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, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
is the largest positive integer that divides both 
and 
. Integers and are coprime if 
.
, which doesn't have right angle, let 
be its circumcenter. Circle 
intersects 
and 
at 
and 
for the second time, respectively. Similarly, circle 
intersects 
and 
at 
and 
, and circle 
intersects 
and 
at 
and 
for the second time, respectively. Let 
and 
be circumcenters of triangle 
and 
, respectively. Prove that 
are collinear.
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.
be an acute triangle with 
, its orthocenter and it's circumcenter. Let be a point on the parallel through to such that 
and and are on different sides of . Denote by 
the intersection of the circumcircle of and 
, where 
is the reflection of over , 
the midpoint of 
, the intersection of 
and the parallel through to 
, the intersection of 
and the perpendicular through to 
and a point on such that 
, where 
is the midpoint of 
. Prove that 
lie on a line.
+1 w
be the set of real numbers. Determine all functions 
that satisfy the equation![\[f(x^2f(x+y))=f(xyf(x))+xf(x)^2\]](http://latex.artofproblemsolving.com/b/4/b/b4b1e22e937394550f13cfdab142ca5bf4a26aea.png)
for all real numbers 
and 
.
