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such that the polynomial 
has 8 linear factors of the form 
with 
for 
. Find all possible values of the constant 
.
be real numbers such that ![\[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]](http://latex.artofproblemsolving.com/3/d/0/3d015292c249a69a44d5f923092201d7a271b896.png)
Find the smallest possible value of 
.
positive integers, less than or equal to 
, such that the difference between any 
elements in the set is at least 
.
which verify relation![\[
\left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/3/a/2/3a235e8be4c61c32f376999e61d64973db21dc75.png)
is a finite set, let denote the number of elements in . Call an ordered pair 
of subsets of 
if 
for each 
, and 
for each 
. How many admissible ordered pairs of subsets 
are there? Prove your answer.
be a continuous function such that 
for all 
. Show that 
for 
.
be a function differentiable at 0 with 
. Find
Show that the following statements are equivalent:
there exist 
such that 
there exist 
such that 
be a positive sequence and let 
be a constant such that ![\[ \sum_{k=1}^{n-1} \lambda^2_k < K \lambda^2_n \;(n=1,2,...).\]](http://latex.artofproblemsolving.com/5/3/c/53ccb27390a368e1a74afae1da4f7566ad86af8a.png)
Prove that there exists a constant 
such that ![\[ \sum_{k=1}^{n-1} \lambda_k < K' \lambda_n \;(n=1,2,...).\]](http://latex.artofproblemsolving.com/e/7/4/e744b1b1179f879dbea84a53622afc5ca8e45fe2.png)
such that 
. Call 
the linear, constant, and quadratic coefficients respectively of 
. Clearly, 
for all 
by polynomial multiplication. It is easy to show that 
for all using induction since 
. This combined with the fact that 
and 
gives us 
for all . Additionally, 
for all 
and since 
, 
for all . This combined with and gives 
. We will show next with induction that 
for all 
.
which is true since 
is linear.
, 
. By I.H., 
as desired. By principle of mathematical induction, the formula holds true for all .
and 
if 
in 
in terms of 
