from , prove that 
and are collinear.
, such that 
for all integers 
.
of positive integers, with 
prime, such that 
is a perfect square. inscribed in 
. Two points 
lie on such that 
are isogonal in 
. is the midpoint of 
. 
lies on 
such that 
. 
intersects at . Prove that the line through parallel to 
passes through the Euler center of triangle .
be a trapezoid inscribed in circle , 
. Let is the symmetric point of across , 
intersects at . Let is midpoint of and is orthocenter of triangle . On 
take a point so that 
. Prove that: 
coincide
be a trapezoid inscribed in circle , . Let is the symmetric point of across , intersects at . Let is midpoint of and is orthocenter of triangle . On take a point so that . Prove that: coincide
,
,
.![\[\sum_{i=1}^{n}\frac{1}{1+x_{i}}\cdot \sum_{i=1}^{n}x_{i}\leq \sum_{i=1}^{n}\frac{x_{i}^{k+1}}{1+x_{i}}\cdot \sum_{i=1}^{n}\frac{1}{x_{i}^{k}}\]](http://latex.artofproblemsolving.com/2/6/4/264d110e2a40c7f787c650aa8b8dc30af5d28263.png)
such that![\[
f(x^4 + 5y^4 + 10z^4) = f(x)^4 + 5f(y)^4 + 10f(z)^4
\]](http://latex.artofproblemsolving.com/3/0/8/308079e75e1acd9a9f896885387b67a7b4020c16.png)
for all 
.
