such that 
.
such that 
and 
.
.
such that the inequality below holds for any positive real numbers 
:![\[ \frac ab + \frac bc + \frac ca - 3 \geq k \left( \frac a{b + c} + \frac b{c + a} + \frac c{a + b} - \frac 32 \right).
\]](http://latex.artofproblemsolving.com/2/5/c/25c95e3e933649ed55bdd49ee446475223d8c871.png)
![$f \in R[X]$](http://latex.artofproblemsolving.com/b/5/6/b56c526a97b7f343d66e1317d98611980543ff21.png)
has an odd degree and it is given that 
is co-prime with 
and
Prove that 
there exists a positive integer N such that 
for every positive integer 
,
we denote the sum of digits of a positive integer 
.
be a sequence of rational numbers given by 
and for all 
we have 
.
can be partitioned into 
parallelograms. Prove that it can also be partitioned into 
parallelograms.
Y by
In a soccer championship 
teams are subscribed. Because of the extremely large number of teams the usual rules of the championship are modified as follows:
a) any two teams can play against one each other at most one game;
b) from any
teams, 
of them play against one each other.
How many days are necessary to make such a championship, knowing that each team can play at most one game per day?
teams are subscribed. Because of the extremely large number of teams the usual rules of the championship are modified as follows:a) any two teams can play against one each other at most one game;
b) from any
teams, 
of them play against one each other.How many days are necessary to make such a championship, knowing that each team can play at most one game per day?
