1. Do there exist such integers $a, b$ and $c$ that the discriminant of the quadratic equation $a x^{2}+b x+c=0$ is equal to 2019?
Solution: Suppose the discriminant of the given equation is equal to the number 2019. Then we can write: $\mathrm{b}^{2}-4 \mathrm{ac}=2019$, and $\mathrm{b}^{2}-1225=4 \mathrm{ac}+794$ or $(\mathrm{b}-35) \cdot(\mathrm{b}+35)=2(2 \mathrm{ac}+397)$. Note that $\mathrm{b}-35$ and $\mathrm{b}+35$ are numbers of the same parity, so their product, if even, is divisible by 4. The right side of the last equation is an even number not divisible by 4. A contradiction is obtained, so the assumption is false.

Answer: no.