We have defined a binary operation, denoted by $*$, on the set of integers, for which the following properties hold:

1) $x * 0=x$ for all integers $x$;
2) $0 * y=-y$ for all integers $y$;
3) $((x+1) * y)+(x *(y+1))=3(x * y)-x y+2 y$ for all integers $x, y$.

Determine the result of the operation $19 * 90$.
Solution. We prove by complete induction on $y$ that if $y \geq 0$, then

$$
x * y=x(y+1)-y
$$

According to the 1st condition, (1) holds for any $x$ when $y=0$. Now let $y \geq 0$, and assume that (1) holds for all $x$. From the 3rd condition,

$$
x *(y+1)=3(x * y)-x y+2 y-[(x+1) * y]
$$

The value of the right-hand side, according to the induction hypothesis, is

$$
3[x(y+1)-y]-x y+2 y-[(x+1)(y+1)-y]
$$

After rearranging, we get

$$
x y+2 x-(y+1)=x(y+2)-(y+1)
$$

which is precisely the form of $x *(y+1)$ according to statement (1). This completes the inductive proof.

We can now easily answer the question of the problem:

$$
19 * 90=19 \cdot 91-90=1639
$$

Remarks. 1. In the solution, we did not use the 2nd condition; the desired value is already determined by the 1st and 3rd conditions. The found form also satisfies the 2nd condition, so the operation in question indeed exists.

2. Although it is not necessary for answering the question of the problem, it can be proven by further inductive reasoning that from the 1st and 3rd conditions, for all integer pairs $x, y$,

$$
x * y=x(y+1)-y
$$

Based on the work of Csekő Zoltán (Szolnok, Verseghy F. Gymnasium, 2nd grade)