Be real!

Consider all sets of 100 real numbers \( r_i \) whose sum is positive.

Define \( t_{ij} = r_i + r_j \) for \( i < j \). There are \( { 100 \choose 2 } = 4950 \) such pairs.

What is the maximum number of \( t_{ij} \) which are non-positive?

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