1. The equation $x^{2}-a x+2022=0$ has 2 positive integer roots. What is the smallest value of $a$ for which this is possible?
Solution. According to the theorem converse to Vieta's theorem, we have: $x_{1}+x_{2}=a$, $x_{1} \cdot x_{2}=2022$. Note that the product of the roots can be factored into two factors in 4 ways: $1 \cdot 2022, 2 \cdot 1011, 3 \cdot 674, 6 \cdot 337$. The smallest sum of the factors is $6+337=343$. Answer: 343.