Problem 5. Solve the equation $x^{5}-[x]=5$, where $[x]$ is the integer part of the number $x$ (the greatest integer not exceeding $x$).
Problem 5. Solve the equation

$x^{5}-[x]=5$, where $[x]$ is the integer part of the number $x$ (the greatest integer not exceeding $x$).

| Solution | Criteria |
| :---: | :---: |
| ![](https://cdn.mathpix.com/cropped/2024_05_06_0c6dedf8af9404787004g-2.jpg?height=401&width=939&top_left_y=389&top_left_x=208) | 1 point for the correct answer.  Solution is done by enumeration, but not all cases $-2-6$ points.  Solution is correct but contains arithmetic errors - 6 points |
| equations $x^{5}+1=5, x^{5}=5, x^{5}-1=5$.  solutions $x=\sqrt[5]{4}, x=\sqrt[5]{5}, x=\sqrt[5]{6}$. Their integer parts  are equal to 1. Therefore, the answer is only  the number $x=\sqrt[5]{6}$.  Answer: $x=\sqrt[5]{6}$ |  |