# Problem 1.

A rule is given by which each pair of integers $X$ and $Y$ is assigned a number $X \nabla Y$. (The symbol «»» means applying the rule to the numbers $X$ and $Y$.) It is known that for any integers $X, Y$ the following properties hold:

1) $X \nabla 0=X$
2) $X \nabla(Y-1)=(X \nabla Y)-2$
3) $X \nabla(Y+1)=(X \nabla Y)+2$.

Find the formula that describes the given rule, and solve the equation

$$
X \nabla X=-2019
$$

#
# Solution.

Let's start writing down in order the result of applying the operation $X \nabla Y$ for $Y=0,1,2, \ldots$, using property 3.

$$
\begin{aligned}
& X \nabla 0=0, \\
& X \nabla 1=X \nabla(0+1)=(X \nabla 0)+2=X+2, \\
& X \nabla 2=X \nabla(1+1)=(X \nabla 1)+2=X+4, \\
& X \nabla 3=X \nabla(2+1)=(X \nabla 2)+2=X+6
\end{aligned}
$$

and so on. It is clear that the result can be calculated using the formula $x+2 y$.

Similarly, writing down the results of applying the operation $X \nabla Y$ for $Y=-1,-2,-3, \ldots$ (using property 2) also leads to the formula $x+2 y$.

It remains to solve the equation

$$
X \nabla X=X+2 X=3 X=-2019
$$

Answer. $X \nabla Y=X+2 Y$, the solution to the equation is $X=-673$