1. Solve the equation: $(x-2016)(x-2017)(x-2018)=(x-2017)(x-2018)(x-2019)$.
Answer: 2017 and 2018.

## Solution:

## First Method.

Move everything to the left side of the equation and transform:

$(x-2016)(x-2017)(x-2018)-(x-2017)(x-2018)(x-2019)=0$

$(x-2017)(x-2018) \cdot((x-2016)-(x-2019))=0$

$(x-2017)(x-2018) \cdot 3=0$

The obtained equation has two roots: 2017 and 2018.

## Second Method.

It is easy to verify that 2017 and 2018 are roots (for these values of the argument, both the left and right sides of the equation become 0). If $x \neq 2017$ and $x \neq 2018$, then $x-2017 \neq 0$ and $x-2018 \neq 0$, so we can divide both sides by them, leaving $x-2016=x-2019-$ this equation has no roots (which means that the original equation has no other roots besides 2017 and 2018).

## Third Method.

Plan for solving: expand the brackets, move everything to one side - you will get a quadratic equation, solve it (naturally, this plan should be implemented, not just outlined).

## Criteria:

Both roots are found, but there is no justification for why there are no others - 2 points.

Only one root is found, or none at all, or extra roots are found - 0 points.

When dividing by $(x-2017)(x-2018)$, it is not mentioned why we do not divide by 0 - the problem is not solved.