AoPS Academy
be a triangle, and let 
be its altitudes. Let 
be its orthocenter, and let 
and 
be the circumcenters of triangles 
and 
. Let 
be the second intersection of the circumcircles of triangles 
and 
. Prove that the lines 
, 
, and 
-median of 
are concurrent.
such that 
divides 
.
be a triangle with circumcenter 
and circumcircle 
. Let 
be the foot of the altitude from to 
. Let 
and 
be points on the circumcircles of triangles 
and 
, respectively, such that , , and are collinear. Prove that if the circumcircle of triangle 
is tangent to at 
, then 
.
+1 w
. Find all functions 
such that for all 
,![\[
f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)).
\]](http://latex.artofproblemsolving.com/f/9/3/f93b7526ff4338eea89a1c59bc009aa001ffc15d.png)
,
,
, 
, 
is not greater than 
. and 
be points on a plane such that 
, where 
is a positive integer. Let 
be the set of all points such that 
, where 
is a real number. The path that traces is continuous, and the value of is minimized. Prove that is rational for all positive integers 
to the cross product 
reveals that a negative determinant [of 
] corresponds to a matrix which reverses the direction of the cross product of two vectors.[/quote]
and 
, but am confused on how to prove this statement for general vectors 
and 
although its supposed to be a quick comparison.
and 
Prove that
.
.
be the sequence of positive integers such that 
and 
for all positive integers 
.
such that the greatest prime divisor of 
is more than 
for all 
.