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.
, the points 
and 
are taken such that the point lies between the points 
and , and 
. On segments 
and 
, points 
and 
are taken so that 
and 
. Prove that 
.
and points and on sides 
and 
, respectively, such that 
. Let 
and 
be intersection points of lines 
and 
with circumcircle 
of triangle 
, and 
and 
intersection points of lines 
and 
with side 
. If 
, prove that circle contains intersection point of angle bisector of 
with
such that 
divides 
for all positive integers 
and 
with 
.
such that 
Follows for all reals 
.
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 
.
line such that, new two pieces' areas and perimeters are equal. Prove that 