[/quote]
,
be a positive integer, and let 
be a subset of 
.
parts is a representation of n as a sum 
,
belong to 
.
parts with 
.![$\sqrt[3]{6n}$](http://latex.artofproblemsolving.com/2/0/1/201489b1f95ad200036698a606f712da7bb465b9.png)
different parts.
other knights where friendship is mutual. Prove that King Arthur can place some of his knights around a round table in such a way that every knight is friends with the 
people adjacent to him and between them there are at least 
friendships of knights that are not adjacent to each other.
is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities 
we define:
A 
between 
,
,
and 
for every 
;
A 
between 
A 
between 
be the total number of pairs of cities in the country that are connected by direct flights. In terms of 
,
.
.
are fixed positive integers.
be the set of all functions 
with the following properties:
the following equality holds: 
such that
,
.
such that:
,
bisects neither the angle 
nor the angle 
.
lies inside ![\[\angle PBC=\angle DBA\quad\text{and}\quad \angle PDC=\angle BDA.\]](http://latex.artofproblemsolving.com/c/2/0/c20761f3eadd054958f40259f3d1c05f26279783.png)
Prove that 
.
and 
meet at a point 
be the point of the line 
,
.
and 
.
lie on the same circle.
for at least one 
for every positive integers 
and 
;
for all 
.
be positive real numbers such that 
.
knowing that the denominators are positive real numbers.
was chosen such that 
and 
.
.
balls to 
boxes as she wishes. After that, Aslı and Zehra make alternated moves which consists of taking a ball in any wanted box starting with Aslı. One who takes the last ball from any box takes that box to herself. What is the maximum number of boxes can Aslı guarantee to take herself regardless of Zehra's moves?
,
.
and the external angle bisector of 
and 
,
.
![\[f(xf(y) + x) = xy + f(x)\]](http://latex.artofproblemsolving.com/9/9/f/99f580ebc50846e6bc2c004667559922749a4dfa.png)
Y by jatekos101, Adventure10, Mango247
The king receives 
couples around his round table having 
seats. For 
, the members of the 
th couple want to be seated at distance 
from each other (ie. separated by exactly 
chairs) during dinner.
If is not prime, this is not always possible. (A counterexample is given by a set of couples
wanting all to be seated at distance
for a non-trivial divisor of ).
A computer-program checked that there are no restrictions on the distances for a prime number 
. This leads to the conjecture that it is always possible to have a seating arrangement if 
is a prime number.
A stronger conjecture is the following: It seems always possible to have a seating arrangement if all distances 
are invertible in the ring 
.
Can somebody give a proof of the first, or even better, of the second conjecture?
Remarks: The invertible elements modulo act on the set of distances. This can be used
to reduce the number of cases.
In particular, it is easy to proof the conjecture if there are at least
different values for the distances or at most 
different values for the distances.
couples around his round table having 
seats. For 
, the members of the 
th couple want to be seated at distance 
from each other (ie. separated by exactly 
chairs) during dinner.If
is not prime, this is not always possible. (A counterexample is given by a set of coupleswanting all to be seated at distance
for a non-trivial divisor of ).A computer-program checked that there are no restrictions on the distances for
a prime number 
. This leads to the conjecture that it is always possible to have a seating arrangement if 
is a prime number.A stronger conjecture is the following: It seems always possible to have a seating arrangement if all
distances 
are invertible in the ring 
.Can somebody give a proof of the first, or even better, of the second conjecture?
Remarks: The invertible elements modulo
act on the set of distances. This can be usedto reduce the number of cases.
In particular, it is easy to proof the conjecture if there are at least
different values for the distances or at most 
different values for the distances.
,
