AoPS Academy
. Determine the number of sequences 
of integers such that 
for all 
.
+1 w
for any positiv integer number 
.?
(
) be non-negative real numbers fulfilling ![\[\sum_{k=1}^na_k^2+{\left(n^2-3n+1\right)}\prod_{k=1}^na_k\geqslant{\left(n-1\right)}^2\text.\]](http://latex.artofproblemsolving.com/b/0/4/b04e7a65130d8a03a82014fcac1c9b31dfb7da54.png)
Prove or disprove: ![\[\frac1{n-1}\sum_{1\leqslant i<j\leqslant n}{\left(a_i-a_j\right)}^2\geqslant{\left({\left(n^2-2n-1\right)}\sum_{k=1}^na_k-n{\left(n-1\right)}{\left(n-3\right)}\right)}{\left(n-\sum_{k=1}^na_k\right)}\textnormal.\]](http://latex.artofproblemsolving.com/6/a/8/6a84740ee93278e04b315d45ad5079e69aeac3c1.png)
be a positive integer and let 
be positive integers. Define for all 
, 
to be the least occurring integer in 
(if the smallest number of occurrences is shared between two distinct integers 
, let be the smaller of the two).
is eventually periodic.
, for
We get: 
has three distinct real roots 
, 
and 
. The polynomial 
has roots ![$\sqrt[3]{\alpha}$](http://latex.artofproblemsolving.com/4/f/5/4f57cc54c79420c509385c423ffa42820cef3109.png)
, ![$\sqrt[3]{\beta}$](http://latex.artofproblemsolving.com/d/4/0/d40f4039c320468d9974a9504917b56eaf73c947.png)
, ![$\sqrt[3]{\gamma}$](http://latex.artofproblemsolving.com/7/8/1/78171ba6cd442d8156fd70b3d845075e3d82a926.png)
. Find the integer closest to 
.
, find the maximum value of 
.
. But my doubt is can't we apply other inequality to get the maximum result? or did they us titu's lemma because the given information can fit only in this lemma?
compute the minimum value of the sum:
given that
be real numbers such that 
and 
. Prove that
Equality holds when 
where 
,some algebraic manipulation gives 
for positive integers.On the other hand we clearly have 
.
and 
is a permutation of 
where 
are positive integers.Because the stuff which is showen above one must have 
.
gives clearly no solution.For 
one have 
which gives the triples 
and 
.For 
there is clearly no 
which satisfies the condition,done.
Prove that
Let 
Prove that
P![$$\sqrt[3]{ \frac{1}{a} + 7b} + \sqrt[3]{\frac{1}{b} + 7c} + \sqrt[3]{\frac{1}{c} + 7a } \leq \frac{6}{abc}$$](http://latex.artofproblemsolving.com/c/1/3/c138888009b00e9925179f76371f43421b7d0b37.png)
s
such that 
