[/quote]
,
be a natural number. Prove that,![\[ \left\lfloor \frac{n}{1} \right\rfloor+ \left\lfloor \frac{n}{2} \right\rfloor + \cdots + \left\lfloor \frac{n}{n} \right\rfloor + \left\lfloor \sqrt{n} \right\rfloor \]](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 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
.
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 mueller.25, amar_04, Adventure10
In an acute-angled isosceles triangle 
, altitudes 
and 
intersect the line passing through the vertex 
and parallel to the line 
, at points 
and 
. Let 
be the midpoint of side , and 
the altitude. Lines 
and 
intersect line 
at points 
and 
. Prove that the circles circumscribed around triangles 
and a circle with a diameter intersect at one point.
, altitudes 
and 
intersect the line passing through the vertex 
and parallel to the line 
, at points 
and 
. Let 
be the midpoint of side , and 
the altitude. Lines 
and 
intersect line 
at points 
and 
. Prove that the circles circumscribed around triangles 
and a circle with a diameter intersect at one point.This post has been edited 1 time. Last edited by parmenides51, Feb 25, 2024, 7:46 AM
and 
and 
.
is tangent to the 
is tangent to 
.
lies on one circle 
lie on 
.
lie on 
so 
lie on 
.
