AoPS Academy
such that:
with all 
is real number
be the set of integers. Determine all functions 
such that, for all integers 
and 
, 
Proposed by Liam Baker, South Africa
be a positive integer. Find all real solutions 
to the system:![\[a_1^2 + a_1 - 1 = a_2\]](http://latex.artofproblemsolving.com/a/4/e/a4e26b83c929b82800576bac11e12de0c3aa1cf6.png)
![\[ a_2^2 + a_2 - 1 = a_3\]](http://latex.artofproblemsolving.com/9/9/5/995e8eec93828b82901b0639463f907f5ff95d08.png)
![\[\hspace*{3.3em} \vdots \]](http://latex.artofproblemsolving.com/e/e/1/ee15e1fefe2ad550d3c9b1c51207eaf66fbfa41f.png)
![\[a_{n}^2 + a_n - 1 = a_1\]](http://latex.artofproblemsolving.com/7/c/7/7c727d1f2044c1bf7c80981e1074bb3cde807b8f.png)
of positive integers is called central if for every positive integer , the arithmetic mean of the first 
terms of the sequence is equal to .
of positive integers such that for every central sequence 
there are infinitely many positive integers with 
.
and the lateral area 
of a right circular cone satisfy the inequality![\[\left( \frac{6V}{\pi}\right)^2 \leq \left( \frac{2S}{\pi \sqrt 3}\right)^3\]](http://latex.artofproblemsolving.com/0/0/b/00b3159e5d7df1db22fb27adf8092506aef037b2.png)
be the radius of the base, 
the vertical height so that 
is the slant height.
and 
.![\[ \left(\frac{6V}{\pi}\right)^{2}\leq\left(\frac{2S}{\pi\sqrt 3}\right)^{3}\]](http://latex.artofproblemsolving.com/8/3/9/8391e079f4f41be23a711694f03693c21eff1d79.png)
![\[ \iff (2r^h)^{2}\le \left(\frac{2r\sqrt{r^2+h^2}}{\sqrt 3}\right)^{3}\]](http://latex.artofproblemsolving.com/2/6/9/26973598348f32cf98cd45371dfe35d1ded20733.png)
![\[ \iff 3\sqrt{3}rh^2 \le 2(\sqrt{r^2+h^2})^3\]](http://latex.artofproblemsolving.com/7/6/b/76bd1061e2be6bf8261005edf993c461c5733c1a.png)
![\[ 27r^2h^4 \le 4(r^2+h^2)^3\]](http://latex.artofproblemsolving.com/d/4/5/d4572373a408b52fb0f2cd382771ecc02306b520.png)
and 
. Then after expanding the inequality becomes![\[15ab^2 \le 4a^3+12a^2b+4b^3 \]](http://latex.artofproblemsolving.com/e/2/b/e2b1c2bafc62895090d068150133b596e9675b2a.png)
. Hence equality occurs when 
, i.e. 
. In this case the slant height is equal to 
.
