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.
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 
.
are constructed on diameter 
of a semicircle of radius 
. The centers of the small semicircles divide 
into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?
be 
, and 
. What is the sum 
?
through 
, shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play? such that 
is also prime.
vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color? regions of a three-set Venn Diagram with 
colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and reflection.
(the constant 
) .Prove that
are interior angles of a triangle, the condition 
from which it follows that
where 
,
,
and 
with 
.
,
,
thereby
(the constant 
from which it follows that
Since 
,
,
when 
;
when 
,
.
is 
,
and 
,
,
.
.
) .Then
(the constant 
(the constant 
from which it follows that
Since that 
thereby 
,
.
has no maximum. 
from which it follows that
where 
and 
with 
.
.
is acute. This means that 
.
.
(the constant 
(the constant 
Where 
Where 
Where 
Prove that
Where 
and 
Prove that 
Let 
Prove that 
Let 
Prove that 
Let 
Prove that 
Let 
Prove that 
Let 
Prove that 
Let 
Prove that 
Let 
Prove that 
Prove that 
Let 
Prove that 
.![$abc\geq 3\sqrt[3]{abc}+2$](http://latex.artofproblemsolving.com/7/a/b/7ab2f6a8d1099cb193ad35dfce08897d7eb25f33.png)
.
,
which is equivalent to 
which asserts 
or 
.![$(ab+bc+ca)^{2}\geq 3abc(a+b+c)\geq 9abc\sqrt[3]{abc}=144$](http://latex.artofproblemsolving.com/a/2/c/a2c95a4971acdf865fcaf77902f9242f512ba420.png)
.
![$$ab+bc+ca\geq 3\sqrt[3]{(abc)^2}\geq12$$](http://latex.artofproblemsolving.com/2/c/9/2c988b04ff2bc784b81538dd2cbe02628a7cfd41.png)
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Let 
and 
Prove that
such that 
Prove that
be non-negative numbers such that 
In 
. Prove that
and 
Prove that
and 
prove that:
Ukraine2008？
Prove that
Let 
P
Prove that
we are done!
Let 
,
,
be positive real numbers such that 
.
Let 
be positive real numbers such that 
![\[\frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}\leq 1.\]](http://latex.artofproblemsolving.com/6/9/b/69b62a5866ba96e43b0c59f89daae9710a896c45.png)
.Prove that
,
,
by Muirhead 
.
be positive real numbers . Prove that
Where 
be positive real numbers such that 
Prove that
