## Task $7 / 84$

A man says: "The birth year of my grandson is the product $x y$ of two natural numbers $x$ and $y$. In the year $x^{2}$, he will be $x$ years old."

In which year was the grandson born?
According to the grandfather's statement, $x^{2}=x y+x$. Since (obviously) $x \neq 0$, it follows that $y=x-1$. Furthermore, $1984 \geq x y=x(x-1)=x^{2}-x$,

$$
45>\sqrt{1984} \geq \sqrt{x(x-1)} \approx x
$$

Therefore, values for $x$ in the vicinity of 45 are in question. Values $x>45, y>44$ lead to $x y \geq 2070 > 1984$ and are thus excluded.

Values $x \leq 44, y \leq 43$ result in $x y \leq 1982$ and are excluded for biological reasons (the still living grandfather would have an unusually high age in our country). Therefore, $x=45, y=44$ and the birth year of the grandson is $x y=1980$. If we allow the grandfather to be more than 120 years old, then 1892 would also be possible (but this is excluded by the wording of the problem "... he will ... be").