[ Irrational Equations $]$ [ Monotonicity and Boundedness

Solve the equation $2017 x^{2017}-2017+x=\sqrt[2017]{2018-2017 x}$.

#
The function $f(x)=2017 x^{2017}-2017+x$ is increasing, while the function $g(x)=\sqrt[2017]{2018-2017 x}$ is decreasing. Therefore, the equation $f(x)=g(x)$ has no more than one root. However, it is obvious that $f(1)=g(1)$.

## Answer

$x=1$.

Author: Volienkov S.G.

A sheet of paper has the shape of a circle. Can five segments be drawn on it, each connecting two points on the boundary of the sheet, such that among the parts into which these segments divide the sheet, there are a pentagon and two quadrilaterals?

## Solution

For example, see the figure. Two quadrilaterals and a pentagon are highlighted in color.

![](https://cdn.mathpix.com/cropped/2024_05_06_93ee318ee1723cbdd653g-16.jpg?height=443&width=443&top_left_y=476&top_left_x=819)

## Answer

Yes, it is possible.