[/quote]
,
satisfy:
![$[(x-2)^2+(y-4)^2][ (x-1)^2 + (y-2)^2] = 36$](http://latex.artofproblemsolving.com/b/1/0/b108dc2cac5b7491fb83418eb36c7184f6d5d30c.png)
such that
holds for all 
.
such that:
with all 
be a triangle, and let 
,
,
be collinear points such that 
,
,
.
,
,
are the reflections of 
,
,
,
is a nondegenerate triangle, then its circumcenter lies on the circumcircle of 
be three positive real numbers satisfying 
Prove that
There is a very elegant proof :-D Could anyone think of it?
such that 
and has exactly 
then
be tangent to the side 
.
on 
on 
and 
,
lies on the circumcircle of triangle 
be the circumcircle of triangle 
is the midpoint of 
be the second intersection of 
and 
,
be a transversal of 
be intersections of 
with 
be a tangent line of 
is cyclic
be positive integers such that 
and 
.
.
and 
Prove that
at 
,
perpendicular to 
be the intersections of 
with 
,
be the projections of 
onto line 
be the second intersections of 
with the incircle 
be the intersection of 
and 
.
bisects segment 
.
Y by
a) Prove that when there is no face of a convex polyhedron triangle, there are at least eight of its vertices, from which it follows exactly three edges . (There are exactly eight such vertices in a cube).
b) Prove that when from each vertex of a convex polyhedron it turns out at least four edges, there are at least eight of its faces, each of which is a triangle. (The octahedron has exactly eight such faces).
b) Prove that when from each vertex of a convex polyhedron it turns out at least four edges, there are at least eight of its faces, each of which is a triangle. (The octahedron has exactly eight such faces).
