11.3. How many points (different from the origin) are there on the parabola $y=x^{2}$, such that the tangent at these points intersects both coordinate axes at points with integer coordinates not exceeding 2020 in absolute value?
Answer: 44. Solution. The equation of the tangent to the parabola $y=x^{2}$ at the point $\left(x_{0}, y_{0}\right)$, where $y_{0}=x_{0}^{2}$, is $y-y_{0}=2 x_{0}\left(x-x_{0}\right)$. From this, we find the coordinates of the points of intersection of the tangent with the axes, namely $x_{1}=\frac{x_{0}}{2}, y_{1}=-x_{0}^{2}$. Thus, $x_{0}$ must be an even number not exceeding $\sqrt{2020}$ in absolute value. Since $44<\sqrt{2020}<45$, there are 22 positive values of $x_{0}$, and considering the symmetric negative values, there will be twice as many.