5. The unit square in the first quadrant of the coordinate plane ( $0 \leq x, y \leq 1$ ) is divided into squares with side length $2 \cdot 10^{-4}$. How many nodes of this partition (inside the unit square) lie on the parabola $y=x^{2}$?
Answer: 49. Note. The nodes of the partition have coordinates of the form ( $i / 5000, j / 5000$ ), where $i$, $j=1,2, \ldots, 4999$. The condition that a given node lies on the parabola is $\frac{j}{5000}=\left(\frac{i}{5000}\right)^{2}$, i.e., $i^{2}=j \cdot 5^{4} \cdot 2^{3}$. Therefore, the number $i$ must have the form $i=5^{2} \cdot 2^{2} \cdot k=100 k$, where $k-$ is any natural number less than $\frac{5000}{100}=50$.

## "Future Researchers - Future of Science" Mathematics Olympiad (Final Round)  Variant 2

## 9th Grade