4. [5] As is known, a quadratic equation has no more than two roots. Can the equation $\left[x^{2}\right]+p x+q=0$ for $p \neq 0$ have more than 100 roots? ([ $\left.x^{2}\right]$ denotes the greatest integer not exceeding $x^{2}$.)

Alexei Tolpygo
Answer: can.

Solution. Consider, for example, the equation $\left[x^{2}\right]-100 x+2500=0$. It has 199 roots of the form $50+\frac{k}{100}$ (where $k=-99,-98, \ldots, 99$). Indeed,

$$
\left[\left(50+\frac{k}{100}\right)^{2}\right]=\left[2500+k+\left(\frac{k}{100}\right)^{2}\right]=2500+k=100 \cdot\left(50+\frac{k}{100}\right)-2500
$$

Idea. The line $y=100 x-2500$ is tangent to the parabola $y=x^{2}$ at the point (50,2500).

Remark. Let's informally explain how one could come up with the solution to the problem.

Since $\left[x^{2}\right]=x^{2}-\left\{x^{2}\right\}$, the original equation can be rewritten as $x^{2}+p x+q=\left\{x^{2}\right\}$. We will solve it graphically: find the intersections of the graphs of the parabola and the fractional part of the square. The graph of the fractional part $y=\{x\}$ consists of a series of uniformly spaced inclined half-intervals:

![](https://cdn.mathpix.com/cropped/2024_05_06_a853c196fe23a0ced394g-2.jpg?height=181&width=1604&top_left_y=1747&top_left_x=301)

Similarly, the graph of $y=\left\{x^{2}\right\}$ consists of pieces of the parabola: we cut the parabola $y=x^{2}$ with horizontal lines of the form $y=n$, where $n=0,1,2, \ldots$, into pieces and shift each piece down parallel to the x-axis. But these pieces are not uniformly spaced; they become more frequent as we move away from zero (since as $x$ approaches infinity, the ordinate increases by 1 for a smaller and smaller (approaching 0) increment in $x$:

![](https://cdn.mathpix.com/cropped/2024_05_06_a853c196fe23a0ced394g-2.jpg?height=181&width=1604&top_left_y=2208&top_left_x=304)

But then any equation of the form $(x-a)^{2}=\left\{x^{2}\right\}$ with a sufficiently large $a$ will do: in the neighborhood of its vertex, the parabola $y=(x-a)^{2}$ will intersect many pieces of the graph of $y=\left\{x^{2}\right\}$.