AoPS Academy
of degree 
has exactly 
real roots, counted with multiplicity. We know that the coefficient of 
is 
, all the roots are less than or equal to , and 
. What values can 
take?
lies inside triangle 
such that 
and 
. Point 
is the midpoint of segment 
. Point 
lies on segment 
with 
. Prove that 
.
satisfy,
Where {
} is a permutation of . Prove that 
such that for all real numbers 
and 
,![\[f(xf(y))+f(y)=f(x+y)+f(xy).\]](http://latex.artofproblemsolving.com/d/2/f/d2ff6ed448cf39e7ec9bce6b944965d5e89b9878.png)
be a convex hexagon with 
and 
.
which is equidistant from sides 
and 
.
and 
are the centers of mass of 
and 
, show that 
.
be a convex hexagon with and .
which is equidistant from sides and . and are the centers of mass of and , show that .
, 
, to form an equilateral triangle; the only possible point is its center.
, , to form an equilateral triangle; the only possible point is its center. is the incenter.
,
is the triangle formed by lines 
.
are the centroids of triangles 
.
is similar to three triangles above.
.
,
,
,
is similar to the previous three (this is probably the only statement ever where complex numbers give the cleanest proof). But spotting it requires you to think synthetically about the configuration. I'm not sure how a "blind" complex numbers approach would work here. To be fair, if it weren't for the complex numbers part, this problem absolutely wouldn't have landed here.