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 positive integer larger than 
. Build an infinite set 
of subsets of 
having the following properties: distinct sets of have exactly one common element;
distinct sets of have void intersection.
and 
be positive integers such that 
divides 
. Prove that 
.
where 
are nine distinct integers. Prove that there exists an integer 
such that for all integers 
the number 
is divisible by a prime number greater than 20.
be the lengths of the sides of a triangle. Prove that![\[\frac{1}{(a-4b)^2}+\frac{1}{(b-4c)^2}+\frac{1}{(c-4a)^2} \geqslant \frac{1}{ab+bc+ca}.\]](http://latex.artofproblemsolving.com/7/4/b/74b7aa5621bc52977d9acd448d74293dc27a8e00.png)
of points in the plane is balanced if, for any two different points 
and 
in , there is a point 
in such that 
. We say that is centre-free if for any three different points , and in , there is no points 
in such that 
.
, there exists a balanced set consisting of 
points. for which there exists a balanced centre-free set consisting of points.
let 
be the number whose 
-th digit after the decimal point is the 
-th digit after the decimal point of 
. Show that if is rational then so is 
.
from the set of positive integers into itself is defined by the equality ![\[f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}.\]](http://latex.artofproblemsolving.com/2/a/5/2a5a20395450d37d0352ab75e9e4875ab7803805.png)
for every two relatively prime 
.
the equation 
has a solution.
such that the equation has a unique solution.
be an even integer and let 
be real numbers satisfying 
.
such that 
, , , 
be four points such that no three are collinear and is not the orthocenter of 
. Let , 
, 
be the orthocenters of 
, 
, 
, respectively. Suppose that the lines 
, 
, 
are pairwise distinct and are concurrent. Show that the four points , , , lie on a circle.
and 
be the circumcenter and orthocenter, respectively, of an acute scalene triangle . The perpendicular bisector of 
intersects 
and 
at 
and 
respectively. Let 
denote the intersection of the circumcircles of triangles 
and 
other than .
and 
analogously by repeating this construction two more times. Prove that , , , and are concyclic.
from the vertex 
, the perpendiculars 
, 
are drawn, 
on planes dividing dihedral angles at edges 
, 
, 
in half. Prove that the plane 
is parallel to the plane 
.
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
.
