Long time people
This sure is nice and I wouldn't have been able to solve it without first solving the version.

Problem (Romanian Mathematical Olympiad Grade XI) :
Let be a real matrix, such that , for all . Prove that for all non-negative real numbers we have
Solution :
Let
Obviously is a polynomial.
So we have to show that

But the above result will follow directly once we establish that all the coefficients of are positive.
obviously.

I will call the kind of matrices asked in the question as ski matrix.
To show this I will use induction on the order of the matrix.
Let the result be true for all such ski matrices.

Now consider any such ski matrix and call it .
Then,
But obviously since is a sum of determinants of ski matrices of the order , all it's coefficients by the induction hypothesis are positive and hence so are it's integrals (except the constants which we have already established as zero).

Hence has all it's coefficients positive and hence by Cauchy Schwartz inequality, we are done.