AoPS Academy
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.
opposite the vertex 
be tangent to the side 
at the point 
. Define the points 
on 
and 
on 
analogously, using the excircles opposite 
and 
, respectively. Suppose that the circumcentre of triangle 
lies on the circumcircle of triangle . Prove that triangle is right-angled.
is a function for which the expression 
for all 
is always a perfect square. Prove that 
for all 
.
and 
, Jaqueline has a ruler that allows tracing the line 
. Jaqueline also has a special object that allows the construction of a circle of diameter . Also, always when two circles (or a circle and a line, or two lines) intersect, she can mark the points of the intersection with a pencil and trace more lines and circles using these dispositives by the points marked. Initially, she has an acute scalene triangle . Show that Jaqueline can construct the incenter of .
and 
intersect at points 
and 
. Circle 
externally touches and at points 
and 
respectively. Let 
be the midpoint of the chord cut by the line 
on circles . Prove that 
.
and intersect at points and . Circle externally touches and at points and respectively. Let be the midpoint of the chord cut by the line on circles . Prove that .
to 
and 
Resp. By radical axis the tangents and 
is cyclic, but since 
is midpoint of the chord we get that 
so 
is cyclic, then the conclusion follows because 
.