AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
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.
such that
holds for all 
.
such that:
with all 
is real number
be a triangle, and let 
, 
, and 
be collinear points such that 
, 
, and 
. Points 
, 
, and 
are the reflections of , , and over 
, 
, and 
, respectively. Prove that if 
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 is a multiple of 
and has exactly positive factors.
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.
be the circumcircle of triangle , 
is the midpoint of and 
be the second intersection of 
and . Tangent line of at intersects at 
, let 
be a transversal of and 
be intersections of 
with . Let 
be a tangent line of . Prove that 
is cyclic
be positive integers such that 
and 
. Find the maximum possible value of 
.
have the equal area.
. We need to prove that 
and 
, which implies 
,
,
, we can easily solve the problem. Assume that 
are different, multiply three equations yields
, which is obviously wrong., which implies 
is the centroid of triangle 
![$[(x-2)^2+(y-4)^2][ (x-1)^2 + (y-2)^2] = 36$](http://latex.artofproblemsolving.com/b/1/0/b108dc2cac5b7491fb83418eb36c7184f6d5d30c.png)
then
and 
be points on the sides 
and 
respectively of a triangle 
and 
meet at a point 
.