[/quote]
,
of each imon 
.
become entangled if and only if the original imons 
are entangled, and each copy 
members from party 
,
members from party 
,
members from party 
,
members from party 
,
or more people involved).
that if they were to be considered the number of members from the four parties, it is possible that, after some time, all of the kidnapped people belong to the same party, under the same rules imposed by the wizard.
at 
meets sides 
and 
again at 
and 
respectively. Let 
be the circumcentre of 
is perpendicular to 
.
for which 
is the square of an integer
of natural numbers with 
of 
if the product of the elements of 
![\[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\]](http://latex.artofproblemsolving.com/e/7/f/e7fb8c20a43535b5aaed5ab254f1ef043263c62b.png)
true?
be an integer greater than zero. Then, the value of the sum of the reciprocals of the cubes of the roots of the equation![\[
mx^4 + 8x^3 - 139x^2 - 18x + 9 = 0
\]](http://latex.artofproblemsolving.com/a/3/a/a3a9bca412e927632e4921c51f2a6593409031d9.png)
is equal to:
for which there exists an even natural number 
such that the number![\[
(a - 1)(a^2 - 1)\cdots(a^n - 1)
\]](http://latex.artofproblemsolving.com/9/c/a/9caf4eeb82ff46b5ba55ab4b6bc28f0cace586ec.png)
is a perfect square.
and 
be on segment 
of an acute triangle 
such that 
and 
.
be the points on 
and 
,
and 
.
and 
is on the circumference of triangle 
,![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](http://latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
be triangle, inscribed in parabola. Tangents in points 
forms triangle 
.
)
such that the equality 
holds for all pairs of real numbers 
.
is an obvious solution. Now, let 
.
or 
.
.
Y by Adventure10, Mango247
A king in a tournament is a player 
such that for every player 
either beats or there exists 
such that beats and beats . Describe the set of integers 
such that there exists a tournament with players in which every player is a king.
Other recent problems involving kings in tournaments:
Existence
If 2 then at least 3
If only 1, then undefeated
such that for every player 
either beats or there exists 
such that beats and beats . Describe the set of integers 
such that there exists a tournament with players in which every player is a king.Other recent problems involving kings in tournaments:
Existence
If 2 then at least 3
If only 1, then undefeated
.
.
with 
and 
with the following properties (we say one vertex "beats" another if the edge connecting those vertices leads from the one who beat to the one who was beaten):
,
are in a cycle of length three. Therefore, we can get from any of 
with 
.
,
,
as follows:
.
beats 
beats 
or 
.
possiblities if you wanted to.
