59th Ukrainian National Mathematical Olympiad

It is easy to write down all the pairs of the numbers, that will successively appear on the board: $$a,b\to b+a,b-a\to 2a,2b\to 2(b+a),2(b-a)\to 2a^2,2b^2\dots$$

As we see, after the appearance of the first four numbers: $a,b,b+a,b-a$ – all the other numbers that may appear on the board are even. Since 2019 is an odd number, then the number 2019 must appear among these first four numbers.

Case 1: $b=2019$.

Case 2: $a=2019<b$.

Case 3: $b-a=2019<b$.

Case 4: $b+a=2019$, then $2b>b+a=2019\Rightarrow 2b\ge 2020\Rightarrow b\ge 1010$. Thus the smallest possible value of $b$ is $b=1010$. Then if $b=1010$ and $a=1009$ the number 2019 can appear on the board.