it is an interesting problem. the result as stated is not quite correct (the reason it that not all real eigenvalues can be >=n, it is only the largest eigenvalue that is guaranteed to be >=n. this can be seen from the simplest case in which a_{ij} all equal to 1, in this case A is only rank-1, and it can have only one positive real eigenvalue, all the rest are zero eigenvalues which clearly cannot be >=n)
I have provided a detailed proof in the attached file (you may notice that I reword the problem to make it simpler, also, I assign it as an excise with a given answer). please check. thanks.
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