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, and let 
denote the sum of the positive divisors of 
. Prove that the 
smallest positive integer relatively prime to is at least , and determine for which equality holds. to be 
-smooth if all of its prime factors are 
. Let 
denote the number of -smooth integers upto 
. Let 
be the least quadratic non-residue mod 
. Show that there are at least quadratic residues mod up to .
is a non-negative arithmetic function and
is convergent for some 
, then prove that
, find the smallest integer 
such that any graph with 
vertices and at least edges must contain at least two disjoint cycles (cycles with no common vertices).
and 
, let all integer points on the number line 
be vertices. Define a directed graph 
as follows: for any integer point 
, 
, 
. For integers 
, let ![\( G[k, l] \)](http://latex.artofproblemsolving.com/6/3/a/63a5df1124231155f3240a3e21e24d293d0e8db3.png)
denote the subgraph of with vertices limited to the interval ![\([k, l]\)](http://latex.artofproblemsolving.com/2/9/f/29f08f190960a99d062fde7f2128bcb635657f36.png)
. Find the largest positive integer 
such that for any integer 
, the subgraph ![\( G[r, r + \alpha - 1] \)](http://latex.artofproblemsolving.com/5/1/a/51afcceb73daae97f8fc61456bf043b73b2689d7.png)
of is acyclic. Clarify the structure of subgraphs and ![\( G[r, r + \alpha] \)](http://latex.artofproblemsolving.com/8/8/7/887e55f5636b842e7b64c9c03982b0374cdf1c9e.png)
(i.e., how many connected components and what each component is like).
of a graph be partitioned into parts 
, with 
. If there is an edge between any two vertices only when they belong to different parts, the graph is called a complete -partite graph, denoted as 
. Let and be positive integers, , 
. Consider the complete 
-partite graph 
.-partite graph., for all , find the maximum number of edges in a complete -partite graph 
where no more than one circle is disjoint.
be a median in an acute-angled triangle 
. A point 
is chosen on the line through 
tangent to the circumcircle of 
so that 
. The segments 
and meet at 
. Prove that the circumcenter of 
lies on the line 
.
be a cyclic quadrilateral for which 
. Construct two squares 
and 
externally on the sides 
and 
respectively.
meets 
at 
, 
meets 
at 
and 
denotes circumcenter of . Prove that 
and 
meets again on 
.
be an acute triangle, with 
. Let be the midpoint of the arch 
that does not contain 
and let 
be the midpoint of . Let 
be the 
-excenter and -excenter of , respectively. Let 
be the reflection of with respect to . Prove that the points 
are concyclic.
be a convex pentagon with 
and 
For each point in the plane, define 
The least possible value of 
can be expressed as 
where 
and are positive integers and is not divisible by the square of any prime. Find 
be the set of positive integers. A function 
satisfies the equation ![\[\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}\]](http://latex.artofproblemsolving.com/2/e/7/2e73ff2f75b103e9d6a4004a42e4ed7f4ee64a67.png)
for all positive integers . Given this information, determine all possible values of 
.
