MathNet 2019-part-05 — 184 problems (EN–VI) Question 164

The cells of the $2 \times 2019$ table are to be filled with real numbers (one number in each cell) so that the following rules will be satisfied. The first row should contain $2019$ pairwise distinct real numbers; the second row should be a permutation of the first row. Each column should contain two distinct real numbers whose sum is a rational number.
Find the greatest possible number of irrational numbers the first row may contain.