I am trying solving Romania TST 2014 problem #3 of day 3 http://www.artofproblemsolving.com/Forum/viewtopic.php?p=3717806&sid=523a7da94c84b9ea1123a6b2f6302d34#p3717806
After finding out the gradient of both sides to apply Lagrange's multiplier method, it turns into a eigenvalue problem of a symmetric tridiagonal matrix whose $n^{th}$ diagonal is $2n^2$ and whose $(i,i+1)-th$ elements $-ij$ which is also the $(i+1,i)-th$ element and $0$ anywhere other than the diagonal, superdiagonal, subdiagonal.
That matrix's characterstic equation is in turn the orthogonal polynomials satisfying
$f_n (x) = (x-2n^2) f_{n-1}(x)- n^2 (n-1)^2 f_{n-2}(x)$
with $ f_1 (x) = x-2 , f_2 (x) = x^2 - 10x +12 $
I know that those orthogonal polynomials have $n$ distinct real zeros, which can be verified through two different ways 1. eigenvalues of real symmetric matrix 2. Sturm-Liouvile Theory.
But I have no ideas how to proceed from here. I know that In the case of Chebyshef or other famous special polynomials there are explicit formulas for the roots of those polynomials, but is there any hint where I can get a hint?
Thanks in advance.