3. Solve the equation $2021 x=2022 \cdot \sqrt[2022]{x^{2021}}-1 .(10$ points $)$
Solution. $x \geq 0$. Transform the equation to the form:

![](https://cdn.mathpix.com/cropped/2024_05_06_66ff5449eae3e24affe1g-21.jpg?height=88&width=497&top_left_y=573&top_left_x=414)

![](https://cdn.mathpix.com/cropped/2024_05_06_66ff5449eae3e24affe1g-21.jpg?height=114&width=470&top_left_y=660&top_left_x=424)

But by the relation for the arithmetic mean and the geometric mean

$x+x+\ldots+x+1$

![](https://cdn.mathpix.com/cropped/2024_05_06_66ff5449eae3e24affe1g-21.jpg?height=91&width=1496&top_left_y=854&top_left_x=423)
the inequality holds for $x=1$.

Answer. $x=1$.