Problem 8.3. How many lines exist that pass through the point $(0,2019)$ and intersect the parabola $y=x^{2}$ at two points with integer coordinates on the $y$-axis?

Answer: 9.
Solution. A vertical line obviously does not fit. All lines, different from the vertical one and passing through the point $(0,2019)$, are given by the equation $y=k x+2019$ for some $k$. Let such a line intersect the parabola at points $\left(a, a^{2}\right)$ and $\left(b, b^{2}\right)$, where $a^{2}$ and $b^{2}$ are integers. Without loss of generality, we will assume $a<b$. We write the system of equations

$$
\left\{\begin{array}{l}
a^{2}=k a+2019 \\
b^{2}=k b+2019
\end{array}\right.
$$

Subtracting one equation from the other, we get $a^{2}-b^{2}=k(a-b)$, and since $a \neq b$, we conclude that $k=a+b$. Substituting the obtained $k$ into any of the equations of the system, we arrive at the relation $-a b=2019$.

Note that this equality is equivalent to the fact that the points $\left(a, a^{2}\right)$ and $\left(b, b^{2}\right)$ lie on the same line. This can be verified by substituting $k=a+b$ and $2019=-a b$ into the system above.

Squaring the equation, we get $a^{2} b^{2}=2019^{2}$, where $a^{2}$ and $b^{2}$ are non-negative integers (but $a$ and $b$ are not necessarily integers!). Considering the prime factorization $2019^{2}=3^{2} \cdot 673^{2}$, we have 9 options for decomposing the number $2019^{2}$ into the product $a^{2} \cdot b^{2}$ (taking into account the order of the factors). Since $a$ and $b$ have different signs, and $a<b$, each such option corresponds to a unique solution $a=-\sqrt{a^{2}}, b=\sqrt{b^{2}}$ and, accordingly, one sought line.

Remark. A similar idea underlies the well-known geometric construction - the Matiyasevich-Steckin sieve.

## Criteria

The highest suitable criterion is used:

76 . Any correct solution to the problem.

3 6. The equation equivalent to the equality $-a b=2019$ from the solution is obtained, but the further calculation of the number of lines is done incorrectly for some reason.

2 6. The equation equivalent to the equality $k=a+b$ from the solution is obtained.

## 16 . The correct answer is given.