AoPS Academy
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 the Fibonacci sequence, defined by 
, and for all positive integers 
, 
. Prove that the following inequality takes place for all positive integers :
.
