[/quote]
,
![\[ \left\lfloor \frac{n}{1} \right\rfloor+ \left\lfloor \frac{n}{2} \right\rfloor + \cdots + \left\lfloor \frac{n}{n} \right\rfloor + \left\lfloor \sqrt{n} \right\rfloor \]](http://latex.artofproblemsolving.com/5/6/0/560a9568b2c09512cac1d317a011931a439e6a02.png)
and the sets 
If 
has exactly three elements, prove that 
cannot be an integer.
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?
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 
.
.
Y by Davi-8191, mathematicsy, Adventure10
Let 
be the area of the triangle 
. A non-regular convex polygon 
is called guayaco if exists a point 
in its interior such that ![\[A(P_1OP_2) = A(P_2OP_3) = \cdots = A(P_nOP_1).\]](//latex.artofproblemsolving.com/a/5/e/a5e3a48dd716a4c643d446af8fd8e903bbff8eb8.png)
Show that, for every integer 
, a guayaco polygon of 
sides exists.
be the area of the triangle 
. A non-regular convex polygon 
is called guayaco if exists a point 
in its interior such that ![\[A(P_1OP_2) = A(P_2OP_3) = \cdots = A(P_nOP_1).\]](http://latex.artofproblemsolving.com/a/5/e/a5e3a48dd716a4c643d446af8fd8e903bbff8eb8.png)
Show that, for every integer 
, a guayaco polygon of 
sides exists.This post has been edited 1 time. Last edited by adrian97, Aug 18, 2017, 4:48 AM
Reason: Fix typo.
Reason: Fix typo.
,
,
,
congruent isosceles triangles with an angle of 
in such a way that they add up to a flat angle. Then add 2 rectangle triangles with matching areas at the bottom to finish the polygon:
