Estonian Math Competitions

Let $a=\overline{x7}$ and $b=\overline{yd}$, where $d$ is the final digit of $b$ and $x,y$ are the numbers $a$ and $b$ without their final digits. Then $a=10x+7$ and $b=10y+d$. From the given information, we compile the system of equations $$\{\begin{array}{l}x+10y+d=2022,\\ y+10x+7=5000.\end{array}(1)$$ Subtracting the first equation from the second one, we get the equation $$9(x-y)-d=2971.(2)$$ As $x,y$ are integers, $9(x-y)$ is divisible by 9. As the number 2971 gives a remainder of 1 upon division by 9, the number $d$ must give a remainder of 8; the only option for this is $d=8$. Substituting this into (2), we get $x-y=331$, which we can substitute into either equation of (1) to get $x=484$ and $y=153$. Therefore $a+b=4847+1538=6385$.