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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 has at least two complex non-real roots.
there exists a positive integer N such that 
for every positive integer 
, where by 
we denote the sum of digits of a positive integer 
.
be a sequence of rational numbers given by 
and for all 
we have 
. Prove that all the terms of the sequence are integers.
can be partitioned into 
parallelograms. Prove that it can also be partitioned into 
parallelograms.
be an integer and 
rational numbers with the property that for any irrational numbers 
, 
, there exist the positive integers 
such that 
Prove that 
for all 
.
